Heller triangulated categories

Matthias Künzer

March 11, 2009

Abstract

Let E be a Frobenius category. Let \underline{\underline{E}} denote its stable category. The shift functor on \underline{\underline{E}} induces, by pointwise application, an inner shift functor on the category of acyclic complexes with entries in \underline{\underline{E}}. Shifting a complex by 3 positions yields an outer shift functor on this category. Passing to the quotient modulo split acyclic complexes, Heller remarked that inner and outer shift become isomorphic, via an isomorphism satisfying still a further compatibility. Moreover, Heller remarked that a choice of such an isomorphism determines a Verdier triangulation on \underline{\underline{E}}, except for the octahedral axiom. We generalise the notion of acyclic complexes such that the accordingly enlarged version of Heller’s construction includes octahedra.

Contents

0 Introduction  3
0.1 Heller’s idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.1.1 Stable Frobenius categories and an isomorphism between outer and inner shift . . . . . . 3
0.1.2 The stable Frobenius case models a general definition of Puppe triangulations . . . . . . . 4
0.1.3 From Puppe to Verdier and beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.2 Definition of Heller triangulated categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.2.1 A diagram shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.2.2 Heller triangulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
0.2.3 Strictly exact functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
0.2.4 Enlarge to simplify . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
0.3 A result to begin with . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
0.4 A quasicyclic category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
0.5 Some remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
0.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
0.7 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1 Definition of a Heller triangulated category  10
1.1 Periodic linearly ordered sets and their strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Heller triangulated categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.1 The stable category of pretriangles \underline{C^{+}(P^{#})} . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.1.1 Definition of \underline{C^{+}(P^{#})} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.1.2 Naturality of \underline{C^{+}(P^{#})} in P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1.3 Naturality of \underline{C^{+}(P^{#})} in C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.2 Folding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.2.1 Some notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.2.2 The folding operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.2.3 An example: folding from Δ̄_{5}^{#} to Δ̄_{3}^{#} . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.3 A definition of Heller triangulated categories and strictly exact functors . . . . . . . . . . 18

MSC 2000: 18E30.

2 Some equivalences  20
2.1 Some notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.1 Some posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.2 Fixing parametrisations κ^{△}, κ^{▽} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.3 The categories Ĉ^{+,∗}(Δ̄_{n}^{△}), C^{+}(Δ̄_{n}^{△}) etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.4 Reindexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Density of the restriction functor from Δ̄_{n}^{#} to Δ̇_{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Upwards and downwards spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.2.1 A stability under pointwise pushouts and pullbacks . . . . . . . . . . . . . . . . 23
2.2.2.2 Upwards and downwards resolution . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.2.3 Both-sided resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Fullness of the restriction functor from Δ̄_{n}^{#} to Δ̇_{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 The equivalence between \underline{C^{+}(Δ̄_{n}^{#})} and \underline{C(Δ̇_{n})} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Auxiliary equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5.1 Factorisation into two equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5.2 Cutting off the last object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.3 Not quite an equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Verification of Verdier’s axioms  33
3.1 Restriction from C^{+, ϑ=1}(Δ̄_{n}^{#}) to C(Δ̇_{n}) is dense and full . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 An omnibus lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Turning n-triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Application to the axioms of Verdier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5 n-triangles and strictly exact functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 A remark on spectral sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 The stable category of a Frobenius category is Heller triangulated  41
4.1 Definition of F^{□}(Δ̄_{n}^{#}), modelling \underline{\underline{F}^{+}(Δ̄_{n}^{#})} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 Folding for F^{□}(Δ̄_{n}^{#}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3 Some 1-epimorphic functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4 Construction of ϑ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5 Exact functors induce strictly exact functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5 Some quasicyclic categories  52
5.1 The category C^{+}(Δ̄_{n}^{#}) is Frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.1.1 The category A^{0}(Δ̄_{n}^{#}) is Frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.1.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.1.1.2 The periodic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.1.1.3 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.1.2 The subcategory C^{+}(Δ̄_{n}^{#}) ⊆ C^{0}(Δ̄_{n}^{#}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1.3 Two examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 A quasicyclic category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3 A biquasicyclic category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
A Some general lemmata  60
A.1 An additive lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
A.2 Exact categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
A.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
A.2.2 Embedding exact categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
A.2.3 Frobenius categories : definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
A.3 Kernel-cokernel-criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
A.4 An exact lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A.5 Some abelian lemmata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A.6 On Frobenius categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
A.6.1 Some Frobenius-abelian lemmata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
A.6.2 Decomposing split diagrams in intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.6.3 A Freyd category reminder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.6.4 Heller operators for diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A.7 Pointwise pullback and pushout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
A.8 1-epimorphic functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

0 Introduction

0.1 Heller’s idea (^{1})

0.1.1 Stable Frobenius categories and an isomorphism between outer and inner shift

Let E be a Frobenius category, i.e. an exact category with enough bijective objects. For instance, the category of complexes with values in an additive category, equipped with pointwise split exact sequences, is a Frobenius category.

Let \underline{\underline{E}} denote the stable category of E; cf. §0.3. Assume that \underline{\underline{E}} has split idempotents.

A complex with entries in \underline{\underline{E}} is acyclic if any Hom functor turns it into an acyclic complex of abelian groups. Let \underline{\underline{E}}^{+}(Δ̄_{2}^{#}) denote the category of acyclic complexes with entries in \underline{\underline{E}} (^{2}). Let \underline{\underline{\underline{E}}^{+}(Δ̄_{2}^{#})} denote the homotopy category of the category \underline{\underline{E}}^{+}(Δ̄_{2}^{#}) of acyclic complexes; that is, the quotient category of acyclic complexes modulo split acyclic complexes.

There is a shift automorphism T on \underline{\underline{E}}. It induces a first, inner shift automorphism T^{+}(Δ̄_{2}^{#}) on \underline{\underline{E}}^{+}(Δ̄_{2}^{#}) by pointwise application.

There is also a shift automorphism T_{2} on the diagram Δ̄_{2}^{#}. It induces a second, outer shift automorphism \underline{\underline{E}}^{+}(T_{2}) on \underline{\underline{E}}^{+}(Δ̄_{2}^{#}), shifting a complex by three positions.

Both outer and inner shift induce automorphisms

\underline{\underline{\underline{E}}^{+}(T_{2})} resp. \underline{T^{+}(Δ̄_{2}^{#})} on \underline{\underline{\underline{E}}^{+}(Δ̄_{2}^{#})} .

Heller remarked that these functors are isomorphic. But there is no a priori given isomorphism. So he chose an isomorphism

\underline{\underline{\underline{E}}^{+}(T_{2})} \overline{}^{ϑ_{2}}_{∼} \underline{T^{+}(Δ̄_{2}^{#})} ,

satisfying, for technical reasons, still a further compatibility.

Then he remarked that the choice of such an isomorphism ϑ_{2} determines a triangulation on \underline{\underline{E}} in the sense of Puppe [28, Sec. 2]; that is, it satisfies all the axioms of Verdier [31, Def. 1-1] except possibly for the octahedral axiom. Namely, as distinguished triangles we take acyclic complexes on which outer and inner shift coincide (i.e., which are “3-periodic up to shift”) and on which ϑ_{2} is the identity.

Whether this observation now fathoms Puppe triangulations remains to be discussed. Whenever two objects are isomorphic but lack a nature-given isomorphism, it is at any rate not unusual to pick an isomorphism. Once a suitable isomorphism between our shift functors chosen, a Puppe triangulation ensues. In nontechnical terms, we may let the relation between the two shifts govern the Puppe triangulations. This is a possible point of view, which we shall adopt and put into a larger framework; cf. §0.2.2.

Heller used this construction to parametrise Puppe triangulations on \underline{\underline{E}}. The non-uniqueness of such a Puppe triangulation on \underline{\underline{E}}, and hence the impossibility of an intrinsic definition of distinguished triangles, thus can be regarded as rooted in the possible nontriviality of the automorphism group of the inner shift functor \underline{T^{+}(Δ̄_{2}^{#})}, or, by choice, of the outer shift functor \underline{\underline{\underline{E}}^{+}(T_{2})}. This is to be seen in contrast to the intrinsic characterisation of short exact sequences in an abelian category.

1Heller formulated his idea using Freyd categories. We will rephrase it using complexes, for this is the language we will use below. Cf. §§ 0.2.2, 0.2.4.

2The notation using the diagram Δ̄_{2}^{#} is chosen to fit into a larger framework; see §0.2.2 for more details.

0.1.2 The stable Frobenius case models a general definition of Puppe triangulations

A weak kernel in an additive category is defined by the universal property of a kernel, except for the uniqueness of the induced morphism; dually a weak cokernel.

A weakly abelian category is an additive category in which each morphism has a weak kernel and a weak cokernel, and in which each morphism is a weak kernel and a weak cokernel. For instance, the stable category \underline{\underline{E}} appearing in §0.1.1 is a weakly abelian category.

Let C be a weakly abelian category with split idempotents carrying a shift automorphism T. Now Heller’s construction yields an alternative, equivalent definition of a Puppe triangulation on (C, T) as being an isomorphism

\underline{C^{+}(T_{2})} \overline{}^{ϑ_{2}}_{∼} \underline{T^{+}(Δ̄_{2}^{#})}

satisfying still a further compatibility. In other words, a Puppe triangulated category can be defined to be such a triple (C, T, ϑ_{2}).

0.1.3 From Puppe to Verdier and beyond

In a Puppe triangulated category, Verdier’s octahedral axiom [31, Def. 1-1] does not seem to hold in general (^{3}).

In a Verdier triangulated category, in turn, it seems to be impossible to derive the existence of the two extra triangles in a particular octahedron described in [3, 1.1.13], or to distinguish crosses as in [16, App.].

Moreover, to define a K-theory simplicial set of a triangulated category, one is inclined to take objects as 1-simplices, distinguished triangles as 2-simplices, distinguished octahedra as 3-simplices, etc.

So we enlarge the framework, generalising from C^{+}(Δ̄_{2}^{#}) to C^{+}(Δ̄_{n}^{#}), as described next in §0.2.

0.2 Definition of Heller triangulated categories

0.2.1 A diagram shape

Given n ⩾ 0, we let Δ_{n} := {i ∈ Z : 0 ⩽ i ⩽ n}, considered as a linearly ordered set. Let Δ̄_{n} be the periodic prolongation of Δ_{n}, consisting of Z copies of Δ_{n} put in a row. This is a periodic linearly ordered set; that is, a linearly ordered set equipped with a shift automorphism i \overline{}i^{+1}. For instance, Δ̄_{2} = \{…, 2^{−1}, 0, 1, 2, 0^{+1}, … \}, equipped with i \overline{}i^{+1}. Let Δ̄ be the category consisting of periodic linearly ordered sets of the form Δ̄_{n} as objects, and of monotone shiftcompatible maps as morphisms.

3The author lacks an example of a category that is Puppe but not Verdier triangulated, but strongly suspects that such an example exists, i.e. that the octahedral axiom is not a consequence of Puppe’s axioms; cf. Question 1.6. In any case, such a deduction is unknown.

Let Δ̄_{n}(Δ_{1}) denote the category of morphisms in Δ̄_{n}, i.e. the category of Δ̄_{n}-valued diagrams of shape Δ_{1}. Given α, β ∈ Δ̄_{n} such that α ⩽ β, the object (α \overline{}β) in Δ̄_{n}(Δ_{1}) is abbreviated by β/α.

Let Δ̄_{n}^{#} be the full subcategory of Δ̄_{n}(Δ_{1}) that consists of objects β/α within a single period, i.e. such that β^{−1} ⩽ α ⩽ β ⩽ α^{+1}. For instance,

Δ̄_{2}^{#}  =  \xymatrix{ &  &  & 0^{+1}/0^{+1} \ar[r] & ⋯ \\  &  & 2/2 \ar[r] & 0^{+1}/2 \ar[u] \ar[r] & ⋯ \\  & 1/1 \ar[r] & 2/1 \ar[u] \ar[r] & 0^{+1}/1 \ar[u] \ar[r] & ⋯ \\ 0/0 \ar[r] & 1/0 \ar[u] \ar[r] & 2/0 \ar[u] \ar[r] & 0^{+1}/0 \ar[u] \\ ⋮ \ar[u] & ⋮ \ar[u] & ⋮ \ar[u]}

0.2.2 Heller triangulations

Let C be a weakly abelian category; cf. §0.1.2. A sequence X ^{\overline{}^{f}} Y ^{\overline{}^{g}} Z in C is called exact at Y if f is a weak kernel of g, or, equivalently, if g is a weak cokernel of f. A commutative quadrangle in C whose diagonal sequence is exact at the middle object is called a weak square.

Let C^{+}(Δ̄_{n}^{#}) be the category of C-valued diagrams of shape Δ̄_{n}^{#} with a zero at α/α and at α^{+1}/α for each α ∈ Δ̄_{n}, and such that the quadrangle on (γ/α, δ/α, γ/β, δ/β) is a weak square whenever δ^{−1} ⩽ α ⩽ β ⩽ γ ⩽ δ ⩽ α^{+1}. Let \underline{C^{+}(Δ̄_{n}^{#})} be the quotient of C^{+}(Δ̄_{n}^{#}) modulo the full subcategory of diagrams therein that consist entirely of split morphisms.

For instance, C^{+}(Δ̄_{2}^{#}) is the category of C-valued acyclic complexes; and \underline{C^{+}(Δ̄_{2}^{#})} is its quotient modulo split acyclic complexes, i.e. the homotopy category of C-valued acyclic complexes.

Furthermore, suppose given an automorphism T on C. We obtain two shift functors on \underline{C^{+}(Δ̄_{n}^{#})}, the inner shift given by pointwise application of T, and the outer shift induced by a diagram shift j/i \overline{}i^{+1}/j.

A Heller triangulation on (C, T) is a tuple of isomorphisms ϑ = (ϑ_{n})_{n⩾0}, where ϑ_{n} is an isomorphism from the outer to the inner shift on \underline{C^{+}(Δ̄_{n}^{#})}. This tuple is required to be compatible with the functors induced by periodic monotone maps between Δ̄_{n} and Δ̄_{m}, where m, n ⩾ 0. Moreover, it is required to be compatible with an operation called folding, which emerges from the fact that a weak square

[figure: X; f; Y; X′; f′; x; +; Y′; y]

entails a folded weak square

[figure: 0; Y; X′; _{(_{x f′})}; +; X ⊕Y′ .; _{(\begin{matrix}f \\ −y \end{matrix} )}]

A Heller triangulated category is a triple (C, T, ϑ) as just described, often just denoted by C.

An n-triangle in a Heller triangulated category C is an object X of C^{+}(Δ̄_{n}^{#}) that is periodic in the sense that outer shift and inner shift coincide on X, and that satisfies Xϑ_{n} = 1. The usual properties of 2-triangles generalise to n-triangles.

If C is a Heller triangulated category in which idempotents split, then, taking the 2-triangles as the distinguished triangles, it is also triangulated in the sense of Verdier [31, Def. 1-1]; see Proposition 3.6.

0.2.3 Strictly exact functors

An additive functor C ^{\overline{}^{F}} C′ between Heller triangulated categories (C, T, ϑ) and (C′, T′, ϑ′) is called strictly exact if, firstly, it respects weak kernels, or, equivalently, weak cokernels; if, secondly, F T′ = T F; and if, thirdly, the functor

\underline{C^{+}(Δ̄_{n}^{#})} ^{\overline{}^{\underline{F^{+}(Δ̄_{n}^{#})}}} \underline{C^{′+}(Δ̄_{n}^{#})} ,

induced by pointwise application of F, satisfies \underline{F^{+}(Δ̄_{n}^{#})} ⋆ ϑ′_{n} = ϑ_{n} ⋆ \underline{F^{+}(Δ̄_{n}^{#})} for n ⩾ 0.

0.2.4 Enlarge to simplify

Let Δ̇_{n} := \{i ∈ Z : 1 ⩽ i ⩽ n\}. We have an embedding Δ̇_{n} [figure] Δ̄_{n}^{#} via α \overline{}α/0. Let C be a weakly abelian category. Let C(Δ̇_{n}) denote the category of C-valued diagrams of shape Δ̇_{n}. Let \underline{C(Δ̇_{n})} be the quotient of C(Δ̇_{n}) modulo the full subcategory of split diagrams. Restriction induces an equivalence

(∗)  \underline{C^{+}(Δ̄_{n}^{#})}  \overline{}^{(−)|Δ̇n}_{∼}  \underline{C(Δ̇_{n})} ,

which is also a useful technical tool; cf. Proposition 2.6.

At first sight, one might be inclined to prefer \underline{C(Δ̇_{n})} over \underline{C^{+}(Δ̄_{n}^{#})}. It contains smaller diagrams and has a less elaborate definition. By transport of structure along (∗), one obtains an outer shift on \underline{C(Δ̇_{n})} as well. By pointwise application of the shift functor on C, we also obtain an inner shift on \underline{C(Δ̇_{n})}. These could be compared in order to write down a definition of Heller triangulated categories.

So why then did we prefer to use \underline{C^{+}(Δ̄_{n}^{#})} in our definition of Heller triangulated categories in §0.2.2? Working with \underline{C(Δ̇_{n})}, the indirect definition of the outer shift would cause problems. In practice, one would have to pass the equivalence (∗) back and forth. The “blown-up variant” \underline{C^{+}(Δ̄_{n}^{#})} of \underline{C(Δ̇_{n})} carries a directly defined outer shift functor and is thus easier to work with.

There is a further equivalence \underline{C(Δ̇_{n})} ^{\overline{}^{∼}} \underline{Ĉ(Δ̇_{n−1})}, where Ĉ denotes the Freyd category of C, i.e. the universal abelian category containing C, and where \underline{Ĉ(Δ̇_{n−1})} is the quotient of Ĉ(Δ̇_{n−1}) modulo split diagrams with entries in C; cf. Proposition 2.10. Originally, Heller worked with \underline{Ĉ(Δ̇_{n−1})} for n = 2, i.e. with Ĉ/C.

0.3 A result to begin with

Let E be a Frobenius category. We define its stable category \underline{\underline{E}} to be the quotient category of the category of purely acyclic complexes with values in the bijective objects of E, modulo the subcategory of split acyclic such complexes.

Then \underline{\underline{E}} is equivalent to the classical stable category \underline{E} of E, defined as the quotient category of E modulo bijective objects. But \underline{\underline{E}} carries a shift automorphism T (invertible), whereas \underline{E} carries, in general, only a shift autoequivalence (invertible up to isomorphism). In this sense, \underline{\underline{E}} is a “strictified version” of \underline{E}.

Theorem (Corollary 4.7, Corollary 4.9). Given a Frobenius category E, there exists a Heller triangulation ϑ on (\underline{\underline{E}}, T). An exact functor E ^{\overline{}^{E}} E′ between Frobenius categories that sends all bijective objects of E to bijective objects of E′ induces a strictly exact functor \underline{\underline{E}} ^{\overline{}^{\underline{\underline{E}}}} \underline{\underline{E}}′.

The Verdier triangulated version of this theorem is due to Happel [11, Th. 2.6].

0.4 A quasicyclic category

Let C be a Heller triangulated category.

A quasicyclic category is a contravariant functor from Δ̄^{∘} to the (1-)category of categories. Letting qcyc_{n} C be the subcategory of isomorphisms in C^{+}(Δ̄_{n}^{#}) for n ⩾ 0, we obtain a quasicyclic category qcyc_{•} C. There is a quasicyclic subcategory qcyc_{•}^{ϑ=1} C that consists only of n-triangles and their isomorphisms instead of all objects in C^{+}(Δ̄_{n}^{#}) and their isomorphisms (^{4}).

Restricting qcyc_{•}^{ϑ=1} C along the functor Δ^{∘} [figure] Δ̄^{∘} of “periodic prolongation”, this yields a simplicial category, hence a topological space; depending functorially on C. This space is the author’s tentative proposal for the definition of the K-theory of C; cf. [27, Rem. 63]. Of course, this definition still needs to be justified by results one expects of such a K-theory, which has not yet been attempted.

0.5 Some remarks

A comparison of our theory to the derivator approach and related constructions in [4], [14, chap. V.1], [13], [10], [18], [7] and [23] would be interesting. One might ask whether the base category of a triangulated derivator in the sense of [23] carries a Heller triangulation; and if so, whether morphisms of triangulated derivators give rise to strictly exact functors.

Our approach differs from the derivator approach in that we consider a single category C with shift and an “exactness structure”, i.e. a Heller triangulation, on it. The categories C^{+}(Δ̄_{n}^{#}) needed to define this “exactness structure” on C consist of veritable C-valued diagrams; cf. §0.2. In particular, a “structure preserving map” between two such categories C and C′, i.e. a strictly exact functor, is a single additive functor C ^{\overline{}^{F}} C′ compatible with the “exactness structures” imposed on C and on C′. In contrast, a “structure preserving map” of triangulated derivators is a compatible family of additive functors.

The generalised triangles in [3, 1.1.14] are, in our language, n-pretriangles for which the 2-pretriangle obtained by restriction along any periodic monotone map Δ̄_{2} \overline{}Δ̄_{n} is a 2-triangle. An n-triangle is such a generalised triangle, but the converse does not hold in general, as pointed out to me by A. Neeman. For an example, see [21, §2].

4Here, “ ϑ=1 ” is a mere symbol that should evoke the definition of n-triangles via ϑ.

It is conceivable that the concept of Heller triangulated categories is essentially equivalent to a direct axiomatisation via n-triangles, as worked out independently by G. Maltsiniotis [24] and myself [20]. To compare these approaches, on the one hand, the Heller triangulated category should be closed in the sense of [21, §A.2, Def. 13]; on the other hand, the n-triangles should be stable under folding as in Lemma 3.4.(2).

Concerning the motivation to consider triangulated categories at all, and in particular derived categories, conceived by Grothendieck, we refer the reader to the introduction of the thesis of Verdier [30]; cf. also [15] and [33, p. 26].

0.6 Acknowledgements

I thank A. Wiedemann for an introduction to derived categories. I thank A. Beligiannis for directing me to Heller’s parametrisation of Puppe triangulations, and for helpful remarks. I thank B. Keller for the hint how to “strictify” the classical stable category of a Frobenius category using acyclic complexes. I thank A. Neeman for the hint leading to the folding operation, and for corrections.

More than once I returned to A. Heller’s original construction [12, p. 53–54], the reference not only for the basic idea, but also for arguments perfectly extendable to the more general framework used here.

0.7 Notations and conventions

(i) The disjoint union of sets X and Y is written X ⊔ Y.

(ii) Given a, b, c ∈ Z, the assertion a ≡_{c} b is defined to hold if there exists a z ∈ Z such that a − b = cz.

(iii) For a, b ∈ Z, we denote by [a, b] := {z ∈ Z : a ⩽ z ⩽ b} the integral interval. Similarly, we let [a, b[ := {z ∈ Z : a ⩽ z < b}, ]a, b] := {z ∈ Z : a < z ⩽ b}, Z_{⩾0} := {z ∈ Z : z ⩾ 0} and Z_{⩽0} := {z ∈ Z : z ⩽ 0}.

(iv) All categories are supposed to be small with respect to a sufficiently big universe.

(v) Given a category C, and objects X, Y in C, we denote the set of morphisms from X to Y by _{C}(X, Y), or simply by (X, Y), if unambiguous.

(vi) Given a poset P, we frequently consider it as a category, letting _{P}(x, y) contain one element y/x if x ⩽ y, and letting it be empty if x ⩽̸ y, where x, y ∈ Ob P = P.

(vii) Given n ⩾ 0, we denote by Δ_{n} := [0, n] the linearly ordered set with ordering induced by the standard ordering on Z. Let Δ̇_{n} := Δ_{n} ∖ {0} = [1, n], considered as a linearly ordered set.

(viii) Maps act on the right. Composition of maps, and of more general morphisms, is written on the right, i.e. ^{\overline{}^{a}} ^{\overline{}^{b}} = ^{\overline{}^{ab}} .

(ix) Functors act on the right. Composition of functors is written on the right, i.e. ^{\overline{}^{F}} ^{\overline{}^{G}} = ^{\overline{}^{F G}} . Accordingly, the entry of a transformation a between functors at an object X will be written Xa.

The reason for this convention is that we will mainly consider functors of type “restriction to a subdiagram” or “shift”, and such operations are usually written on the right.

(x) A functor is called strictly dense if its map on the objects is surjective. It is called dense if its induced map on the isoclasses is surjective.

(xi) Given transformations [figure: C; F; G; C′; F′; G′; C′′ ,; a; a′] we write a ⋆ a′ for the transformation from F F′ to GG′

given at X ∈ Ob C by X(a ⋆ a′) := (XF a′)(XaG′) = (XaF′)(XGa′). In this context, we also write the object F for the identity 1_{F} on this object, i.e. e.g. X(F ⋆ a′) = X(1_{F} ⋆ a′) = (XF)a′.

(xii) The inverse of an isomorphism f is denoted by f^{−}. Note that if we denote an iterated shift automorphism f \overline{}f^{+1} by f \overline{}f^{+z} for z ∈ Z, then we have to distinguish f^{−} (inverse isomorphism if f is an isomorphism) and f^{−1} (inverse of the shift functor applied to f).

(xiii) In an exact category, pure monomorphy is indicated by X [figure] Y, pure epimorphy by X \overline{}Y.

(xiv) A morphism in an additive category A is split if it is isomorphic, in A(Δ_{1}), to a morphism of the form

X ⊕ Y ^{\overline{}^{( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} )}} Y ⊕ Z. A morphism being split is indicated by [figure: X; Y] (not to be confused with monomorphy). Accordingly, a morphism being a split monomorphism is indicated by [figure: X; •; Y] , a morphism being a split epimorphism by [figure: X; Y] . Cf. §A.2.1.

(xv) We say that idempotents split in an additive category A if every endomorphism e in A that satisfies e^{2} = e is split.

(xvi) The category of functors and transformations from a category D to a category C is denoted by D, C or by C(D). To objects in C(D), we also refer to as diagrams on D with values or entries in C.

(xvii) If C and D are categories, and X ∈ Ob C(D), we usually write (d ^{\overline{}^{a}} e)X =: (X_{d} ^{\overline{}^{X_{a}}} X_{e}) for a morphism

d ^{\overline{}^{a}} e in D. If the morphism a is unambiguously given by the context, we also use small letters to

write (X_{d} ^{\overline{}^{x}} X_{e}) := (X_{d} ^{\overline{}^{X_{a}}} X_{e}) (similarly X′_{d} ^{\overline{}^{x′}} X′_{e}, Ỹ_{d} ^{\overline{}^{ỹ}} Ỹ_{e}, …)

(xviii) Let Add denote the 2-category of additive categories.

(xix) Given an additive category A and a full additive subcategory B ⊆ A, we denote by A/B the quotient of A by B, having as objects the objects of A and as morphisms equivalence classes of morphisms of A; where two morphisms f and f′ are equivalent, written f ≡_{B} f′, if their difference factors over an object of B.

(xx) In an exact category, an object P is called projective if (P, −) turns pure epimorphisms into epimorphisms. An object I is called injective if (−, I) turns pure monomorphisms into epimorphisms. It is called bijective if it is injective and projective. See §A.2 for details.

(xxi) In an additive category, a morphism K ^{\overline{}^{i}} X is called a weak kernel of a morphism X ^{\overline{}^{f}} Y if for

every morphism T ^{\overline{}^{t}} X with tf = 0 there exists a morphism T ^{\overline{}^{t′}} K with t′i = t. A weak cokernel is defined dually. An additive category is called weakly abelian if every morphism has a weak kernel and a weak cokernel, and is a weak kernel and a weak cokernel.

(xxii) The Freyd category of a weakly abelian category C is written Ĉ. See §A.6.3 for details.

(xxiii) In an abelian category, a commutative quadrangle

\xymatrix{X \ar[r]^{f} \ar[d]_{x} & Y \ar[d]^{y} \\ X′ \ar[r]_{f′} & Y′}

is called a square if its diagonal sequence X ^{\overline{}^{(^{x f} )}} X′ ⊕ Y ^{\overline{}^{( \begin{matrix}f′ \\ −y \end{matrix} )}} Y′ is short exact. Being a square is indicated by a box sign “ □ ” in the quadrangle. The quadrangle (X, Y, X′, Y′) is called a weak square if its diagonal sequence is exact in the middle; cf. Definition A.9. Being a weak square is indicated by a “ + ”-sign in the quadrangle. In an exact category, (X, Y, X′, Y′) is a pure square if it has a pure short exact diagonal sequence. Being a pure square is indicated by a box sign “ □ ” in the quadrangle. In a weakly abelian category, (X, Y, X′, Y′) is a weak square if it is a weak square in the Freyd category of that weakly abelian category.

(xxiv) In an abelian category, given a morphism X ^{\overline{}^{f}} Y, we sometimes denote its kernel by K_{f}, and its cokernel by C_{f}.

1 Definition of a Heller triangulated category

1.1 Periodic linearly ordered sets and their strips

Without further comment, we consider a poset D as a category, whose set of objects is given by D, and for which #_{D}(α, β) = 1 if α ⩽ β, and #_{D}(α, β) = 0 otherwise. If existent, i.e. if α ⩽ β, the morphism from α to β is denoted by β/α. A full subposet of a category is a full subcategory that is a poset. In particular, a full subposet of a poset is just a full subcategory of that poset.

A periodic poset is a poset P together with an automorphism T : P ^{\overline{}^{∼}} P, α \overline{}α T =: α^{+1}. Likewise, we denote α T^{m} =: α^{+m} resp. α T^{−m} =: α^{−m} for m ∈ Z_{⩾0}. By abuse of notation, we denote a periodic poset (P, T) simply by P.

A morphism of periodic posets P ^{\overline{}^{p}} P′ is a monotone map p of the underlying posets such that (α^{′+1})p = ((α′)p)^{+1} for all α′ ∈ P′. The category of periodic posets shall be denoted by 𝔭𝔭.

A periodic linearly ordered set is a periodic poset the underlying poset of which is linearly ordered, i.e. such that #(_{D}(α, β) ∪ _{D}(β, α)) = 1 for all α, β ∈ D.

To any linearly ordered set D we can attach a periodic linearly ordered set D̄ by letting D̄ := D × Z, and (α, z) ⩽ (β, w) if z < w or if (z = w and α ⩽ β in D). We let (α, z)^{+1} := (α, z + 1). Sending D \overline{}D̄, α \overline{}(α, 0), and identifying D with its image, we obtain (α, z) = α^{+z}, and the latter is the notation we will usually use. The periodic linearly ordered set D̄ is called the periodic repetition of D. Likewise, the functor D \overline{}D̄ from the category of linearly ordered sets to the category of periodic linearly ordered sets is called periodic repetition.

Let Δ be the full subcategory of the category of linearly ordered sets defined by Ob Δ := {Δ_{n} : n ∈ Z_{⩾0}}.

Let Δ̄ be the full subcategory of the category of periodic linearly ordered sets defined by Ob Δ̄ := {Δ̄_{n} : n ∈ Z_{⩾0}} (^{5}).

The reason for considering periodic linearly ordered sets is that the functor periodic repetition from Δ to Δ̄ is dense and faithful but not full. We will require a naturality of a certain construction with respect to P ∈ Ob Δ̄, which is stronger than setting P = D̄ and requiring naturality with respect to D ∈ Ob Δ.

Given n ⩾ 0, the underlying linearly ordered set of Δ̄_{n} is isomorphic to Z via α^{+z} \overline{}α + (n + 1)z. We use this isomorphism to define the operation

Δ̄_{n} × Z \overline{}Δ̄_{n} , (α^{+z}, x) \overline{}α^{+z} + x := (\overline{α + x})^{+(z+\underline{α+x})} ,

where we write k = (n + 1)\underline{k} + \overline{k} with \underline{k} ∈ Z and \overline{k} ∈ [0, n] for k ∈ Z. For instance, if n = 3, then 2^{+1} + 7 = 1^{+3}.

To a periodic linearly ordered set P, we attach the poset

P^{#} := {β/α ∈ P (Δ_{1}) : β^{−1} ⩽ α ⩽ β ⩽ α^{+1}}

as a full subposet of P (Δ_{1}), called the strip of P. A morphism therein from β/α to δ/γ is written δ/γ//β/α, which is unique if it exists, i.e. if α ⩽ γ and β ⩽ δ.

5The category Δ̄ is isomorphic to the category L defined by Elmendorf in [6].

The strip P^{#} carries the automorphism β/α \overline{}(β/α)^{+1} := α^{+1}/β in 𝔭𝔭, where β/α ∈ P^{#}.

If P = Δ̄_{n}, we also write β/α ^{\overline{}^{T_{n}}} (β/α)^{+1}.

This construction defines a functor

[figure: Δ̄; (−)^{#}; 𝔭𝔭; P; P^{#}]

which sends a morphism P ^{\overline{}^{p}} P′ in Δ̄ to

[figure: P^{#}; p^{#}; P^{′#}; β′p/α′p; β′/α′]

In fact, p^{#} is welldefined, since if β^{′−1} ⩽ α′ ⩽ β′ ⩽ α^{′+1}, then (β′p)^{−1} ⩽ α′p ⩽ β′p ⩽ (α′p)^{+1}. Moreover, p^{#} is monotone and compatible with shift.

Example 1.1 The periodic poset Δ̄_{2}^{#}, i.e. the strip of the periodic repetition of Δ_{2}, can be displayed as

\xymatrix{ &  &  &  &  & 1^{+1}/1^{+1} \ar[r] & ⋯ \\  &  &  &  & 0^{+1}/0^{+1} \ar[r] & 1^{+1}/0^{+1} \ar[u] \ar[r] & ⋯ \\  &  &  & 2/2 \ar[r] & 0^{+1}/2 \ar[u] \ar[r] & 1^{+1}/2 \ar[u] \ar[r] & ⋯ \\  &  & 1/1 \ar[r] & 2/1 \ar[u] \ar[r] & 0^{+1}/1 \ar[u] \ar[r] & 1^{+1}/1 \ar[u] \\  & 0/0 \ar[r] & 1/0 \ar[u] \ar[r] & 2/0 \ar[u] \ar[r] & 0^{+1}/0 \ar[u] \\ 2^{−1}/2^{−1} \ar[r] & 0/2^{−1} \ar[u] \ar[r] & 1/2^{−1} \ar[u] \ar[r] & 2/2^{−1} \ar[u] \\ ⋮ \ar[u] & ⋮ \ar[u] & ⋮ \ar[u]}

1.2 Heller triangulated categories

Suppose given a weakly abelian category C; cf. Definition A.26. From §1.2.1.3 on, we assume it to be equipped with an automorphism

\begin{matrix}C & \overline{}^{T}_{∼} & C \\ (X ^{\overline{}^{u}} Y) & \overline{} & (X T ^{\overline{}^{u T}} Y T) =: (X^{+1} ^{\overline{}^{u^{+1}}} Y^{+1}) . \end{matrix}

Similarly, we denote (X T^{m} ^{\overline{}^{u T^{m}}} Y T^{m}) =: (X^{+m} ^{\overline{}^{u^{+m}}} Y^{+m}) for m ∈ Z.

Recall that its Freyd category Ĉ is an abelian Frobenius category, and that the image of C in Ĉ, identified with C, is a sufficiently big subcategory of bijectives; cf. §A.6.3.

1.2.1 The stable category of pretriangles \underline{C^{+}(P^{#})}

1.2.1.1 Definition of \underline{C^{+}(P^{#})}

Concerning the Freyd category Ĉ of C, cf. §A.6.3. Concerning the notion of a weak square in Ĉ, see Definition A.9. A weak square in C is a weak square in Ĉ that has all four objects in Ob C. Applying Remark A.27, we obtain an elementary way to characterise weak squares as having a diagonal sequence with first morphism being a weak kernel of the second; or, equivalently, with second morphism being a weak cokernel of the first.

Given a periodic linearly ordered set P, we let C^{+}(P^{#}) be the full subcategory of C(P^{#}) defined by

Ob C^{+}(P^{#}) := \{ X ∈ Ob C(P^{#}) : \begin{matrix}1) & X_{α/α} = 0 and X_{α^{+1}/α} = 0 for all α ∈ P. \\ 2) & For all δ^{−1} ⩽ α ⩽ β ⩽ γ ⩽ δ ⩽ α^{+1} in P, \\ & the quadrangle \\ X_{γ/β} & x & [figure] & X_{δ/β} & X_{γ/α} & x & [figure] & x & [figure] & + & X_{δ/α} & x & [figure] \\ & is a weak square (as indicated by +). \end{matrix} \} .

Note that we do not require that (X_{α^{+1}/γ} ^{\overline{}^{x}} X_{β^{+1}/δ}) = (X_{γ/α} ^{\overline{}^{x}} X_{δ/β})^{+1} for γ/α, δ/β ∈ P^{#} with γ/α ⩽ δ/β.

An object of C^{+}(P^{#}) is called a P -pretriangle. Given n ⩾ 0, an object of C^{+}(Δ̄_{n}^{#}), i.e. a Δ̄_{n}-pretriangle, is also called an n-pretriangle.

Roughly put, an n-pretriangle is a diagram on the strip Δ̄_{n}^{#} of the periodic repetition Δ̄_{n} of Δ_{n} consisting of weak squares with zeroes on the boundaries.

Example 1.2 A 0-pretriangle consists of zero objects. A 1-pretriangle is just a sequence …, X_{0/1^{−1}}, X_{1/0}, X_{0^{+1}/1}, … of objects of C, decorated with some zero objects. A 2-pretriang-le is a complex in C which becomes acyclic in Ĉ – for short, which is acyclic –, decorated with some zero objects.

A morphism in C is split in Ĉ if and only if it factors in Ĉ into a retraction followed by a coretraction. Equivalently, its image, taken in Ĉ, is bijective as an object of Ĉ.

Let C^{+, split}(P^{#}) be the full subcategory of C^{+}(P^{#}) defined by

Ob C^{+, split}(P^{#}) := \{X ∈ Ob C^{+}(P^{#}) : \begin{matrix}X_{γ/α} ^{\overline{}^{x}} X_{δ/β} is split in Ĉ \\ for all γ/α, δ/β ∈ P^{#} with γ/α ⩽ δ/β \end{matrix} \}

We denote the quotient category by

\underline{C^{+}(P^{#})} := C^{+}(P^{#})/C^{+, split}(P^{#}) ,

called the stable category of P -pretriangles.

Example 1.3 We have \underline{C^{+}(Δ̄_{0}^{#})} = \underline{C^{+}(Δ̄_{1}^{#})} = 0. The category \underline{C^{+}(Δ̄_{2}^{#})} can be regarded as the homotopy category of the category of acyclic complexes with entries in C.

1.2.1.2 Naturality of \underline{C^{+}(P^{#})} in P

Suppose given periodic linearly ordered sets P, P′, and a morphism P^{#} ^{\overline{}^{q}} P^{′#} of periodic posets such that either (P = P′ and q = T, the shift functor on P^{#}) or q = p^{#} for some morphism P ^{\overline{}^{p}} P′ of periodic linearly ordered sets.

Recall that if P = Δ̄_{n}, then we write alternatively T_{n} for the shift functor T on Δ̄_{n}^{#}.

We obtain an induced functor

[figure: C^{+}(P^{#}); C^{+}(q); C^{+}(P^{′#}); X; X(C^{+}(q)) := q X ,]

given by composition of q, followed by X.

In particular, the shift T on P^{#} induces a functor

[figure: C^{+}(P^{#}); C^{+}(T); C^{+}(P^{#}); X; [X]^{+1} := X(C^{+}(T)) ,]

called the outer shift. Note that if P = Δ̄_{n}, then [X]_{β/α}^{+1} = X_{(β/α)^{+1}} = X_{α^{+1}/β} for β/α ∈ Δ̄_{n}^{#}. On the stable category, this functor induces a functor

[figure: \underline{C^{+}(P^{#})}; \underline{C^{+}(T)}; \underline{C^{+}(P^{#})}; X; [X]^{+1},]

likewise called the outer shift.

Given a morphism P ^{\overline{}^{p}} P′ in Δ̄, we obtain an induced morphism P^{#} ^{\overline{}^{p^{#}}} P^{′#}, and hence an induced functor usually abbreviated by

[figure: C^{+}(P^{#}); p^{#} := C^{+}(p^{#}); C^{+}(P^{′#}); X; Xp^{#} := X(C^{+}(p^{#})) .]

Likewise on the stable categories; we abbreviate \underline{p}^{#} := \underline{C^{+}(p^{#})}.

So altogether, we have defined Xp^{#} := p^{#}X (Xp^{#}: operation induced by p, applied to X ; p^{#}X: composition of p^{#} and X), which is a bit unfortunate, but convenient in practice.

Given a morphism P ^{\overline{}^{p}} P′ in Δ̄ and X ∈ Ob C^{+}(P^{#}), we have

[X]^{+1}p^{#} = [Xp^{#}]^{+1} ,

natural in X. Likewise on the stable categories.

Given P, P′ ∈ Ob Δ̄, a functor C^{+}(P^{#}) ^{\overline{}^{F}} C^{+}(P^{′#}) is called strictly periodic if

[XF]^{+1} = [X]^{+1}F ,

natural in X. Likewise on the stable categories.

1.2.1.3 Naturality of \underline{C^{+}(P^{#})} in C

An additive functor C ^{\overline{}^{F}} C′ is called subexact if the induced additive functor Ĉ ^{\overline{}^{F̂}} Ĉ′ is an exact functor of abelian categories; cf. §A.6.3. Alternatively, it is subexact if and only if it preserves weak kernels, or, equivalently, weak cokernels; cf. Remark A.27.

Suppose given a subexact functor C ^{\overline{}^{F}} C′ and P ∈ Ob Δ. We obtain an induced functor

[figure: C^{+}(P^{#}); F^{+}(P^{#}); C^{′+}(P^{#}); X; X F^{+}(P^{#}) ,]

where, writing Y := X F^{+}(P^{#}), we let

(Y_{β/α} ^{\overline{}^{y}} Y_{δ/γ}) := (X_{β/α}F ^{\overline{}^{xF}} X_{δ/γ}F)

for β/α, δ/γ ∈ P^{#} with β/α ⩽ δ/γ.

In particular, the automorphism C ^{\overline{}^{T}} C induces an automorphism

[figure: C^{+}(P^{#}); T^{+}(P^{#}); C^{+}(P^{#}); X; [X^{+1}] := X( T^{+}(P^{#})) ,]

called the inner shift. Note that if P = Δ̄_{n}, then [X^{+1}]_{β/α} = X_{β/α}^{+1} for β/α ∈ Δ̄_{n}^{#}.

On the stable category, this induces an automorphism

[figure: \underline{C^{+}(P^{#})}; \underline{T^{+}(P^{#})}; \underline{C^{+}(P^{#})}; X; [X^{+1}] ,]

likewise called the inner shift.

1.2.2 Folding

The following construction arose from a hint of A. Neeman, who showed me a multitude of 2-triangles in an n-triangle similar to the two 2-triangles explained in [3, 1.1.13]; cf. Definition 1.5.(ii) below.

1.2.2.1 Some notation

Given P = (P, T) ∈ Ob Δ̄, we denote by 2P the periodic poset (P, T^{2}).

Given a linearly ordered set D, we let ρ⊔D be the linearly ordered set having as underlying set {ρ} ⊔ D; and as partial order ρ ⩽_{ρ⊔D} α for all α ∈ D, and α ⩽_{ρ⊔D} β if α, β ∈ D and α ⩽_{D} β.

Roughly put, 2P is P with doubled period, and ρ ⊔ D is D with an added initial object ρ.

Let n ⩾ 0. We have an isomorphism of periodic linearly ordered sets

2Δ̄_{n}  ^{\overline{}^{∼}}  Δ̄_{2n+1}
k^{+l}  \overline{}\{ \begin{matrix}k^{+l/2} & for l ≡_{2} 0 \\ (k + n + 1)^{+(l−1)/2} & for l ≡_{2} 1 \end{matrix}

and an isomorphism of linearly ordered sets

[figure: ρ ⊔ Δ_{n}; ∼; Δ_{n+1}; k; k + 1 for k ∈ [0, n]; ρ; 0 .]

In order to remain inside Δ̄ resp. inside Δ, we use these isomorphisms as identifications.

Then P \overline{}2P is natural in P and therefore defines an endofunctor of Δ̄, and D \overline{}ρ ⊔ D is natural in D and therefore defines an endofunctor of Δ.

Given a linearly ordered set D, we will need to consider the periodic posets 2D̄ and \overline{ρ ⊔ D}, formed using periodic repetition.

1.2.2.2 The folding operation

Let n ⩾ 0. Let the strictly periodic functor

[figure: C^{+}((2Δ̄_{n})^{#}); 𝔣_{n}; C^{+}(\overline{ρ ⊔ Δ_{n}}^{#}); X; X𝔣_{n}]

be determined on objects by the following data. Writing Y = X𝔣_{n}, we let

\begin{aligned}(Y_{α/ρ} ^{\overline{}^{y}} Y_{β/ρ}) & := & (X_{α^{+1}/α} ^{\overline{}^{x}} X_{β^{+1}/β}) \\ (Y_{β/ρ} ^{\overline{}^{y}} Y_{β/α}) & := & (X_{β^{+1}/β} ^{\overline{}^{(^{x x})}} X_{β^{+1}/α^{+1}} ⊕ X_{α^{+2}/β}) \\ (Y_{β/α} ^{\overline{}^{y}} Y_{δ/γ}) & := & (X_{β^{+1}/α^{+1}} ⊕ X_{α^{+2}/β} ^{\overline{}^{(\begin{matrix}x & 0 \\ 0 & x \end{matrix})}} X_{δ^{+1}/γ^{+1}} ⊕ X_{γ^{+2}/δ}) \\ (Y_{δ/γ} ^{\overline{}^{y}} Y_{ρ^{+1}/γ}) & := & (X_{δ^{+1}/γ^{+1}} ⊕ X_{γ^{+2}/δ} ^{\overline{}^{(\begin{matrix}x \\ −x \end{matrix})}} X_{γ^{+2}/γ^{+1}}) \end{aligned}

for α, β, γ, δ ∈ Δ_{n} with α ⩽ β, with γ ⩽ δ and with β/α ⩽ δ/γ. The remaining morphisms are given by composition.

Note that X ∈ Ob C^{+}((2Δ̄_{n})^{#}), so e.g. X_{β^{+1}/β} ≠ 0 is possible, whereas X_{β^{+2}/β} = 0 for β ∈ Δ_{n}.

We claim that X𝔣_{n} is an object of C^{+}(\overline{ρ ⊔ Δ_{n}}^{#}).

In fact, by Lemma A.14, applied in the abelian category Ĉ, we are reduced to considering the quadrangles of Y on (γ/ρ, δ/ρ, γ/β, δ/β) for β, γ, δ ∈ Δ_{n} with β ⩽ γ ⩽ δ; on (γ/α, δ/α, γ/β, δ/β) for α, β, γ, δ ∈ Δ_{n} with α ⩽ β ⩽ γ ⩽ δ; and on (γ/α, ρ^{+1}/α, γ/β, ρ^{+1}/β) for α, β, γ ∈ Δ_{n} with α ⩽ β ⩽ γ.

The quadrangle of Y on (γ/α, δ/α, γ/β, δ/β) is a weak square as the direct sum of two weak squares.

For the remaining quadrangles to be treated, Lemma A.17 reduces us to considering the quadrangles of Y on (α/ρ, β/ρ, α/α, β/α), on (β/ρ, ρ^{+1}/ρ, β/α, ρ^{+1}/α) and on (β/α, ρ^{+1}/α, β/β, ρ^{+1}/β) for α, β ∈ Δ_{n} with α ⩽ β. These are in fact weak squares, as ensues from Lemma A.18 and its dual assertion. This proves our claim.

This construction of Y = X𝔣_{n} is functorial in X.

To prove that the folding operation passes to the stable categories, we have to show that for an object X of C^{+, split}((2Δ̄_{n})^{#}), the folded object X𝔣_{n} is in C^{+, split}(\overline{ρ ⊔ Δ_{n}}^{#}). Denote Y := X𝔣_{n}. Since Y_{α/ρ} ^{\overline{}^{y}} Y_{β/ρ} is split in Ĉ for all α, β ∈ Δ_{n} with α ⩽ β, it suffices to prove the following lemma.

Lemma 1.4 Suppose given m ⩾ 0 and Z ∈ Ob C^{+}(Δ̄_{m}^{#}) such that Z_{α/0} ^{\overline{}^{z}} Z_{β/0} is split in Ĉ for all α, β ∈ Δ_{m} with 0 < α ⩽ β. Then Z ∈ Ob C^{+, split}(Δ̄_{m}^{#}).

Proof. Consider the morphism Z_{γ/α} ^{\overline{}^{z}} Z_{δ/β} for γ/α ⩽ δ/β in Δ̄_{m}^{#}. We have to show that it is split in Ĉ, i.e. that its image, taken in Ĉ, is bijective there. Unless α ⩽ β ⩽ γ ⩽ δ ⩽ α^{+1}, this morphism is zero, hence split in Ĉ. If this condition holds, it is the diagonal morphism of the weak square (Z_{γ/α}, Z_{δ/α}, Z_{γ/β}, Z_{δ/β}).

So by Lemma A.19, applied in the abelian category Ĉ, we see that it suffices to show that the (horizontal) morphism Z_{β/α} ^{\overline{}^{z}} Z_{γ/α} is split in Ĉand that the (vertical) morphism Z_{γ/α} ^{\overline{}^{z}} Z_{γ/β} is split in Ĉ for all α, β, γ in Δ̄_{m} with γ^{−1} ⩽ α ⩽ β ⩽ γ ⩽ α^{+1}.

The long exact sequence

⋯ \overline{}Z_{α/β^{−1}}  \overline{}Z_{α/γ^{−1}}  \overline{}Z_{β/γ^{−1}}  \overline{}Z_{β/α} \overline{}Z_{γ/α} \overline{}Z_{γ/β} \overline{}Z_{α^{+1}/β} \overline{}⋯

in Ĉ shows that it suffices to show that the morphism Z_{β/α} ^{\overline{}^{z}} Z_{γ/α} is split in Ĉ for all 0 ⩽ α ⩽ β ⩽ γ < 0^{+1}. In fact, first of all we may assume that 0 ⩽ α < 0^{+1}, so that 0 ⩽ α ⩽ β ⩽ γ ⩽ α^{+1} < 0^{+2}. Hence either 0 ⩽ α ⩽ β ⩽ γ < 0^{+1}, or 0 ⩽ γ^{−1} ⩽ α ⩽ β < 0^{+1}, or 0 ⩽ β^{−1} ⩽ γ^{−1} ⩽ α < 0^{+1}.

Now we may assume that 0 < α and apply Lemma A.19 to the weak square (Z_{β/0}, Z_{γ/0}, Z_{β/α}, Z_{γ/α}), in which Z_{β/0} ^{\overline{}^{z}} Z_{γ/0} is split in Ĉ by assumption, in which Z_{β/0} ^{\overline{}^{z}} Z_{β/α} is split in Ĉ since Z_{α/0} ^{\overline{}^{z}} Z_{β/0} is split in Ĉ by assumption, and in which Z_{γ/0} ^{\overline{}^{z}} Z_{γ/α} is split in Ĉ since Z_{α/0} ^{\overline{}^{z}} Z_{γ/0} is split in Ĉ by assumption.

So the folding operation passes to an operation

[figure: \underline{C^{+}((2Δ̄_{n})^{#})}; ^{\underline{𝔣}^{n}}; \underline{C^{+}(\overline{ρ ⊔ Δ_{n}}^{#})}; X; X\underline{𝔣}_{n}]

on the stable categories.

1.2.2.3 An example: folding from Δ̄_{5}^{#} to Δ̄_{3}^{#}

Let D = Δ_{2}. Note that 2Δ̄_{2} ≃ Δ̄_{5}. Let X ∈ Ob C^{+}((2Δ̄_{2})^{#}), depicted as follows.

[figure: 0; X_{1^{+1}/0^{+1}}; x; +; X_{2^{+1}/0^{+1}}; x; +; X_{0^{+2}/0^{+1}}; x; +; X_{1^{+2}/0^{+1}}; x; +; X_{2^{+2}/0^{+1}}; +; 0; 0; X_{0^{+1}/2}; x; +; X_{1^{+1}/2}; x; x; +; X_{2^{+1}/2}; x; x; +; X_{0^{+2}/2}; x; x; +; X_{1^{+2}/2}; x; +; 0; 0; X_{2/1}; x; +; X_{0^{+1}/1}; x; x; +; X_{1^{+1}/1}; x; x; +; X_{2^{+1}/1}; x; x; +; X_{0^{+2}/1}; x; +; 0; 0; X_{1/0}; x; +; X_{2/0}; x; x; +; X_{0^{+1}/0}; x; x; +; X_{1^{+1}/0}; x; x; +; X_{2^{+1}/0}; x; +; 0; 0; X_{0/2^{−1}}; x; +; X_{1/2^{−1}}; x; x; +; X_{2/2^{−1}}; x; x; +; X_{0^{+1}/2^{−1}}; x; x; +; X_{1^{+1}/2^{−1}}; x; +; 0; +; +; +; +; +]

Note that ρ ⊔ Δ_{2} ≃ Δ_{3}. Folding turns X into X𝔣_{2} ∈ Ob C^{+}(\overline{ρ ⊔ Δ_{2}} ^{#}), depicted as follows.

[figure: 0; 0; X_{2^{+2}/2^{+1}}; +; 0; X_{2^{+1}/1^{+1}} ⊕X_{1^{+2}/2}; _{( \begin{matrix}x \\ −x \end{matrix} )}; +; X_{1^{+2}/1^{+1}}; x; +; 0; X_{1^{+1}/0^{+1}} ⊕X_{0^{+2}/1}; _{( \begin{matrix}x & 0 \\ 0 & x \end{matrix} )}; +; X_{2^{+1}/0^{+1}} ⊕X_{0^{+2}/2}; _{( \begin{matrix}x \\ −x \end{matrix} )}; _{( \begin{matrix}x & 0 \\ 0 & x \end{matrix} )}; +; X_{0^{+2}/0^{+1}}; x; +; 0; X_{0^{+1}/0}; x; +; X_{1^{+1}/1}; x; _{(_{x x})}; +; X_{2^{+1}/2}; _{(_{x x})}; +; 0; 0; X_{2/2^{−1}}; _{(_{x x})}; +; X_{2/0}⊕X_{0^{+1}/2^{−1}}; _{( \begin{matrix}x & 0 \\ 0 & x \end{matrix} )}; _{( \begin{matrix}x \\ −x \end{matrix} )}; +; X_{2/1}⊕X_{1^{+1}/2^{−1}}; _{( \begin{matrix}x \\ −x \end{matrix} )}; +; 0; +; +; +]

1.2.3 A definition of Heller triangulated categories and strictly exact functors

Recall that C is a weakly abelian category, and that T = (−)^{+1} is an automorphism of C.

Suppose given n ⩾ 0. We have introduced the automorphisms

\underline{C^{+}(Δ̄_{n}^{#})}  \overline{}^{\underline{C^{+}(T_{n})}}_{∼}  \underline{C^{+}(Δ̄_{n}^{#})}  (outer shift; §1.2.1.2)
X  \overline{}[X]^{+1}

\underline{C^{+}(Δ̄_{n}^{#})}  \overline{}^{\underline{T^{+}(Δ̄_{n}^{#})}}_{∼}  \underline{C^{+}(Δ̄_{n}^{#})}  (inner shift; §1.2.1.3)
X  \overline{}[X^{+1}] .

The outer shift shifts the whole diagram X ∈ Ob \underline{C^{+}(Δ̄_{n}^{#})} one step downwards – the object X_{α^{+1}/β} is the entry of [X]^{+1} at position β/α.

The inner shift applies the given shift automorphism (−)^{+1} of C entrywise to a diagram X ∈ Ob \underline{C^{+}(Δ̄_{n}^{#})}.

Furthermore, we write [X^{+a}]^{+b} := X T^{+}(Δ̄_{n}^{#})^{a} C^{+}(T_{n})^{b} = XC^{+}(T_{n})^{b} T^{+}(Δ̄_{n}^{#})^{a} for a, b ∈ Z_{⩾0} and X ∈ Ob C^{+}(Δ̄_{n}^{#}); similarly for a, b ∈ Z. Likewise in the stable case.

Definition 1.5

(i) A Heller triangulation on (C, T) is a tuple of isomorphisms of functors

ϑ = ( \underline{C^{+}(T_{n})} \overline{}^{ϑ_{n}}_{∼} \underline{T^{+}(Δ̄_{n}^{#})} )_{n⩾0} = ( [−]^{+1} \overline{}^{ϑ_{n}}_{∼} [−^{+1}] )_{n⩾0}

such that

(∗)  \underline{p}^{#} ⋆ ϑ_{m} = ϑ_{n} ⋆ \underline{p}^{#}

for all n, m ⩾ 0 and all periodic monotone maps Δ̄_{n} ^{\overline{}^{p}} Δ̄_{m} in Δ̄, and such that

(∗∗)  \underline{𝔣}_{n} ⋆ ϑ_{n+1} = ϑ_{2n+1} ⋆ \underline{𝔣}_{n}

for all n ⩾ 0.

Note that given n ⩾ 0, the isomorphism ϑ_{n} consists of isomorphisms

[X]^{+1} \overline{}^{Xϑ_{n}}_{∼} [X^{+1}]

in the stable category \underline{C^{+}(Δ̄_{n}^{#})} of n-pretriangles, where X runs over the set Ob \underline{C^{+}(Δ̄_{n}^{#})} of n-pretriangles.

Condition (∗) asserts that the following diagram commutes in Add for all n, m ⩾ 0 and all

periodic monotone maps Δ̄_{n} ^{\overline{}^{p}} Δ̄_{m} in Δ̄.

[figure: \underline{C^{+}(Δ̄_{n}^{#})}; \underline{p}^{#} = \underline{C^{+}(p^{#})}; [−]^{+1}; [−^{+1}]; \underline{C^{+}(Δ̄_{m}^{#})}; [−]^{+1}; [−^{+1}]; \underline{C^{+}(Δ̄_{n}^{#})}; \underline{p}^{#} = \underline{C^{+}(p^{#})}; \underline{C^{+}(Δ̄_{m}^{#})}; ϑ_{n}; ϑ_{m}]

Condition (∗∗) asserts that the following diagram commutes in Add for all n ⩾ 0.

[figure: \underline{C^{+}(Δ̄_{2n+1}^{#})}; ^{\underline{𝔣}^{n}}; [−]^{+1}; [−^{+1}]; \underline{C^{+}(Δ̄_{n+1}^{#})}; [−]^{+1}; [−^{+1}]; \underline{C^{+}(Δ̄_{2n+1}^{#})}; _{\underline{𝔣}_{n}}; \underline{C^{+}(Δ̄_{n+1}^{#})}; ϑ_{2n+1}; ϑ_{n+1}]

(ii) Given a Heller triangulation ϑ on (C, T), we use the following terminology.

(1) The triple (C, T, ϑ) forms a Heller triangulated category, usually just denoted by C.

(2) Given n ⩾ 0, an n-triangle is an object X of C^{+}(Δ̄_{n}^{#}) for which [X^{+1}] = [X]^{+1} in Ob C^{+}(Δ̄_{n}^{#}), and for which

Xϑ_{n} = 1_{[X]^{+1}} = 1_{[X^{+1}]} (equality in \underline{C^{+}(Δ̄_{n}^{#})}) .

A morphism of n-triangles is a morphism X ^{\overline{}^{u}} Y in C^{+}(Δ̄_{n}^{#}) between n-triangles X and Y such that [u]^{+1} = [u^{+1}]. The category of n-triangles and morphisms of n-triangles is denoted by C^{+, ϑ=1}(Δ̄_{n}^{#}).

In the notation C^{+, ϑ=1}(Δ̄_{n}^{#}), the index “ ϑ=1 ” is to be read as a symbol, not as an actual equation. The subcategory of n-triangles C^{+, ϑ=1}(Δ̄_{n}^{#}) in the category of n-pretriangles C^{+}(Δ̄_{n}^{#}) is not full in general.

(iii) An additive functor C ^{\overline{}^{F}} C′ between Heller triangulated categories (C, T, ϑ) and (C′, T′, ϑ′) is called strictly exact if the following conditions hold.

(1) F T′ = T F.
(2) F is subexact; cf. §1.2.1.3.

(3) We have

(∗∗∗)  ϑ_{n} ⋆ \underline{F^{+}(Δ̄_{n}^{#})} = \underline{F^{+}(Δ̄_{n}^{#})} ⋆ ϑ′_{n}

for all n ⩾ 0.

Such a functor F is called strictly exact because of the equality in (1). Condition (∗∗∗) asserts that the following diagram commutes in Add for all n ⩾ 0.

[figure: \underline{C^{+}(Δ̄_{n}^{#})}; \underline{F^{+}(Δ̄_{n}^{#})}; [−]^{+1}; [−^{+1}]; \underline{C^{′+}(Δ̄_{n}^{#})}; [−]^{+1}; [−^{+1}]; \underline{C^{+}(Δ̄_{n}^{#})}; \underline{F^{+}(Δ̄_{n}^{#})}; \underline{C^{′+}(Δ̄_{n}^{#})}; ϑ_{n}; ϑ′_{n}]

To summarise Definition 1.5 roughly, a Heller triangulation is an isomorphism ϑ from the outer shift to the inner shift, varying with Δ_{n}, and compatible with folding. An n-triangle is a periodic n-pretriangle at which ϑ is an identity. A strictly exact functor respects the weakly abelian structure and is compatible with shift and ϑ.

Note that if ϑ is a Heller triangulation on (C, T), so is −ϑ.

Definition 1.5 would make sense for periodic, but not necessarily linearly ordered posets, generalising Δ̄_{n}. But then it is unknown whether, and, it seems to the author, not very probable that the stable category of a Frobenius category is triangulated in this generalised sense. More specifically, it seems to be impossible to generalise Proposition 2.5 below accordingly, which is the technical core of our approach.

Question 1.6 Does there exist an additive functor C ^{\overline{}^{F}} C′ between Heller triangulated categories that, in Definition 1.5.(iii), satisfies (1) and (2), but (3) only for n ⩽ 2 ? If F is an identity, this amounts to asking for the existence of two Heller triangulations ϑ and ϑ′ on (C, T), C weakly abelian, T automorphism of C, such that ϑ_{n} = ϑ′_{n} only for n ⩽ 2.

2 Some equivalences

Suppose given n ⩾ 0. Suppose given a weakly abelian category C, together with an automorphism T : C \overline{}C, X \overline{}X^{+1}. Concerning the Freyd category Ĉ of C, we refer to §A.6.3.

We shall show in Proposition 2.6 that the functor \underline{C(Δ̄_{n}^{#})} \overline{}\underline{C(Δ̇_{n})}, induced on the stable categories by restriction from Δ̄_{n}^{#} to Δ̇_{n} := [1, n], is an equivalence.

2.1 Some notation

2.1.1 Some posets

Let Δ̇_{n} := Δ_{n} ∖ {0} = [1, n], considered as a linearly ordered set. We have an injection Δ̇_{n} \overline{}Δ̄_{n}^{#}, i \overline{}i/0, and identify Δ̇_{n} with its image in Δ̄_{n}^{#}.

We define two subposets of Δ̄_{n}^{#} by

\begin{aligned}Δ̄_{n}^{△} & := & \{β/α ∈ Δ̄_{n}^{#} : 0 ⩽ α\} \\ Δ̄_{n}^{▽} & := & \{β/α ∈ Δ̄_{n}^{#} : α ⩽ 0\} . \end{aligned}

Then Δ̇_{n} = (Δ̄_{n}^{△} ∩ Δ̄_{n}^{▽}) ∖ {0/0, 0^{+1}/0}.

[figure: Δ̄_{n}^{△}; Δ̄_{n}^{▽}; Δ̇_{n}]

2.1.2 Fixing parametrisations κ^{△}, κ^{▽} There exists a bijective morphism Δ̄_{n}^{△} \overline{}Z_{⩾0} of posets (“refining the partial to a linear order”). We fix such a morphism and denote by Z_{⩾0} ^{\overline{}^{κ^{△}}} Δ̄_{n}^{△} its inverse (as a map of sets; in general, κ^{△} is not monotone). So whenever κ^{△}(ℓ) ⩽ κ^{△}(ℓ′), then ℓ ⩽ ℓ′. In particular, κ^{△}(0) = 0/0. There exists a bijective morphism Δ̄_{n}^{▽} \overline{}Z_{⩽0} of posets. We fix such a morphism and denote by Z_{⩾0} ^{\overline{}^{κ^{▽}}} Δ̄_{n}^{▽} its inverse (as a map of sets). So whenever κ^{▽}(ℓ) ⩽ κ^{▽}(ℓ′), then ℓ ⩽ ℓ′. In particular, κ^{▽}(0) = 0^{+1}/0.

2.1.3 The categories Ĉ^{+,∗}(Δ̄_{n}^{△}), C^{+}(Δ̄_{n}^{△}) etc. Let A be an abelian category, and let B ⊆ A be a full subcategory. Let E ⊆ Δ̄_{n}^{#} be a full subposet.

For example, for E we may take the subposets Δ̄_{n}^{△}, Δ̄_{n}^{▽} or Δ̄_{n}^{△} ∩ Δ̄_{n}^{▽,+1} of Δ̄_{n}^{#}.

Moreover, for example, we may take A = Ĉ and for B either Ĉ or C.

Let B^{+,∗}(E) be the full subcategory of B(E) defined by

Ob B^{+,∗}(E) := \{ X ∈ Ob B(E) : \begin{matrix}For all δ^{−1} ⩽ α ⩽ β ⩽ γ ⩽ δ ⩽ α^{+1} in Δ̄_{n} \\ such that γ/α, γ/β, δ/α and δ/β are in E, \\ the quadrangle \\ X_{γ/β} & x & [figure] & X_{δ/β} & X_{γ/α} & x & [figure] & x & [figure] & + & X_{δ/α} & x & [figure] \\ is a weak square (as indicated by +). \end{matrix} \} .

The symbol ∗ should remind us of the fact that we still allow X_{α/α} resp. X_{α^{+1}/α} to be arbitrary for α ∈ Δ̄_{n} such that α/α ∈ E resp. α^{+1}/α ∈ E. In turn, let B^{+}(E) be the full subcategory of B^{+,∗}(E) defined by

Ob B^{+}(E) := \{ X ∈ Ob B^{+,∗}(E) : \begin{matrix}X_{α/α} = 0 for α ∈ Δ̄_{n} such that α/α ∈ E , and \\ X_{α^{+1}/α} = 0 for α ∈ Δ̄_{n} such that α^{+1}/α ∈ E. \end{matrix} \} .

2.1.4 Reindexing Given a subposet E ⊆ Δ̄_{n}^{#}, we have a reindexing equivalence

[figure: C(E); ∼; C(E^{+1}); X; X^{(−1)}; X^{(+1)}; X]

defined by

(X^{(−1)})_{β/α} := X_{(β/α)^{−1}} = X_{α/β^{−1}} ,

where β/α ∈ E^{+1}; and inversely by

(X^{(+1)})_{β/α} := X_{(β/α)^{+1}} = X_{α^{+1}/β} ,

where β/α ∈ E. This equivalence restricts to an equivalence between C^{+}(E) and C^{+}(E^{+1}).

For instance, if E = Δ̄_{n}^{#}, then X^{(+1)} = [X]^{+1}. The outer shift and reindexing will play different roles, and so we distinguish in notation.

2.2 Density of the restriction functor from Δ̄_{n}^{#} to Δ̇_{n}

2.2.1 Upwards and downwards spread

Let the upwards spread S^{△} be defined by

[figure: Ĉ(Δ̇_{n}); S^{△}; Ĉ^{+}(Δ̄_{n}^{△}); X; XS^{△} ,]

where XS^{△} is given by

(XS^{△})_{0/0} := 0
(XS^{△})_{β/0} := X_{β}  for β ∈ Δ̇_{n}
(XS^{△})_{β/α} := Cokern(X_{α} ^{\overline{}^{x}} X_{β}) for α, β ∈ Δ̇_{n} with α ⩽ β
(XS^{△})_{β/α} := 0  for α, β ∈ Δ̄_{n} with 0^{+1} ⩽ β ⩽ α^{+1} ⩽ β^{+1} ,

the diagram being completed with the induced morphisms between the cokernels and zero morphisms elsewhere.

This construction is functorial in X. The functor S^{△} is left adjoint to the restriction functor from Ĉ^{+}(Δ̄_{n}^{△}) to Ĉ(Δ̇_{n}), with unit being the identity, i.e. X = XS^{△}|Δ̇_{n}.

Dually, let the downwards spread S^{▽} be

[figure: Ĉ(Δ̇_{n}); S^{▽}; Ĉ^{+}(Δ̄_{n}^{▽}); X; XS^{▽} ,]

where XS^{▽} is given by

(XS^{▽})_{0^{+1}/0} := 0
(XS^{▽})_{α/0} := X_{α}  for α ∈ Δ̇_{n}
(XS^{▽})_{α/β^{−1}} := Kern(X_{α} ^{\overline{}^{x}} X_{β}) for α, β ∈ Δ̇_{n} with α ⩽ β
(XS^{▽})_{α/β^{−1}} := 0  for α, β ∈ Δ̄_{n} with α^{−1} ⩽ β^{−1} ⩽ α ⩽ 0 ,

the diagram being completed with the induced morphisms between the kernels and zero morphisms elsewhere.

This construction is functorial in X. The functor S^{▽} is right adjoint to the restriction functor from Ĉ^{+}(Δ̄_{n}^{▽}) to Ĉ(Δ̇_{n}), with counit being the identity, i.e. XS^{▽}|Δ̇_{n} = X.

2.2.2 Resolutions

2.2.2.1 A stability under pointwise pushouts and pullbacks

Let E ⊆ Δ̄_{n}^{#} be a full subposet. Moreover, assume that E is a convex subposet, i.e. that whenever given ξ, ζ ∈ E and η ∈ Δ̄_{n}^{#} such that ξ ⩽ η ⩽ ζ, then η ∈ E.

An element δ/β ∈ E is on the left boundary of E if we may conclude from γ/β ∈ E and γ ⩽ δ that γ = δ. It is on the lower boundary of E if we may conclude from δ/α ∈ E and α ⩽ β that α = β.

An element γ/α ∈ E is on the right boundary of E if we may conclude from δ/α ∈ E and γ ⩽ δ that γ = δ. It is on the upper boundary of E if we may conclude from γ/β ∈ E and α ⩽ β that α = β.

Let A be an abelian category. Concerning pointwise pullbacks and pointwise pushouts, we refer to §A.7.

Lemma 2.1 Suppose given ε ∈ E and an object X of A^{+,∗}(E).

(1) Given a monomorphism X_{ε} ^{\overline{}^{x′}} X′ in A, the pointwise pushout X↑^{x′} of X along x′ is an object of A^{+,∗}(E) again.

(2) Given an epimorphism X_{ε} ^{\overline{}^{x′}} X′ in A, the pointwise pullback X↑_{x′} of X along x′ is an object of Ob A^{+,∗}(E) again.

(3) Suppose that ε is on the left boundary or on the lower boundary of E. Given a morphism X_{ε} ^{\overline{}^{x′}} X′ in A, the pointwise pushout X↑^{x′} of X along x′ is an object of A^{+,∗}(E) again.

(4) Suppose that ε is on the right boundary or on the upper boundary of E. Given a morphism X_{ε} ^{\overline{}^{x′}} X′ in A, the pointwise pullback X ↑_{x′} of X along x′ is an object of Ob A^{+,∗}(E) again.

Proof.  Ad (1).  First we remark that by Lemma A.15, the quadrangle

(X_{β/α} , X_{δ/γ} , (X↑^{x′})_{β/α} , (X↑^{x′})_{δ/γ}) is a pushout for ε ⩽ β/α ⩽ δ/γ in E.

We have to show that the quadrangle of X↑^{x′} on (γ/α, δ/α, γ/β, δ/β), where δ^{−1} ⩽ α ⩽ β ⩽ γ ⩽ δ ⩽ α^{+1} in Δ̄_{n}, is a weak square, provided its corners have indices in E. Using Lemmata A.14, A.16 and convexity of E, we are reduced to the case ε ⩽ γ/α. In this case, the assertion follows by Lemma A.15.

Ad (3). Here we need only Lemma A.14 and convexity of E to reduce to the case ε ⩽ γ/α, the rest of the argument is as in (1). Hence the morphism x′ may be arbitrary.

2.2.2.2 Upwards and downwards resolution

Remark 2.2

(1) Given a direct system X_{0} \overline{}X_{1} \overline{}X_{2} \overline{}⋯ in Ĉ^{+,∗}(Δ̄_{n}^{△}) such that its restriction to any finite full subposet E ⊆ Δ̄_{n}^{△} eventually becomes constant, then the direct limit \underrightarrow{lim}_{i} X_{i} exists in Ĉ^{+,∗}(Δ̄_{n}^{△}).

(2) Given an inverse system X_{0} \overline{}X_{1} \overline{}X_{2} \overline{}⋯ in Ĉ^{+,∗}(Δ̄_{n}^{▽}) such that its restriction to any finite subposet E ⊆ Δ̄_{n}^{▽} eventually becomes constant, then the inverse limit \underleftarrow{lim}_{i} X_{i} exists in Ĉ^{+,∗}(Δ̄_{n}^{▽}).

For k ⩾ 0, we let

[figure: Ob Ĉ^{+,∗}(Δ̄_{n}^{△}); R_{k}^{△}; Ob Ĉ^{+,∗}(Δ̄_{n}^{△}); X; X↑^{x′(k)} ,]

where

x′(k) := \{ \begin{matrix}(X_{κ^{△}(k)} \overline{}0) & if κ^{△}(k) ∈ \{α/α, α^{+1}/α\} for some α ∈ Δ̄_{n} with 0 ⩽ α \\ (X_{κ^{△}(k)} ^{\overline{}^{ι}} X_{κ^{△}(k)}I) & if κ^{△}(k) = β/α for some α, β ∈ Δ̄_{n} with 0 ⩽ α < β < α^{+1} \end{matrix}

Define the upwards resolution map by

Ob Ĉ^{+,∗}(Δ̄_{n}^{△}) ^{\overline{}^{R^{△}}} Ob C^{+}(Δ̄_{n}^{△})
X \overline{}XR^{△} := \underrightarrow{lim}_{m} XR_{0}^{△} ⋯ R_{m}^{△} ,

the direct system being given by the transition morphisms

XR_{0}^{△} ⋯ R_{m}^{△} ^{\overline{}^{i}} (XR_{0}^{△} ⋯ R_{m}^{△})R_{m+1}^{△} .

We have XR^{△} = X for X ∈ Ob C^{+}(Δ̄_{n}^{△}).

Note that we apparently cannot turn the upwards resolution into a functor unless we are in a particular case in which the map I on objects can be turned into a functor.

Dually, for k ⩾ 0, we let

[figure: Ob Ĉ^{+,∗}(Δ̄_{n}^{▽}); R_{k}^{▽}; Ob Ĉ^{+,∗}(Δ̄_{n}^{▽}); X; X↑_{x′′(k)} ,]

where

x′′(k) := \{ \begin{matrix}(X_{κ^{▽}(k)} \overline{}0) & if κ^{▽}(k) ∈ \{α/α, α^{+1}/α\} for some α ∈ Δ̄_{n} with α ⩽ 0 \\ (X_{κ^{▽}(k)} ^{\overline{}^{π}} X_{κ^{▽}(k)}P) & if κ^{▽}(k) = β/α for some α, β ∈ Δ̄_{n} with α^{−1} < β^{−1} < α ⩽ 0 \end{matrix}

Define the downwards resolution map by

Ob Ĉ^{+,∗}(Δ̄_{n}^{▽}) ^{\overline{}^{R^{▽}}} Ob C^{+}(Δ̄_{n}^{▽})
X \overline{}XR^{▽} := \underleftarrow{lim}_{m} XR_{0}^{▽} ⋯ R_{m}^{▽} ,

the inverse system being given by the transition morphisms

XR_{0}^{▽} ⋯ R_{m}^{▽} ^{\overline{}^{p}} (XR_{0}^{▽} ⋯ R_{m}^{▽})R_{m+1}^{▽} .

We have XR^{▽} = X for X ∈ Ob C^{+}(Δ̄_{n}^{▽}).

Lemma 2.3

(1) Given a morphism Y ^{\overline{}^{g}} X in Ĉ^{+}(Δ̄_{n}^{△}) with X ∈ Ob C^{+}(Δ̄_{n}^{△}), there exists a factorisation

(Y ^{\overline{}^{g}} X) = (Y \overline{}Y R^{△} \overline{}X) .

(2) Given a morphism Y ^{\overline{}^{g}} X in Ĉ^{+}(Δ̄_{n}^{▽}) with X ∈ Ob C^{+}(Δ̄_{n}^{▽}), there exists a factorisation

(Y ^{\overline{}^{g}} X) = (Y \overline{}Y R^{▽} \overline{}X) .

Proof. Ad (1). Since the entries of X are injective in Ĉ and since X_{α/α} = 0 and X_{α^{+1}/α} = 0 for α ⩾ 0, we obtain, using the universal property of the pointwise pushout, a factorisation

(Y ^{\overline{}^{g}} X) = (Y \overline{}Y R_{0}^{△} ⋯ R_{m}^{△} \overline{}X)

for every m ⩾ 0, compatible with the transition morphisms, resulting in a factorisation over Y R^{△} = \underrightarrow{lim}_{m} Y R_{0}^{△} ⋯ R_{m}^{△}.

2.2.2.3 Both-sided resolutions

Let the resolution map

[figure: Ob C(Δ̇_{n}); R; Ob C^{+}(Δ̄_{n}^{#}); X; XR]

be defined by gluing an upper and a lower part along Δ̇_{n} as follows.

\begin{aligned}XR|Δ̄_{n}^{△} & := & XS^{△}R^{△} \\ XR|Δ̄_{n}^{▽} & := & XS^{▽}R^{▽} \end{aligned}

This is welldefined, since XS^{△}R^{△}|Δ̇_{n} = X = XS^{▽}R^{▽}|Δ̇_{n}. In particular, we obtain

XR|Δ̇_{n} = X .

We summarise.

Proposition 2.4 The restriction functor

[figure: C^{+}(Δ̄_{n}^{#}); (−)|Δ̇n; C(Δ̇_{n}); Y; Y |Δ̇_{n}]

is strictly dense, i.e. it is surjective on objects.

2.3 Fullness of the restriction functor from Δ̄_{n}^{#} to Δ̇_{n}

Proposition 2.5 The restriction functors

[figure: C^{+}(Δ̄_{n}^{#}); (−)|Δ̇n; C(Δ̇_{n}); Y; Y |Δ̇_{n}]

and

[figure: C^{+}(Δ̄_{n}^{△}); (−)|Δ̇n; C(Δ̇_{n}); Y; Y |Δ̇_{n}]

are full.

Proof. By duality and gluing along Δ̇_{n}, it suffices to consider the restriction from Δ̄_{n}^{△} to Δ̇_{n}. So suppose given X, Y ∈ Ob C^{+}(Δ̄_{n}^{△}) and a morphism X|Δ̇_{n} ^{\overline{}^{f}} Y |Δ̇_{n}. We have to find a morphism X ^{\overline{}^{f^{△}}} Y such that f^{△}|Δ̇_{n} = f. We construct f_{κ^{△}(ℓ)}^{△} for ℓ ⩾ 0 by induction on ℓ. At κ^{△}(0), we let f_{κ^{△}(0)}^{△} := 1_{0}. Suppose given ℓ ⩾ 1. If κ^{△}(ℓ) ∈ Δ̇_{n}, we let f_{κ^{△}(ℓ)}^{△} := f_{κ^{△}(ℓ)}. If κ^{△}(ℓ) ∈ {α/α, α^{+1}/α} for some α ⩾ 0, we let f_{κ^{△}(ℓ)}^{△} := 1_{0}. If κ^{△}(ℓ) =: β/α with 0 < α < β < α^{+1}, then we let α′ := α − 1 be the predecessor of α in Δ̄_{n}, and we let β′ := β − 1 be the predecessor of β in Δ̄_{n}, using that Δ̄_{n} is linearly ordered. We may complete the diagram

[figure: Y_{β′/α}; y; Y_{β/α}; X_{β′/α}; f_{β′/α}^{△}; x; X_{β/α}; Y_{β′/α′}; y; y; +; Y_{β/α′}; y; X_{β′/α′}; f_{β′/α′}^{△}; x; x; +; X_{β/α′}; f_{β/α′}^{△}; x]

to a commutative cuboid, inserting a morphism X_{β/α} ^{\overline{}^{f_{β/α}^{△}}} Y_{β/α}.

Since we need the restriction functor C^{+}(Δ̄_{n}^{#})^{\overline{}^{(−)|Δ̇n}} C(Δ̇_{n}) to be full, we are not able to generalise from linearly ordered periodic posets to arbitrary periodic posets.

2.4 The equivalence between \underline{C^{+}(Δ̄_{n}^{#})} and \underline{C(Δ̇_{n})}

Let C^{split}(Δ̇_{n}) be the full subcategory of C(Δ̇_{n}) defined by Ob C^{split}(Δ̇_{n}) := \{X ∈ Ob C(Δ̇_{n}) : X_{α} \overline{}X_{β} is split in Ĉ for all α, β ∈ Δ̇_{n} with α ⩽ β\} . We denote the quotient category by

\underline{C(Δ̇_{n})} := C(Δ̇_{n})/C^{split}(Δ̇_{n}) .

Proposition 2.6 The functor

[figure: \underline{C^{+}(Δ̄_{n}^{#})}; (−)|Δ̇n; \underline{C(Δ̇_{n})}; X; X|Δ̇_{n} ,]

induced by restriction from Δ̄_{n}^{#} to Δ̇_{n}, is an equivalence. Proof. By Propositions 2.4 and 2.5, we may invoke Lemma A.1. Moreover, Lemma 1.4 gives the inverse image of Ob C^{split}(Δ̇_{n}) under C^{+}(Δ̄_{n}^{#})^{\overline{}^{(−)|Δ̇n}} C(Δ̇_{n}) as Ob C^{+, split}(Δ̄_{n}^{#}).

Consider a morphism X ^{\overline{}^{f}} X′ in C^{+}(Δ̄_{n}^{#}) such that (X ^{\overline{}^{f}} X′)|Δ̇_{n} is zero in C(Δ̇_{n}). We have to prove that it factors over an object of C^{+, split}(Δ̄_{n}^{#}). Let Y^{△} be the cokernel in Ĉ(Δ̄_{n}^{△}) of the counit X|Δ̇_{n}S^{△} \overline{}X|Δ̄_{n}^{△} at X|Δ̄_{n}^{△}. Note that Y^{△}|Δ̇_{n} = 0. By Lemma A.21, we have Y^{△} ∈ Ob Ĉ^{+}(Δ̄_{n}^{△}). Consider the following diagram in Ĉ^{+}(Δ̄_{n}^{△}).

\xymatrix{X|Δ̇_{n}S^{△} \ar[r] \ar[d]_{0} & X|Δ̄_{n}^{△} \ar[r] \ar[d]_{f|Δ̄^{△}_{n}} & Y^{△} \ar[r] \ar@{.>}[dl] & Y^{△}R^{△} \ar@{.>}[dll] \\ X′|Δ̇_{n}S^{△} \ar[r] & X′|Δ̄_{n}^{△}}

The indicated factorisation

(X|Δ̄_{n}^{△} ^{\overline{}^{f|Δ̄^{△}_{n}}} X′|Δ̄_{n}^{△}) = (X|Δ̄_{n}^{△} \overline{}Y^{△} \overline{}X′|Δ̄_{n}^{△})

ensues from the universal property of the cokernel Y^{△}. By Lemma 2.3.(1), we can factorise further to obtain

(∗^{△})  (X|Δ̄_{n}^{△} ^{\overline{}^{f|Δ̄^{△}_{n}}} X′|Δ̄_{n}^{△}) = (X|Δ̄_{n}^{△} \overline{}Y^{△}R^{△} \overline{}X′|Δ̄_{n}^{△}) .

Dually, we obtain a factorisation

(∗^{▽})  (X|Δ̄_{n}^{▽} ^{\overline{}^{f|Δ̄^{▽}_{n}}} X′|Δ̄_{n}^{▽}) = (X|Δ̄_{n}^{▽} \overline{}Y^{▽}R^{▽} \overline{}X′|Δ̄_{n}^{▽})

for some Y^{▽} ∈ Ob Ĉ^{+}(Δ̄_{n}^{▽}) such that Y^{▽}|Δ̇_{n} = 0. Since Y^{△}R^{△}|Δ̇_{n} = 0 = Y^{▽}R^{▽}|Δ̇_{n}, there is a unique N ∈ Ob C^{+}(Δ̄_{n}^{#}) such that N|Δ̄_{n}^{△} = Y^{△}R^{△} and N|Δ̄_{n}^{▽} = Y^{▽}R^{▽}. By Lemma 1.4, we have N ∈ Ob C^{+, split}(Δ̄_{n}^{#}). Moreover, since both factorisations (∗^{△}) and (∗^{▽}) restrict to the factorisation

(X|Δ̇_{n} ^{\overline{}^{0}} X′|Δ̇_{n}) = (X|Δ̇_{n} \overline{}0 \overline{}X′|Δ̇_{n})

in C(Δ̇_{n}), we may glue to a factorisation

(∗)  (X ^{\overline{}^{f}} X′) = (X \overline{}N \overline{}X′)

that restricts to (∗^{△}) in C^{+}(Δ̄_{n}^{△}) and to (∗^{▽}) in C^{+}(Δ̄_{n}^{▽}).

2.5 Auxiliary equivalences

We shall extend the equivalence \underline{C^{+}(Δ̄_{n}^{#})} ^{\overline{}^{∼}} \underline{C(Δ̇_{n})} to a diagram of equivalences
\underline{C^{+}(Δ̄_{n}^{#})} ^{\overline{}^{∼}} \underline{C^{+}(Δ_{n}^{△▽})} ^{\overline{}^{∼}} \underline{C(Δ̇_{n})} ^{\overline{}^{∼}} \underline{Ĉ(Δ̇_{n−1})} .

2.5.1 Factorisation into two equivalences

Abbreviate Δ̄_{n}^{△,+1} := (Δ̄_{n}^{△})^{+1} ⊆ Δ̄_{n}^{#}, Δ̄_{n}^{▽,+1} := (Δ̄_{n}^{▽})^{+1} ⊆ Δ̄_{n}^{#} and Δ̇_{n}^{+1} := (Δ̇_{n})^{+1} ⊆ Δ̄_{n}^{#}. Abbreviate Δ̄_{n}^{△▽} := Δ̄_{n}^{△} ∩ Δ̄_{n}^{▽,+1} = {β/α : α, β ∈ Δ̄_{n}, 0 ⩽ α ⩽ β ⩽ 0^{+1}} ⊆ Δ̄_{n}^{#}.

[figure: Δ̄_{n}^{△▽}; Δ̇_{n}; Δ̄_{n}^{▽}; Δ̄_{n}^{△,+1}; Δ̇_{n}^{+1}]

Let C^{+, split}(Δ̄_{n}^{△▽}) be the full subcategory of C^{+}(Δ̄_{n}^{△▽}) defined by

Ob C^{+, split}(Δ̄_{n}^{△▽}) := \{X ∈ Ob C^{+}(Δ̄_{n}^{△▽}) : \begin{matrix}X_{γ/α} ^{\overline{}^{x}} X_{δ/β} is split in Ĉ \\ for all γ/α, δ/β ∈ Δ̄_{n}^{△▽} \\ with γ/α ⩽ δ/β \end{matrix} \}

We denote the quotient category by

\underline{C^{+}(Δ̄_{n}^{△▽})} := C^{+}(Δ̄_{n}^{△▽})/C^{+, split}(Δ̄_{n}^{△▽})

Lemma 2.7

(1) The restriction functors

[figure: C^{+}(Δ̄_{n}^{#}); (−)|Δ̄{△▽ \atop n}; C^{+}(Δ̄_{n}^{△▽}); (−)|Δ̇n; C^{+}(Δ̇_{n})]
X  \overline{}X|Δ̄_{n}^{△▽}
Y  \overline{}Y |Δ̇_{n}

are full and strictly dense.

(2) The functors

[figure: \underline{C^{+}(Δ̄_{n}^{#})}; (−)|Δ̄{△▽ \atop n}; \underline{C^{+}(Δ̄_{n}^{△▽})}; (−)|Δ̇n; \underline{C^{+}(Δ̇_{n})}]
X  \overline{}X|Δ̄_{n}^{△▽}
Y  \overline{}Y |Δ̇_{n} ,

induced by restriction, are equivalences. Proof. Ad (1). The composition

[figure: C^{+}(Δ̄_{n}^{#}); C^{+}(Δ̄_{n}^{△▽}); C(Δ̇_{n}); X; X|Δ̄_{n}^{△▽}; X|Δ̇_{n}]

is strictly dense by Proposition 2.4 and full by Proposition 2.5. Therefore, the restriction functor from C^{+}(Δ̄_{n}^{△▽}) to C^{+}(Δ̇_{n}) is full and strictly dense. We claim that the restriction functor from C^{+}(Δ̄_{n}^{#}) to C^{+}(Δ̄_{n}^{△▽}) is strictly dense. Let

[figure: C(Δ̇_{n}^{+1}); S^{△,+1}; Ĉ^{+}(Δ̄_{n}^{△,+1}); X; (X^{(+1)}S^{△})^{(−1)} ,]

cf. §2.1.4. Similarly,

[figure: Ob Ĉ^{+}(Δ̄_{n}^{△,+1}); R^{△,+1}; Ob C^{+}(Δ̄_{n}^{△,+1}); X; (X^{(+1)}R^{△})^{(−1)} .]

Given X ∈ Ob C^{+}(Δ̄_{n}^{△▽}), we may define X′ ∈ Ob C^{+}(Δ̄_{n}^{#}) letting

\begin{aligned}X′|Δ̄_{n}^{△,+1} & := & X|Δ̇_{n}^{+1}S^{△,+1}R^{△,+1} \\ X′|Δ̄_{n}^{△▽} & := & X \\ X′|Δ̄_{n}^{▽} & := & X|Δ̇_{n}S^{▽}R^{▽} . \end{aligned}

We claim that the restriction functor from Δ̄_{n}^{#} to Δ̄_{n}^{△▽} is full. Suppose given X, Y ∈ Ob C^{+}(Δ_{n}^{#}) and a morphism X|Δ̄_{n}^{△▽} ^{\overline{}^{f}} Y |Δ̄_{n}^{△▽}. By Proposition 2.6 and a shift, there exists a morphism X|_{Δ_{n}^{△,+1}} ^{\overline{}^{f^{△}}} Y |_{Δ_{n}^{△,+1}} such that f^{△}|Δ̇_{n}^{+1} = f|Δ̇_{n}^{+1}. By Proposition 2.6 and by duality, there exists a morphism X|_{Δ_{n}^{▽}} ^{\overline{}^{f^{▽}}} Y |_{Δ_{n}^{▽}} such that f^{▽}|Δ̇_{n} = f|Δ̇_{n}. We may define a morphism X ^{\overline{}^{f′}} Y letting

\begin{aligned}f′|Δ̄_{n}^{△,+1} & := & f^{△} \\ f′|Δ̄_{n}^{△▽} & := & f \\ f′|Δ̄_{n}^{▽} & := & f^{▽} . \end{aligned}

Ad (2). The composition

[figure: \underline{C^{+}(Δ̄_{n}^{#})}; \underline{C^{+}(Δ̄_{n}^{△▽})}; \underline{C(Δ̇_{n})}; X; X|Δ̄_{n}^{△▽}; X|Δ̇_{n}]

is an equivalence by Proposition 2.6. Therefore, the functor induced by restriction from Δ̄_{n}^{#} to Δ̄_{n}^{△▽} is faithful. By (1), it is full and dense, and so it is an equivalence. Therefore, also the functor induced by restriction from Δ̄_{n}^{△▽} to Δ̇_{n} is an equivalence.

2.5.2 Cutting off the last object

Putting n = 2, the equivalence given in Lemma 2.10, composed with the equivalence in Proposition 2.6, can be used to retrieve Heller’s original isomorphism, called δ(Δ) in [12, p. 53].

In this section, we suppose that n ⩾ 2. Consider the functor

[figure: C(Δ̇_{n}); K; Ĉ(Δ̇_{n−1}); X; XK := (X↑_{0})|Δ̇_{n−1} ,]

where 0 denotes the morphism 0 ^{\overline{}^{0}} X_{n}; cf. §A.7.

Explicitly, we have (XK)_{i} := Kern(X_{i} ^{\overline{}^{x}} X_{n}) for i ∈ [1, n − 1], taken in Ĉ, equipped with the induced morphisms (XK)_{i} \overline{}(XK)_{j} for i, j ∈ [1, n − 1] with i ⩽ j, fitting into a pullback ((XK)_{i}, (XK)_{j}, X_{i}, X_{j}).

Let C^{split}(Δ̇_{n}) ⊆ C(Δ̇_{n}) be the full subcategory defined by

Ob C^{split}(Δ̇_{n}) := \{X ∈ Ob C(Δ̇_{n}) : (X_{i} \overline{}X_{j}) is split in Ĉ for all i, j ∈ [1, n] with i ⩽ j\} ,

and let \underline{C(Δ̇_{n})} := C(Δ̇_{n})/C^{split}(Δ̇_{n}) and \underline{Ĉ(Δ̇_{n−1})} := Ĉ(Δ̇_{n−1})/C^{split}(Δ̇_{n−1}).

For Y ∈ Ob Ĉ(Δ̇_{n}) and i ∈ [1, n], we let Y R_{i} := Y ↑^{Y_{i}ι}∈ Ob Ĉ(Δ̇_{n}).

We have a resolution map

[figure: Ob Ĉ(Δ̇_{n}); R′_{n}; Ob C(Δ̇_{n}); Y; Y R′_{n} := Y R_{0} ⋯ R_{n} .]

If Y ∈ Ob Ĉ(Δ̇_{n}) consists of monomorphisms, then so does Y R′_{n}, whence Y R′_{n} ∈ Ob C^{split}(Δ̇_{n}).

Given a morphism Y \overline{}Y′ in Ĉ(Δ̇_{n}) with Y′ having bijective entries, this morphism factors over Y [figure] Y R′_{n} by injectivity of the entries of Y′ and by the universal property of the pointwise pushout.

Lemma 2.8 The functor K is dense.

Proof. Suppose given X ∈ Ob Ĉ(Δ̇_{n−1}). Let X′ ∈ Ob Ĉ(Δ̇_{n}) be defined by X′|Δ̇_{n−1} := X and X′_{n} := 0. Then X′R′_{n} ∈ Ob C(Δ̇_{n}) has (X′R′_{n})K ≃ X.

Lemma 2.9 The functor K is full.

Proof. Suppose given X, Y ∈ Ob C(Δ̇_{n}) and a morphism XK ^{\overline{}^{f}} Y K. We claim that there exists a morphism X ^{\overline{}^{f̃}} Y such that f̃K = f. We construct its components f̃_{ℓ} by induction on ℓ.

For ℓ = 1, we obtain a morphism X_{1} ^{\overline{}^{f̃_{1}}} Y_{1} such that ((XK)_{1}, (Y K)_{1}, X_{1}, Y_{1}) commutes, by in-

jectivity of Y_{1} in Ĉ. For ℓ ⩾ 2, we obtain a morphism X_{ℓ} ^{\overline{}^{f̃_{ℓ}}} Y_{ℓ} such that ((XK)_{ℓ}, (Y K)_{ℓ}, X_{ℓ}, Y_{ℓ}) and (X_{ℓ−1}, Y_{ℓ−1}, X_{ℓ}, Y_{ℓ}) commute, by the fact that ((XK)_{ℓ−1}, (XK)_{ℓ}, X_{ℓ−1}, X_{ℓ}) is a weak square and by injectivity of Y_{ℓ}.

Proposition 2.10 The functor K induces an equivalence

[figure: \underline{C(Δ̇_{n})}; \underline{K}; ∼; \underline{Ĉ(Δ̇_{n−1})}; X; XK .]

Proof. Let C̃ ⊆ Ĉ denote the full subcategory of bijective objects in Ĉ. Every object in C̃ is a direct summand of an object in C. Let C̃^{split}(Δ̇_{n−1}) ⊆ C̃(Δ̇_{n−1}) be the full subcategory defined by

Ob C̃^{split}(Δ̇_{n−1}) := \{X ∈ C̃(Δ̇_{n−1}) : (X_{i} \overline{}X_{j}) is split for all i, j ∈ [1, n − 1] with i ⩽ j\} ,

Let Y be an object of C̃^{split}(Δ̇_{n−1}). Then Y R′_{n−1} is an object of C^{split}(Δ̇_{n−1}) that has Y as a direct summand since the identity on Y factors over Y [figure] Y R_{n−1}.

Therefore, any morphism that factors over an object of C̃^{split}(Δ̇_{n−1}) already factors over an object of C^{split}(Δ̇_{n−1}). We infer that

\underline{Ĉ(Δ̇_{n−1})} = Ĉ(Δ̇_{n−1})/C̃^{split}(Δ̇_{n−1}).

Suppose given X ∈ Ob C(Δ̇_{n}). Denote X′ := XK ∈ Ob Ĉ(Δ̇_{n−1}).

We claim that if X ∈ Ob C^{split}(Δ̇_{n}), then X′ ∈ Ob C̃^{split}(Δ̇_{n−1}). First of all, X′_{i} is bijective for i ∈ [1, n − 1], since the image of X_{i} \overline{}X_{n} is bijective, and since X′_{i} is the kernel of this morphism. Now suppose given i, j ∈ [1, n − 1] with i < j. In Ĉ, we let B be the image of X_{i} \overline{}X_{j} and form a pullback (B′, X′_{j}, B, X_{j}). Then there is an induced morphism X′_{i} \overline{}B′ turning (X′_{i}, B′, X_{i}, B) into a commutative quadrangle, which is a pullback by composition to a pullback (X′_{i}, X′_{j}, X_{i}, X_{j}). We insert the common kernel Z of X_{i} \overline{}X_{j} and X′_{i} \overline{}X′_{j}.

[figure: Z; •; X_{i}; B; •; X_{j}; Z; •; X′_{i}; •; B′; •; •; X′_{j}; •]

Hence Z [figure] X′_{i} is split monomorphic, and therefore X′ \overline{}B′ is split epimorphic. Thus B′ is bijective, and so finally B′ [figure] X′_{j} is split monomorphic. This proves the claim.

We claim that if X′ ∈ Ob C̃^{split}(Δ̇_{n−1}), then X ∈ Ob C^{split}(Δ̇_{n}). Suppose given i, j ∈ [1, n − 1] with i < j. We have to show that X_{i} \overline{}X_{j} is split in Ĉ. In Ĉ, we insert the image B of X_{i} \overline{}X_{j} and form a pullback (B′, X′_{j}, B, X_{j}). Since (X′_{i}, B′, X_{i}, B) is a square, and since X′_{i} is bijective, its diagonal sequence is split short exact. Hence B is bijective as a direct summand of X_{i} ⊕ B′, which proves the claim.

Invoking Lemma A.1 to prove the equivalence, it remains to show that given X ^{\overline{}^{f}} Y in C(Δ̇_{n}) such that fK = 0, there exists an object V in C^{split}(Δ̇_{n}) such that there exists a factorisation

(X ^{\overline{}^{f}} Y) = (X \overline{}V \overline{}Y) .

Denote by XK′ ∈ Ob C(Δ̇_{n}) the object that restricts to XK on Δ̇_{n−1} and that has (XK′)_{n} := 0. Let U be the cokernel of XK′ [figure] X and consider the following diagram.

[figure: XK′; •; 0; X; f; U; •; UR_{n}; Y K′; •; Y]

The morphism U \overline{}Y is induced by the universal property of the cokernel. Its factorisation (U \overline{}Y) = (U [figure] UR_{n} \overline{}Y) exists since Y consists of bijective objects.

Since the morphism XK′ [figure] X consists of pullbacks, its cokernel U consists of monomorphisms. Hence so does V := UR_{n}, which is therefore in Ob C^{split}(Δ̇_{n}), as required.

2.5.3 Not quite an equivalence

Let C^{+, periodic}(Δ̄_{n}^{#}) be the subcategory of C^{+}(Δ̄_{n}^{#}) that consists of morphisms X ^{\overline{}^{f}} Y for which

([X^{+1}]^{\overline{}^{[f^{+1}]}} [Y^{+1}]) = ([X]^{+1} ^{\overline{}^{[f]^{+1}}} [Y]^{+1}) ;

which is in general not a full subcategory. The objects C^{+, periodic}(Δ̄_{n}^{#}) are called periodic n-pretriangles, the morphisms are called periodic morphisms of periodic n-pretriangles. Let C^{+, split, periodic}(Δ̄_{n}^{#}) := C^{+, periodic}(Δ̄_{n}^{#}) ∩ C^{+, split}(Δ̄_{n}^{#}).

For instance, if (C, T, ϑ) is a Heller triangulated category, then C^{+, ϑ=1}(Δ̄_{n}^{#}) ⊆ C^{+, periodic}(Δ̄_{n}^{#}) is a full subcategory.

For Y ∈ Ob C(Δ̇_{n}), we define Y S ∈ Ob C^{+}(Δ̄_{n}^{#}) by

\begin{aligned}Y S|Δ̄_{n}^{△} & := & Y S^{△} \\ Y S|Δ̄_{n}^{▽} & := & Y S^{▽} , \end{aligned}

and similarly on morphisms. If Y ∈ Ob C^{split}(Δ̇_{n}), then Y S ∈ Ob C^{+, split}(Δ̄_{n}^{#}) by Lemma 1.4, since Y S|Δ̇_{n} = Y ∈ Ob C^{split}(Δ̇_{n}).

To any X ∈ Ob C^{+}(Δ̄_{n}^{#}) for which X_{β/α} is zero for all but finitely many β/α ∈ Δ̄_{n}^{#} , we can assign its periodification

X̄ := ⨁_{i∈Z}[X^{+i}]^{−i} ∈ Ob C^{+, periodic}(Δ̄_{n}^{#}) ,

and similarly for morphisms between such objects.

If X ∈ Ob C^{+, split}(Δ̄_{n}^{#}), then X̄ ∈ Ob C^{+, split, periodic}(Δ̄_{n}^{#}).

We have the restriction functor

C^{+, split, periodic}(Δ̄_{n}^{#}) ^{\overline{}^{(−)|Δ̇n}} C^{split}(Δ̇_{n})
X  \overline{}X|Δ̇_{n}

which is not faithful in general, as the case n = 2 shows. In the inverse direction, we dispose of the functor

C^{+, split, periodic}(Δ̄_{n}^{#}) ^{\overline{}^{S̄}} C^{split}(Δ̇_{n})
\overline{Y S} =: Y S̄ \overline{}Y .

Lemma 2.11 For X ∈ Ob C^{+, split, periodic}(Δ̄_{n}^{#}), we have X ≃ X|Δ̇_{n} S̄.

Note that we do not claim that 1 ≃ (−)|Δ̇_{n} S̄ as endofunctors of C^{+, split, periodic}(Δ̄_{n}^{#}).

Proof. We have a short exact sequence

X|Δ̇_{n}S|Δ̄_{n}^{△▽}  \overline{}X|Δ̄_{n}^{△▽}  \overline{}[X|Δ̇_{n}S^{+1}]^{−1}|Δ̄_{n}^{△▽}

in Ĉ(Δ̄_{n}^{△▽}), and it suffices to show that it splits. Write C := X|Δ̇_{n}S|Δ̄_{n}^{△▽}.

It suffices to show that there exists a retraction to C \overline{}X|Δ̄_{n}^{△▽}, which we will construct by induction. Suppose given 0 < α < β ⩽ 0^{+1}. We may assume that after restriction of C \overline{}X|Δ̄_{n}^{△▽} to {δ/γ ∈ Δ̄_{n}^{△▽} : δ/γ < β/α}, there exists a retraction. Let α′ := α − 1 be the predecessor of α, and let β′ := β − 1 be the predecessor of β, using that Δ̄_{n} is linearly ordered. It suffices to show that the morphism from the quadrangle (C_{β′/α′}, C_{β′/α}, C_{β/α′}, C_{β/α}) to the quadrangle (X_{β′/α′}, X_{β′/α}, X_{β/α′}, X_{β/α}) has a retraction.

Let (X_{β′/α′}, X_{β′/α}, X_{β/α′}, T) be the pushout in Ĉ. The quadrangle (C_{β′/α′}, C_{β′/α}, C_{β/α′}, C_{β/α}) is a pushout. The induced morphism from (C_{β′/α′}, C_{β′/α}, C_{β/α′}, C_{β/α}) to (X_{β′/α′}, X_{β′/α}, X_{β/α′}, T) has a retraction by functoriality of the pushout. The morphism T \overline{}X_{β/α} induced by pushout is a monomorphism in Ĉ, since (X_{β′/α′}, X_{β′/α}, X_{β/α′}, X_{β/α}) is a weak square. Note that (C_{β/α}, T, X_{β/α}) is a commutative triangle.

The morphism T \overline{}C_{β/α} that is part of the retraction of quadrangles, factors as

(T \overline{}C_{β/α}) = (T [figure] X_{β/α} \overline{}C_{β/α}) ,

since C_{β/α} is injective in Ĉ as a summand of C_{β/0} = X_{β/0}. Now X_{β/α} \overline{}C_{β/α} completes the three morphisms on the other vertices to a retraction of quadrangles from (X_{β′/α′}, X_{β′/α}, X_{β/α′}, X_{β/α}) to (C_{β′/α′}, C_{β′/α}, C_{β/α′}, C_{β/α}) as sought.

3 Verification of Verdier’s axioms

Let (C, T, ϑ) be a Heller triangulated category.

3.1 Restriction from C^{+, ϑ=1}(Δ̄_{n}^{#}) to C(Δ̇_{n}) is dense and full

Let n ⩾ 1. Let

[figure: C(Δ̇_{n}^{+1}); S^{′▽}; Ĉ^{+}(Δ̄_{n}^{▽,+1}); U; US^{′▽} := (U^{(+1)}S^{▽})^{(−1)}]

be the conjugate by reindexing, i.e. a “shifted version” of S^{▽}; cf. §2.1.4. Note that

(US^{′▽})_{β/α} = Kern(U_{0^{+1}/α} ^{\overline{}^{u}} U_{0^{+1}/β})

for α, β ∈ Δ̄_{n} with 0 ⩽ α ⩽ β ⩽ 0^{+1}.

Lemma 3.1 Suppose that idempotents split in C. Given X ∈ Ob C(Δ̇_{n}), there exists an n-triangle X̃ ∈ Ob C^{+, ϑ=1}(Δ̄_{n}^{#}) that restricts to

X̃|Δ̇_{n} = X .

In other words, the restriction functor C^{+, ϑ=1}(Δ̄_{n}^{#}) ^{\overline{}^{(−)|Δ̇n}} C(Δ̇_{n}) is strictly dense.

Proof. Let Y := XR ∈ Ob C^{+}(Δ̄_{n}^{#}); cf. §2.2.2.3. We have an isomorphism [Y]^{+1} \overline{}^{Y ϑ_{n}}_{∼} [Y^{+1}] in \underline{C^{+}(Δ̄_{n}^{#})}. Let [Y^{+1}] ^{\overline{}^{θ}} [Y]^{+1} be a representative in C^{+}(Δ̄_{n}^{#}) of the inverse isomorphism (Y ϑ_{n})^{−} in \underline{C^{+}(Δ̄_{n}^{#})}. Consider the morphism (^{6})

[Y^{+1}]^{−1}|Δ̇_{n}^{+1} ^{\overline{}^{[θ]^{−1}|Δ̇{+1 \atop n}}} [Y]^{+1−1}|Δ̇_{n}^{+1} = Y |Δ̇_{n}^{+1} .

We have an induced pointwise epimorphism

Y |Δ̄_{n}^{△▽} \overline{}Y |Δ̇_{n}^{+1}S^{′▽}|Δ̄_{n}^{△▽} ,

which we may use to form the pullback

[figure: Z; f; [Y^{+1}]^{−1}|Δ̇_{n}^{+1}S^{′▽}|Δ̄_{n}^{△▽}; [θ]^{−1}|Δ̇{+1 \atop n}S^{′▽}|Δ̄{△▽ \atop n}; Y |Δ̄_{n}^{△▽}; Y |Δ̇_{n}^{+1}S^{′▽}|Δ̄_{n}^{△▽}]

in the abelian category Ĉ(Δ̄_{n}^{△▽}), i.e. pointwise. An application of Lemma A.21.(2) to the diagonal sequence of this pullback shows that Z ∈ Ob Ĉ^{+,∗}(Δ̄_{n}^{△▽}). We obtain Z_{α/α} = 0 for all 0 ⩽ α ⩽ 0^{+1}; and we obtain Z_{0^{+1}/0} = 0. Hence we have Z ∈ Ob Ĉ^{+}(Δ̄_{n}^{△▽}).

Suppose given β/α ∈ Δ̄_{n}^{△▽}. We claim that Z_{β/α} ^{\overline{}^{f_{β/α}}} Y_{β/α} represents an isomorphism in Ĉ/C. By Lemma A.24, it suffices to show that

([Y^{+1}]^{−1}|Δ̇_{n}^{+1}S^{′▽})_{β/α} ^{\overline{}^{([θ]^{−1}|Δ̇{+1 \atop n}S^{′▽})_{β/α}}} (Y |Δ̇_{n}^{+1}S^{′▽})_{β/α}

6We recall the convention that the inverse of the outer shift applied to a morphism f is written [f]^{−1}, whereas f^{−} denotes the inverse morphism, if existent.

represents an isomorphism in Ĉ/C. Since evaluation at β/α induces a functor from Ĉ^{+}(Δ̄_{n}^{△▽})/C^{+, split}(Δ̄_{n}^{△▽}) to Ĉ/C, where C^{+, split}(Δ̄_{n}^{△▽}) denotes the full subcategory of C^{+}(Δ̄_{n}^{△▽}) consisting of diagrams all of whose morphisms split (in Ĉ or, equivalently, in C), it suffices to show that

[Y^{+1}]^{−1}|Δ̇_{n}^{+1}S^{′▽}|Δ̄_{n}^{△▽} ^{\overline{}^{[θ]^{−1}|Δ̇{+1 \atop n}S^{′▽}|Δ̄{△▽ \atop n}}} Y |Δ̇_{n}^{+1}S^{′▽}|Δ̄_{n}^{△▽}

represents an isomorphism in Ĉ^{+}(Δ̄_{n}^{△▽})/C^{+, split}(Δ̄_{n}^{△▽}).

Now (−)^{(−1)}S^{′▽}|Δ̄_{n}^{△▽} induces a functor from \underline{C(Δ̇_{n})} to Ĉ^{+}(Δ̄_{n}^{△▽})/C^{+, split}(Δ̄_{n}^{△▽}), since it maps C^{split}(Δ̇_{n}) to C^{+, split}(Δ̄_{n}^{△▽}) by Lemma 1.4, using that idempotents are assumed to split in C. Therefore, it suffices to show that

([Y^{+1}]^{−1}|Δ̇_{n}^{+1})^{(+1)} ^{\overline{}^{([θ]^{−1}|Δ̇{+1 \atop n})^{(+1)}}} (Y |Δ̇_{n}^{+1})^{(+1)}

represents an isomorphism in \underline{C(Δ̇_{n})}. Since ([−]^{−1}|Δ̇_{n}^{+1})^{(+1)} = (−)|Δ̇_{n}, this means that it suffices to show that

[Y^{+1}]|Δ̇_{n} ^{\overline{}^{θ|Δ̇n}} [Y]^{+1}|Δ̇_{n}

represents an isomorphism in \underline{C(Δ̇_{n})}. Since (−)|Δ̇_{n} induces a functor from \underline{C^{+}(Δ_{n}^{#})} to \underline{C(Δ̇_{n})}, it suffices to show that

[Y^{+1}] ^{\overline{}^{θ}} [Y]^{+1}

represents an isomorphism in \underline{C^{+}(Δ̄_{n}^{#})}. This, however, follows by choice of θ. This proves the claim.

Since idempotents are assumed to split in C, we can conclude from the claim that for all β/α in Δ̄_{n}^{△▽}, the entry Z_{β/α} is isomorphic in Ĉ to an object in C, hence without loss of generality, the entry Z_{β/α} is an object of C. So Z ∈ Ob C^{+}(Δ̄_{n}^{△▽}).

We remark that

(∗)  \begin{aligned}Z|Δ̇_{n} & = & Y |Δ̇_{n} & = & X \\ Z|Δ̇_{n}^{+1} & = & [Y^{+1}]^{−1}|Δ̇_{n}^{+1} & = & (X^{+1})^{(−1)} , \end{aligned}

where X^{+1} arises from X by pointwise application of (−)^{+1}. Concerning morphisms, we remark that

(∗∗)  \begin{aligned}(Z ^{\overline{}^{f}} Y |Δ̄_{n}^{△▽})|Δ̇_{n} & = & (X ^{\overline{}^{1_{X}}} X) \\ (Z ^{\overline{}^{f}} Y |Δ̄_{n}^{△▽})|Δ̇_{n}^{+1} & = & ([Y^{+1}]^{−1} ^{\overline{}^{[θ]^{−1}}} Y)|Δ̇_{n}^{+1} . \end{aligned}

In fact, on Δ̇_{n}, the right hand side column of our pullback vanishes; and on Δ̇_{n}^{+1}, the lower row of our pullback is an identity.

Now, (∗) allows to define the periodic prolongation Z̄ ∈ Ob C^{+}(Δ̄_{n}^{#}) of Z ∈ Ob C^{+}(Δ̄_{n}^{△▽}) by Z̄|Δ̄_{n}^{△▽} := Z and by the requirement that [Z̄]^{+1} = [Z̄^{+1}].

We claim that Z̄ϑ_{n} = 1_{[Z̄]^{+1}} in \underline{C^{+}(Δ̄_{n}^{#})}. Let Z̄ ^{\overline{}^{f̂}} Y be an inverse image of Z ^{\overline{}^{f}} Y |Δ̄_{n}^{△▽} under

C^{+}(Δ̄_{n}^{#})^{\overline{}^{(−)|Δ̄{△▽ \atop n}}} C^{+}(Δ̄_{n}^{△▽}); cf. Lemma 2.7.(1). By (∗∗), we get

(∗∗′)  \begin{aligned}(Z̄ ^{\overline{}^{f̂}} Y)|Δ̇_{n} & = & (X ^{\overline{}^{1_{X}}} X) \\ (Z̄ ^{\overline{}^{f̂}} Y)|Δ̇_{n}^{+1} & = & ([Y^{+1}]^{−1} ^{\overline{}^{[θ]^{−1}}} Y)|Δ̇_{n}^{+1} . \end{aligned}

We consider the commutative quadrangle

\xymatrix{[Z̄]^{+1} \ar[r]^{Z̄ϑ_{n}} \ar[d]_{[f̂]^{+1}} & [Z̄^{+1}] \ar[d]^{[f̂^{+1}]} \\ [Y]^{+1} \ar[r]^{Y ϑ_{n}} & [Y^{+1}]}

in \underline{C^{+}(Δ̄_{n}^{#})}. We restrict it to Δ̇_{n} to obtain the commutative quadrangle

\xymatrix{[Z̄]^{+1}|Δ̇_{n} \ar[r]^{Z̄ϑ_{n}|Δ̇n} \ar[d]_{[f̂]^{+1}|Δ̇n} & [Z̄^{+1}]|Δ̇_{n} \ar[d]^{[f̂^{+1}]|Δ̇n} \\ [Y]^{+1}|Δ̇_{n} \ar[r]^{Y ϑ_{n}|Δ̇n} & [Y^{+1}]|Δ̇_{n}}

in \underline{C(Δ̇_{n})}, which, using (∗∗′), can be rewritten as

\xymatrix{X^{+1} \ar[r]^{Z̄ϑ_{n}|Δ̇n} \ar[d]_{θ|Δ̇n} & X^{+1} \ar[d]^{1_{X^{+1}}} \\ [Y]^{+1}|Δ̇_{n} \ar[r]^{Y ϑ_{n}|Δ̇n} & X^{+1} ,}

where we did not distinguish in notation between θ|Δ̇_{n} and its residue class in \underline{C(Δ̇_{n})}, etc. Since θ(Y ϑ_{n}) = 1_{[Y^{+1}]} in \underline{C^{+}(Δ̄_{n}^{#})}, we have θ|Δ̇_{n}(Y ϑ_{n}|Δ̇_{n}) = 1_{X^{+1}} in \underline{C(Δ̇_{n})}. Thus the last quadrangle shows that Z̄ϑ_{n}|Δ̇_{n} = 1_{X^{+1}} = 1_{[Z̄]^{+1}}|Δ̇_{n} in \underline{C(Δ̇_{n})} as well. Since \underline{C^{+}(Δ̄_{n}^{#})} \overline{}^{(−)|Δ̄{△▽ \atop n}}_{∼} \underline{C^{+}(Δ̄_{n}^{△▽})} is an equivalence, we conclude that Z̄ϑ_{n} = 1_{[Z̄]^{+1}} in \underline{C^{+}(Δ̄_{n}^{#})}; cf. Proposition 2.6. This proves the claim; i.e. we have shown that Z̄ is an n-triangle. Since Z̄|Δ̇_{n} = X by (∗), this proves the lemma.

In the proof of Lemma 3.1, we needed the assumption that idempotents split in C in the equivalent form that the residue class functor Ĉ \overline{}Ĉ/C maps precisely the objects isomorphic to objects of C to zero – just as Heller did at that point.

Lemma 3.2 Given n-triangles X and Y and a morphism X|Δ̇_{n} ^{\overline{}^{f}} Y |Δ̇_{n}

in C(Δ̇_{n}), there exists a morphism X ^{\overline{}^{f̃}} Y of n-triangles such that f̃|Δ̇_{n} = f. In other words, the restriction functor C^{+, ϑ=1}(Δ̄_{n}^{#}) ^{\overline{}^{(−)|Δ̇n}} C(Δ̇_{n}) is full.

Proof. Since the restriction functor C^{+}(Δ̄_{n}^{#})^{\overline{}^{(−)|Δ̇n}} C(Δ̇_{n}) is full by Proposition 2.5, we find a morphism X ^{\overline{}^{g}} Y in C^{+}(Δ̄_{n}^{#}) such that g|Δ̇_{n} = f. Let \underline{g} denote the residue class of g in \underline{C^{+}(Δ̄_{n}^{#})}. Since ϑ_{n} is a transformation, we have [\underline{g}]^{+1}(Y ϑ_{n}) = (Xϑ_{n})[\underline{g}^{+1}]. Since X and Y are n-triangles, both Xϑ_{n} and Y ϑ_{n} are identities, and this equality amounts to [\underline{g}]^{+1} = [\underline{g}^{+1}], i.e. the difference [g]^{+1} − [g^{+1}] factors over an object of C^{+, split}(Δ̄_{n}^{#}). Restricting to Δ̇_{n}, the difference

([g]^{+1} − [g^{+1}])|Δ̇_{n} = (g|Δ̇_{n}^{+1})^{(+1)} − f^{+1} factors over an object of C^{split}(Δ̇_{n}). Therefore, g|Δ̇_{n}^{+1} − (f^{+1})^{(−1)} factors over an object Z of C^{split}(Δ̇_{n}^{+1}), say, as

(X|Δ̇_{n}^{+1} ^{\overline{}^{g|Δ̇{+1 \atop n} − (f^{+1})^{(−1)}}} Y |Δ̇_{n}^{+1}) = (X|Δ̇_{n}^{+1} ^{\overline{}^{a}} Z ^{\overline{}^{b}} Y |Δ̇_{n}^{+1}) .

By periodic continuation, it suffices to find a morphism X|Δ̄_{n}^{△▽} ^{\overline{}^{f̆}} Y |Δ̄_{n}^{△▽} in C^{+}(Δ̄_{n}^{△▽}) such that f̆|Δ̇_{n} = f and such that f̆|Δ̇_{n}^{+1} = (f^{+1})^{(−1)}. I.e. we have to find a morphism X|Δ̄_{n}^{△▽} ^{\overline{}^{h}} Y |Δ̄_{n}^{△▽} such that h|Δ̇_{n} = 0 and such that h|Δ̇_{n}^{+1} = ab, for then we may take f̆:= g|Δ̄_{n}^{△▽} − h.

Note that (ZS^{′▽}|Δ̄_{n}^{△▽})|Δ̇_{n} = 0. Note that ZS^{′▽}|Δ̄_{n}^{△▽} is in C^{+}(Δ̄_{n}^{△▽}), hence in C^{+, split}(Δ̄_{n}^{△▽}) by Lemma 1.4.

Since S^{′▽}|Δ̄_{n}^{△▽} is right adjoint to restriction to Δ̇_{n}^{+1}, we have a morphism X|Δ̄_{n}^{△▽} ^{\overline{}^{a′}} ZS^{′▽}|Δ̄_{n}^{△▽} such that a′|Δ̇_{n}^{+1} = a.

Since C^{+}(Δ̄_{n}^{△▽})^{\overline{}^{(−)|Δ̇{+1 \atop n}}} C(Δ̇_{n}^{+1}) is full by the dual and shifted assertion of Lemma 2.7.(1), there is a morphism ZS^{′▽}|Δ̄_{n}^{△▽} ^{\overline{}^{b′}} Y |Δ̄_{n}^{△▽} such that b′|Δ̇_{n}^{+1} = b.

We may take h := a′b′.

In Lemmata 3.1 and 3.2, we do not claim the existence of a coretraction from C(Δ̇_{n}) to C^{+, ϑ=1}(Δ̄_{n}^{#}) to restriction. The construction made in the proof of Lemma 3.2 involves e.g. a choice of a lift b′ of b. Cf. [30, II.1.2.13]. The fullness used in the proof of Lemma 3.2 to lift b, can also be used to lift a. We have used the direct argument and thus seen that the lift a′ of a does not involve a choice.

Remark 3.3 Suppose that idempotents split in C. By Lemmata 3.1 and 3.2, the restriction functor

C^{+, ϑ=1}(Δ̄_{n}^{#}) ^{\overline{}^{(−)|Δ̇n}} C(Δ̇_{n})

is full and strictly dense. By Proposition 2.6, the restriction functor

\underline{C^{+}(Δ̄_{n}^{#})} \overline{}^{(−)|Δ̇n}_{∼}  \underline{C(Δ̇_{n})}

is an equivalence. Denoting by \underline{C^{+, ϑ=1}(Δ̄_{n}^{#})} the image of C^{+, ϑ=1}(Δ̄_{n}^{#}) in \underline{C^{+}(Δ̄_{n}^{#})}, we obtain a

full and strictly dense functor \underline{C^{+, ϑ=1}(Δ̄_{n}^{#})} ^{\overline{}^{(−)|Δ̇n}} \underline{C(Δ̇_{n})}. Since it factors as a faithful embed-

ding \underline{C^{+, ϑ=1}(Δ̄_{n}^{#})} [figure] \underline{C^{+}(Δ̄_{n}^{#})} followed by an equivalence, it is also faithful. We end up with equivalences

\underline{C^{+, ϑ=1}(Δ̄_{n}^{#})} \overline{}^{(−)|Δ̇n}_{∼}  \underline{C(Δ̇_{n})} ,  \underline{C^{+, ϑ=1}(Δ̄_{n}^{#})} ^{\overline{}^{∼}} \underline{C^{+}(Δ̄_{n}^{#})} .

3.2 An omnibus lemma

Suppose given n, m ⩾ 1. Concerning the category C^{+, periodic}(Δ̄_{n}^{#}) of periodic n-pretriangles and its full subcategory C^{+, split, periodic}(Δ̄_{n}^{#}), cf. §2.5.3; concerning the category C^{+, ϑ=1}(Δ̄_{n}^{#}) of n-triangles, cf. Definition 1.5.(ii). Note that

C^{+, ϑ=1}(Δ̄_{n}^{#}) ⊆ C^{+, periodic}(Δ̄_{n}^{#}) ⊆ C^{+}(Δ̄_{n}^{#}) ,

and that the first inclusion is full.

Lemma 3.4

(1) Let X be an n-triangle, and let Δ̄_{n} ^{\overline{}^{p}} Δ̄_{m} be a morphism of periodic linearly ordered sets. Then Xp^{#}, obtained by “restriction along p”, is an m-triangle.

(2) Let X be a (2n + 1)-triangle. Then X𝔣_{n}, obtained by folding, is an (n + 1)-triangle.

(3) The category C^{+, ϑ=1}(Δ̄_{n}^{#}) of n-triangles is a full additive subcategory of the category C^{+, periodic}(Δ̄_{n}^{#}) of periodic n-pretriangles, closed under direct summands.

(4) Suppose given an isomorphism X ^{\overline{}^{f}} Y in C^{+, periodic}(Δ̄_{n}^{#}). If X is an n-triangle, then Y is an n-triangle.

(5) Let X ^{\overline{}^{f}} Y be a morphism in C^{+, periodic}(Δ̄_{n}^{#}) such that f|Δ̇_{n} is an isomorphism. Then f is an isomorphism.

(6) Let X and Y be n-triangles. Suppose given an isomorphism X|Δ̇_{n} \overline{}^{u}_{∼} Y |Δ̇_{n} in C(Δ̇_{n}).

Then there exists an isomorphism X \overline{}^{ũ}_{∼} Y in C^{+, ϑ=1}(Δ̄_{n}^{#}) such that ũ|_{C(Δ̇_{n})} = u.

(7) If X ∈ Ob C^{+, split, periodic}(Δ̄_{n}^{#}), then X is an n-triangle.

Note that Lemma 3.4.(5) applies in particular to n-triangles and a morphism of n-triangles.

Proof. Ad (1). In \underline{C^{+}(Δ̄_{m}^{#})}, we have

(X\underline{p}^{#})ϑ_{m} = (Xϑ_{n})\underline{p}^{#} = (1_{[X]^{+1}})\underline{p}^{#} = 1_{[X\underline{p}^{#}]^{+1}} .

Ad (2). In \underline{C^{+}(Δ̄_{n+1}^{#})}, we have

(X\underline{𝔣}_{n})ϑ_{n+1} = (Xϑ_{2n+1})\underline{𝔣}_{n} = (1_{[X]^{+1}})\underline{𝔣}_{n} = 1_{[X\underline{𝔣}_{n}]^{+1}} .

Ad (3). We have to show that

X, Y ∈ Ob C^{+, ϑ=1}(Δ̄_{n}^{#})  ⟺  X ⊕ Y ∈ Ob C^{+, ϑ=1}(Δ̄_{n}^{#}) .

But since ϑ_{n} is a morphism between additive functors, we have (X ⊕ Y)ϑ_{n} = 1_{[X⊕Y]^{+1}} if and only if Xϑ_{n} = 1_{[X]^{+1}} and Y ϑ_{n} = 1_{[Y]^{+1}}. In fact, (X ⊕ Y)ϑ_{n} identifies with (\begin{matrix}Xϑ_{n} & 0 \\ 0 & Y ϑ_{n} \end{matrix}).

Ad (4). Since f|Δ̇_{n} is an isomorphism in C(Δ̇_{n}), so is its image in \underline{C(Δ̇_{n})}. Hence the image of f in \underline{C^{+}(Δ̄_{n}^{#})} is an isomorphism by Proposition 2.6. Consider the commutative quadrangle

\xymatrix{[X]^{+1} \ar[r]^{[f]^{+1}}_{∼} \ar[d]_{Xϑ_{n}}^{≀} & [Y]^{+1} \ar[d]^{Y ϑ_{n}}_{≀} \\ [X^{+1}] \ar[r]^{[f^{+1}]}_{∼} & [Y^{+1}]}

in \underline{C^{+}(Δ̄_{n}^{#})}. Since [f]^{+1} = [f^{+1}] by assumption, we conclude from Xϑ_{n} = 1_{[X]^{+1}} that Y ϑ_{n} = 1_{[Y]^{+1}}.

Ad (5). It suffices to show that given 0 ⩽ i ⩽ j ⩽ n, the morphism f_{j/i} is an isomorphism in C. In fact, we have a morphism of exact sequences

( f_{i/0}, f_{j/0}, f_{j/i}, f_{i/0}^{+1}, f_{j/0}^{+1} )

in Ĉ, whose entries except possibly f_{j/i} are isomorphisms; hence also f_{j/i} is isomorphic.

Ad (6). This follows by Lemma 3.2 using (5).

Ad (7). We have [X]^{+1} ≃ 0 in \underline{C^{+}(Δ̄_{n}^{#})}, whence Xϑ_{n} = 1_{[X]^{+1}} ∈ _{\underline{C^{+}(Δ̄_{n}^{#})}}([X]^{+1}, [X]^{+1}).

3.3 Turning n-triangles

Let n ⩾ 2.

Lemma 3.5 Given an n-triangle X ∈ Ob C^{+, ϑ=1}(Δ̄_{n}^{#}), we define Y ∈ Ob C^{+, periodic}(Δ̄_{n}^{#}) by letting

(Y_{j/i} ^{\overline{}^{y}} Y_{j′/i′}) := (X_{i^{+1}/j} ^{\overline{}^{x}} X_{i^{′+1}/j′})

for 0 ⩽ i ⩽ j ⩽ n and 0 ⩽ i′ ⩽ j′ ⩽ n such that i ⩽ i′ and j ⩽ j′, and by letting

(Y_{n/i} ^{\overline{}^{y}} Y_{0^{+1}/i}) := (X_{i^{+1}/n} ^{\overline{}^{−x}} X_{i^{+1}/0^{+1}})

for 0 ⩽ i ⩽ n. Then [X]_{−}^{+1} := Y is an n-triangle.

Proof. Let

[figure: 2Δ̄_{n−1}; h_{n}; Δ̄_{n}; i^{+j}; \{ \begin{matrix}(i + 1)^{+j/2} & if j ≡_{2} 0 \\ 0^{+(j+1)/2} & if j ≡_{2} 1 , \end{matrix}]

where i ∈ [0, n − 1] and j ∈ Z. The map h_{n} is a morphism of periodic posets. We claim that

Y = Xh_{n}^{#}𝔣_{n−1} .

Once this claim is shown, we are done by Lemma 3.4.(1, 2).

Note that (Xh_{n}^{#})_{l/k} = X_{lh_{n}/kh_{n}} for k, l ∈ 2Δ̄_{n−1} with k ⩽ l. For 0 ⩽ i ⩽ n and 1 ⩽ j ⩽ 0^{+1}, we obtain

(Xh_{n}^{#}𝔣_{n−1})_{j/i} = \{ \begin{matrix}(Xh_{n}^{#})_{(j−1)^{+1}/(j−1)} & for i = 0 and 1 ⩽ j ⩽ n \\ (Xh_{n}^{#})_{(i−1)^{+2}/(i−1)^{+1}} & for 1 ⩽ i ⩽ n and j = 0^{+1} \\ (Xh_{n}^{#})_{(j−1)^{+1}/(i−1)^{+1}} ⊕ (Xh_{n}^{#})_{(i−1)^{+2}/(j−1)} & for 1 ⩽ i ⩽ j ⩽ n \\ 0 & for i = 0 and j = 0^{+1} \end{matrix}

= \{ \begin{matrix}X_{0^{+1}/j} & for i = 0 and 1 ⩽ j ⩽ n \\ X_{i^{+1}/0^{+1}} & for 1 ⩽ i ⩽ n and j = 0^{+1} \\ X_{i^{+1}/j} & for 1 ⩽ i ⩽ j ⩽ n \\ 0 & for i = 0 and j = 0^{+1} , \end{matrix}

and also the morphisms result as claimed.

3.4 Application to the axioms of Verdier

Recall that (C, T, ϑ) is a Heller triangulated category.

Proposition 3.6 Suppose that idempotents split in C. The tuple (C, T), equipped with the set of 2-triangles as the set of distinguished triangles, is a triangulated category in the sense of Verdier [31, Def. 1-1].

Proof. We number the axioms of Verdier as in loc. cit.

Ad (TR 1). Stability under isomorphism of the set of distinguished triangles follows from Lemma 3.4.(4).

The possible extension of a morphism to a distinguished triangle follows by Lemma 3.1.

The distinguished triangle (X, X, 0) on the identity of an object X in C follows by Lemma 3.4.(7). Alternatively, one can use that each morphism is contained in a distinguished triangle and the fact that a distinguished triangle is a long exact sequence in Ĉ.

Ad (TR 2). Suppose given a distinguished triangle

X ^{\overline{}^{u}} Y ^{\overline{}^{v}} Z ^{\overline{}^{w}} X^{+1} .

By Lemma 3.5, we obtain the distinguished triangle

X^{+1} ^{\overline{}^{u^{+1}}} Y^{+1} ^{\overline{}^{v^{+1}}} Z^{+1} ^{\overline{}^{−w^{+1}}} X^{+2} .

By Lemma 3.4.(1), applied to the morphism Δ̄_{2} \overline{}Δ̄_{2} that sends 0 to 2^{−1}, 1 to 0 and 2 to 1, we obtain the distinguished triangle

Y ^{\overline{}^{v}} Z ^{\overline{}^{−w}} X^{+1} ^{\overline{}^{u^{+1}}} Y^{+1} .

By Lemma 3.4.(4), we obtain the distinguished triangle

Y ^{\overline{}^{v}} Z ^{\overline{}^{w}} X^{+1} ^{\overline{}^{−u^{+1}}} Y^{+1} .

Ad (TR 3). The possible completion of a morphism in C(Δ̇_{2}) to a morphism of distinguished triangles follows from Lemma 3.2.

Ad (TR 4). The octahedral axiom, i.e. the compatibility of forming cones with composition of morphisms, follows from Lemma 3.1, applied to the case n = 3, for by Lemma 3.4.(6), we may arbitrarily choose completions to distinguished triangles.

Note that 3-triangles are particular octahedra, in the language of [3, 1.1.6]. Using 3-triangles, we will now verify the axiom proposed in [3, 1.1.13].

Lemma 3.7 Suppose given a 3-triangle T in Ob C^{+, ϑ=1}((2Δ̄_{1})^{#}), depicted as follows.

[figure: 0; 0; Z^{+1}; 0; Z′′; w′′; +; Y^{+1}; v^{+1}; 0; Y′; v′; +; Z′; w′; z′; +; X^{+1}; u^{+1}; 0; X; u; +; Y; v; y; +; Z; z; +; 0]

Then

T 𝔣_{1} = ( Y ^{\overline{}^{vz}} Z′ ^{\overline{}^{(^{z′ w′})}} Z′′ ⊕ X^{+1}^{\overline{}^{(\begin{matrix}w′′ \\ −u^{+1} \end{matrix} )}} Y^{+1} )

and

T s^{#}𝔣_{1} = (Z^{′−1} ^{\overline{}^{w^{′−1}u}} Y ^{\overline{}^{(^{y v})}} Y′ ⊕ Z ^{\overline{}^{(\begin{matrix}v′ \\ −z \end{matrix} )}} Z′ )

are distinguished triangles, where 2Δ̄_{1} ^{\overline{}^{s}} 2Δ̄_{1} is the morphism of periodic posets determined by 0s = 1^{−1}, 1s = 0, 0^{+1}s = 1 and 1^{+1}s = 0^{+1}. Proof. This follows by Lemma 3.4.(1, 2).

3.5 n-triangles and strictly exact functors

Let C ^{\overline{}^{F}} C′ be a strictly exact functor between Heller triangulated categories (C, T, ϑ) and (C′, T′, ϑ′). Let n ⩾ 0. Lemma 3.8 Given an n-triangle X ∈ Ob C^{+, ϑ=1}(Δ̄_{n}^{#}), the diagram X(F^{+}(Δ̄_{n}^{#})), obtained by pointwise application of F to X, is an n-triangle, i.e. an object of C^{′+, ϑ′=1}(Δ̄_{n}^{#}). Proof. Using F T′ = T F as well as [X]^{+1} = [X^{+1}], we obtain

[XF^{+}(Δ̄_{n}^{#})]^{+1} = X(F^{+}(Δ̄_{n}^{#}))(C^{+}(T_{n})) = X(F^{+}(T_{n}))
= X(C^{+}(T_{n}))(F^{+}(Δ̄_{n}^{#})) = [X]^{+1}(F^{+}(Δ̄_{n}^{#}))
= [X^{+1}](F^{+}(Δ̄_{n}^{#}))  = X(T^{+}(Δ̄_{n}^{#}))(F^{+}(Δ̄_{n}^{#}))
= X((T F)^{+}(Δ̄_{n}^{#}))  = X((F T′)^{+}(Δ̄_{n}^{#}))
= X(F^{+}(Δ̄_{n}^{#}))(T^{′+}(Δ̄_{n}^{#})) = [(X(F^{+}(Δ̄_{n}^{#})))^{+1}] .

Moreover,

X(F^{+}(Δ̄_{n}^{#}))ϑ′_{n} = Xϑ_{n}(F^{+}(Δ̄_{n}^{#})) = 1_{[X]^{+1}}(F^{+}(Δ̄_{n}^{#})) = 1_{[XF^{+}(Δ̄_{n}^{#})]^{+1}} .

3.6 A remark on spectral sequences

Verdier calls certain pretriangles objets spectraux (spectral objects); cf. [30, Sec. II.4]. We shall explain the connection to spectral sequences in our language.

Consider the linearly ordered set Z_{∞} := {−∞} ⊔ Z ⊔ {+∞}. Let Z̄_{∞}^{##} be the subposet of Z̄_{∞}^{#}(Δ_{1}) consisting of those δ/β//γ/α for which

δ^{−1} ⩽ α ⩽ β ⩽ γ ⩽ δ ⩽ α^{+1} ,

where α, β, γ, δ ∈ Z̄_{∞}. A spectral object, in a slightly different sense from [30, II.4.1.2], is an object of C^{+}(Z̄_{∞}^{#}). The spectral sequence functor

[figure: C^{+}(Z̄_{∞}^{#}); E; Ĉ(Z̄_{∞}^{##}); X; XE ,]

is defined by

XE(δ/β//γ/α) := Im(X_{γ/α} \overline{}X_{δ/β})

for δ/β//γ/α ∈ Z̄_{∞}^{##}, equipped with the induced morphisms.

Lemma 3.9 Given α, β, γ, δ, ε ∈ Z̄_{∞} such that

ε^{−1} ⩽ α ⩽ β ⩽ γ ⩽ δ ⩽ ε ⩽ α^{+1} ,

and given X ∈ Ob C^{+}(Z̄_{∞}^{#}), the morphisms appearing in XE form a short exact sequence

XE(ε/β//γ/α) [figure] XE(ε/β//δ/α) \overline{}XE(ε/γ//δ/α) .

Proof. This follows by Lemma A.22, applied to the diagram

( X_{γ/α}, X_{δ/α}, X_{ε/α}, X_{γ/β}, X_{δ/β}, X_{ε/β}, \underbrace{X_{γ/γ}}_{=0}, X_{δ/γ}, X_{ε/γ}) .

Note that we may apply a shift β/α \overline{}α^{+1}/β to the indices, i.e. an outer shift to X, before applying Lemma 3.9, to get another short exact sequence.

The usual exact sequences of spectral sequence terms can be derived from Lemma 3.9. Cf. [30, II.4.2.6], [5, App.].

4 The stable category of a Frobenius category is Heller triangulated

Let F = (F, T, I, ι, P, π) be a functorial Frobenius category; cf. Definition A.5.(3). Let B ⊆ F denote the full subcategory of objects in the image of I, coinciding with the full subcategory of the objects in the image of P; then B is a sufficiently big full subcategory of bijectives in F.

We shall prove in Theorem 4.6 below that the classical stable category \underline{F} carries a Heller triangulation.

4.1 Definition of F^{□}(Δ̄_{n}^{#}), modelling \underline{\underline{F}^{+}(Δ̄_{n}^{#})}

We shall model, in the sense of Proposition 4.5 below, the category \underline{F}^{+}(Δ̄_{n}^{#}) by a category F^{□}(Δ̄_{n}^{#}). Morally, we represent weak squares (+) in \underline{F} by pure squares (□) in F. To do so, we have to represent the zeroes on the boundaries by bijective objects.

Let n ⩾ 0. Concerning the notion of a pure square, see §A.4. Let F^{□}(Δ̄_{n}^{#}) ⊆ F(Δ̄_{n}^{#}) be the full subcategory defined by

Ob F^{□}(Δ̄_{n}^{#}) := \{ X ∈ Ob F(Δ̄_{n}^{#}) : \begin{matrix}1) & X_{α/α} and X_{α^{+1}/α} are in Ob B for all α ∈ Δ̄_{n} \\ 2) & For all δ^{−1} ⩽ α ⩽ β ⩽ γ ⩽ δ ⩽ α^{+1} in Δ̄_{n}, \\ & the quadrangle \\ X_{γ/β} & x & [figure] & X_{δ/β} & X_{γ/α} & x & [figure] & x & [figure] & □ & X_{δ/α} & x & [figure] \\ & is a pure square. \end{matrix} \} .

Given n, m ⩾ 0, a morphism Δ̄_{n} ^{\overline{}^{p}} Δ̄_{m} induces a morphism F^{□}(p^{#}), usually, and by abuse of notation, denoted by p^{#}.

Given an exact functor F ^{\overline{}^{F}} F̃ between functorial Frobenius categories that sends bijectives to bijectives, we obtain an induced functor F^{□}(Δ̄_{n}^{#}) ^{\overline{}^{F^{□}(Δ̄_{n}^{#})}} F̃^{□}(Δ̄_{n}^{#}) by pointwise application of F.

Denote by

F^{□}(Δ̄_{n}^{#}) ^{\overline{}^{M}}  \underline{\underline{F}^{+}(Δ̄_{n}^{#})}
F^{□}(Δ̄_{n}^{#}) ^{\overline{}^{M′}}  \underline{F}^{+}(Δ̄_{n}^{#})
\underline{F}^{+}(Δ̄_{n}^{#}) ^{\overline{}^{M′′}}  \underline{\underline{F}^{+}(Δ̄_{n}^{#})}
F ^{\overline{}^{N}}  \underline{F}
\underline{F}(Δ̇_{n}) ^{\overline{}^{N′}}  \underline{\underline{F}(Δ̇_{n})}

the respective residue class functors, welldefined by Lemma A.31. In particular, M = M′M′′.

4.2 Folding for F^{□}(Δ̄_{n}^{#})

We model, in the sense of Remark 4.1, the folding operation 𝔣 introduced in §1.2.2.

Suppose given n ⩾ 0. Let the periodic functor

[figure: F^{□}((2Δ̄_{n})^{#}); 𝔣̃_{n}; F^{□}(\overline{ρ ⊔ Δ_{n}}^{#}); X; X𝔣̃_{n}]

be determined by the following data. Writing Y := X𝔣̃_{n}, we let

\begin{aligned}(Y_{α/ρ} ^{\overline{}^{y}} Y_{β/ρ}) & := & (X_{α^{+1}/α} ^{\overline{}^{x}} X_{β^{+1}/β}) \\ Y_{ρ/ρ} & := & 0 \\ Y_{ρ^{+1}/ρ} & := & 0 \\ (Y_{β/ρ} ^{\overline{}^{y}} Y_{β/α}) & := & (X_{β^{+1}/β} ^{\overline{}^{(^{x x})}} X_{β^{+1}/α^{+1}} ⊕ X_{α^{+2}/β}) \\ (Y_{β/α} ^{\overline{}^{y}} Y_{δ/γ}) & := & (X_{β^{+1}/α^{+1}} ⊕ X_{α^{+2}/β} ^{\overline{}^{(\begin{matrix}x & 0 \\ 0 & x \end{matrix})}} X_{δ^{+1}/γ^{+1}} ⊕ X_{γ^{+2}/δ}) \\ (Y_{δ/γ} ^{\overline{}^{y}} Y_{ρ^{+1}/γ}) & := & (X_{δ^{+1}/γ^{+1}} ⊕ X_{γ^{+2}/δ} ^{\overline{}^{(\begin{matrix}x \\ −x \end{matrix})}} X_{γ^{+2}/γ^{+1}}) \end{aligned}

for α, β, γ, δ ∈ Δ_{n} with α ⩽ β, with γ ⩽ δ and with β/α ⩽ δ/γ. The remaining morphisms are given by composition.

We claim that X𝔣̃_{n} is an object of F^{□}(\overline{ρ ⊔ Δ_{n}}^{#}).

In fact, by Lemma A.12, we are reduced to considering the quadrangles of Y inside Δ_{n}^{△▽}, i.e. the quadrangles

(i) on (γ/α, δ/α, γ/β, δ/β) for α, β, γ, δ ∈ Δ_{n} with α ⩽ β ⩽ γ ⩽ δ;

(ii) on (γ/ρ, δ/ρ, γ/β, δ/β) for β, γ, δ ∈ Δ_{n} with β ⩽ γ ⩽ δ;

(iii) on (γ/α, ρ^{+1}/α, γ/β, ρ^{+1}/β) for α, β, γ ∈ Δ_{n} with α ⩽ β ⩽ γ;

(iv) and on (β/ρ, ρ^{+1}/ρ, β/α, ρ^{+1}/α) for α, β ∈ Δ_{n} with α ⩽ β.

Another application of loc. cit. reduces case (i) to case (ii) (or (iii)). Still another application of loc. cit. reduces the cases (ii) and (iii) to case (iv). Now the quadrangle in case (iv) is in fact a pure square, as follows from X ∈ Ob F^{□}(Δ̄_{n}^{#}) and the definition of a pure square via its pure short exact diagonal sequence.

The construction of Y is functorial in X.

Remark 4.1 We have 𝔣̃_{n}M′ = M′𝔣_{n}, and thus 𝔣̃_{n}M = M\underline{𝔣}_{n} for n ⩾ 0.

\xymatrix{F^{□}((2Δ̄_{n})^{#}) \ar[r]^{𝔣̃_{n}} \ar[d]_{M′} & F^{□}(\overline{ρ ⊔ Δ_{n}}^{#}) \ar[d]^{M′} \\ \underline{F}^{+}((2Δ̄_{n})^{#}) \ar[r]^{𝔣_{n}} \ar[d]_{M′′} & \underline{F}^{+}(\overline{ρ ⊔ Δ_{n}}^{#}) \ar[d]^{M′′} \\ \underline{\underline{F}^{+}((2Δ̄_{n})^{#})} \ar[r]^{_{\underline{𝔣}_{n}}} & \underline{\underline{F}^{+}(\overline{ρ ⊔ Δ_{n}}^{#})}}

Example 4.2 Let n = 2. Note that 2Δ̄_{2} ≃ Δ̄_{5}. Let X ∈ Ob F^{□}((2Δ̄_{2})^{#}), depicted as follows.

[figure: X_{0^{+1}/0^{+1}}; x; X_{1^{+1}/0^{+1}}; x; □; X_{2^{+1}/0^{+1}}; x; □; X_{0^{+2}/0^{+1}}; x; □; X_{1^{+2}/0^{+1}}; x; □; X_{2^{+2}/0^{+1}}; x; □; X_{0^{+3}/0^{+1}}; X_{2/2}; x; X_{0^{+1}/2}; x; x; □; X_{1^{+1}/2}; x; x; □; X_{2^{+1}/2}; x; x; □; X_{0^{+2}/2}; x; x; □; X_{1^{+2}/2}; x; x; □; X_{2^{+2}/2}; x; X_{1/1}; x; X_{2/1}; x; x; □; X_{0^{+1}/1}; x; x; □; X_{1^{+1}/1}; x; x; □; X_{2^{+1}/1}; x; x; □; X_{0^{+2}/1}; x; x; □; X_{1^{+2}/1}; x; X_{0/0}; x; X_{1/0}; x; x; □; X_{2/0}; x; x; □; X_{0^{+1}/0}; x; x; □; X_{1^{+1}/0}; x; x; □; X_{2^{+1}/0}; x; x; □; X_{0^{+2}/0}; x; X_{2^{−1}/2^{−1}}; x; X_{0/2^{−1}}; x; x; □; X_{1/2^{−1}}; x; x; □; X_{2/2^{−1}}; x; x; □; X_{0^{+1}/2^{−1}}; x; x; □; X_{1^{+1}/2^{−1}}; x; x; □; X_{2^{+1}/2^{−1}}; x; □; □; □; □; □]

Note that the objects on the boundary of the diagram,

\begin{matrix}…, X_{2^{−1}/2^{−1}} , X_{0/0} , X_{1/1} , X_{2/2} , X_{0^{+1}/0^{+1}} , … \\ …, X_{2^{+1}/2^{−1}} , X_{0^{+2}/0} , X_{1^{+2}/1} , X_{2^{+2}/2} , X_{0^{+3}/0^{+1}} , … \end{matrix}

are all supposed to be in Ob B. Note that ρ ⊔ Δ_{2} ≃ Δ_{3}. Folding turns X into X𝔣̃_{2} ∈ Ob F^{□}(\overline{ρ ⊔ Δ_{2}} ^{#}), depicted as follows.

[figure: 0; X_{2^{+1}/2^{+1}} ⊕X_{2^{+2}/2}; _{( \begin{matrix}x \\ −x \end{matrix} )}; X_{2^{+2}/2^{+1}}; □; X_{1^{+1}/1^{+1}} ⊕X_{1^{+2}/1}; _{( \begin{matrix}x & 0 \\ 0 & x \end{matrix} )}; X_{2^{+1}/1^{+1}} ⊕X_{1^{+2}/2}; _{( \begin{matrix}x \\ −x \end{matrix} )}; _{( \begin{matrix}x & 0 \\ 0 & x \end{matrix} )}; □; X_{1^{+2}/1^{+1}}; x; □; X_{0^{+1}/0^{+1}} ⊕X_{0^{+2}/0}; _{( \begin{matrix}x & 0 \\ 0 & x \end{matrix} )}; X_{1^{+1}/0^{+1}} ⊕X_{0^{+2}/1}; _{( \begin{matrix}x & 0 \\ 0 & x \end{matrix} )}; _{( \begin{matrix}x & 0 \\ 0 & x \end{matrix} )}; □; X_{2^{+1}/0^{+1}} ⊕X_{0^{+2}/2}; _{( \begin{matrix}x \\ −x \end{matrix} )}; _{( \begin{matrix}x & 0 \\ 0 & x \end{matrix} )}; □; X_{0^{+2}/0^{+1}}; x; □; 0; X_{0^{+1}/0}; x; _{(_{x x})}; □; X_{1^{+1}/1}; x; _{(_{x x})}; □; X_{2^{+1}/2}; _{(_{x x})}; □; 0; X_{2^{−1}/2^{−1}} ⊕X_{2/2^{−2}}; _{( \begin{matrix}x \\ −x \end{matrix} )}; X_{2/2^{−1}}; _{(_{x x})}; □; X_{2/0}⊕X_{0^{+1}/2^{−1}}; _{( \begin{matrix}x & 0 \\ 0 & x \end{matrix} )}; _{( \begin{matrix}x \\ −x \end{matrix} )}; □; X_{2/1}⊕X_{1^{+1}/2^{−1}}; _{( \begin{matrix}x & 0 \\ 0 & x \end{matrix} )}; _{( \begin{matrix}x \\ −x \end{matrix} )}; □; X_{2/2}⊕X_{2^{+1}/2^{−1}}; _{( \begin{matrix}x \\ −x \end{matrix} )}; □; □; □]

4.3 Some 1-epimorphic functors

Let n ⩾ 0. Concerning 1-epimorphy, cf. §A.8.

Lemma 4.3 The restriction functor

[figure: F^{□}(Δ̄_{n}^{#}); (−)|Δ̇n; F(Δ̇_{n}); X; X|Δ̇_{n}]

is 1-epimorphic.

Proof. We claim that the functor (−)|Δ̇_{n} satisfies the requirements (i, ii) of Corollary A.36, which then implies that it is 1-epimorphic. Suppose given Y ∈ Ob F(Δ̇_{n}). We construct an object Ỹ of F^{□}(Δ̄_{n}^{#}) such that Ỹ |Δ̇_{n} = Y by the following procedure. Write Δ̄_{n}^{△, ·} := Δ̄_{n}^{△} ∖ {0/0} and Δ̄_{n}^{▽, ·} := Δ̄_{n}^{▽} ∖ {0^{+1}/0}; cf. §2.1.1. On Δ̄_{n}^{△, ·}, we proceed by induction to construct a diagram for which, moreover, the morphisms Ỹ_{γ/α} \overline{}Ỹ_{γ/β} are purely monomorphic for all α, β, γ ∈ Δ̄_{n} with 0 ⩽ α ⩽ β ⩽ γ ⩽ α^{+1}, and, moreover, for which Ỹ_{α^{+1}/α} = 0 for all 0 ⩽ α. First of all, let Ỹ |Δ̇_{n} := Y. Assume given ℓ ⩾ 0 such that Ỹ_{κ^{△}(ℓ′)}, together with all diagram morphisms pointing to position κ^{△}(ℓ′), is already constructed for all ℓ′ < ℓ, but such that Ỹ_{κ^{△}(ℓ)} is not yet constructed; cf. §2.1.2. If κ^{△}(ℓ) is of the form α/α for some α ∈ Δ̄_{n} with 0 < α, then choose a pure monomorphism Ỹ_{α/(α−1)} [figure] Ỹ_{α/α} into an object Ỹ_{α/α} of B

We do not necessarily choose Ỹ_{α/(α−1)} ι here.

If κ^{△}(ℓ) = α^{+1}/α for some α ∈ Δ̄_{n} with 0 ⩽ α, then let Ỹ_{α^{+1}/α} := 0. If κ^{△}(ℓ) is of the form β/α for some α, β ∈ Δ̄_{n} with 0 < α < β < α^{+1}, then we let

( Ỹ_{(β−1)/(α−1)} , Ỹ_{(β−1)/α} , Ỹ_{β/(α−1)} , Ỹ_{β/α} )

be a pushout. Recall that by induction assumption, Ỹ_{(β−1)/(α−1)} [figure] Ỹ_{(β−1)/α} is purely monomorphic. So Ỹ_{β/(α−1)} [figure] Ỹ_{β/α} is purely monomorphic as well. On Δ̄_{n}^{▽, ·}, we proceed dually, and finally glue along Δ̇_{n} to obtain the sought Ỹ. Ad (i). The restriction map _{F(Δ̄_{n}^{#})}(Ỹ_{1}, Ỹ_{2}) ^{\overline{}^{(−)|Δ̇n}} _{F(Δ̇_{n})}(Y_{1}, Y_{2}) is surjective for Y_{1}, Y_{2} ∈ Ob F(Δ̇_{n}), as we see by induction, using bijectivity to prolong morphisms and universal properties of occurring pushouts and pullbacks. Ad (ii). Suppose given X ∈ Ob F^{□}(Δ̄_{n}^{#}). Let X′′ := (X|Δ̇_{n})˜ ∈ Ob F^{□}(Δ̄_{n}^{#}). Let X′ ∈ Ob F^{□}(Δ̄_{n}^{#}) be defined by X′|Δ̄_{n}^{△, ·} = X′′|Δ̄_{n}^{△, ·} and by X′|Δ̄_{n}^{▽, ·} = X|Δ̄_{n}^{▽, ·}. There is a morphism X′ \overline{}X that restricts to the identity of X|Δ̄_{n}^{▽, ·} on Δ̄_{n}^{▽, ·}, and hence to the identity of X|Δ̇_{n} on Δ̇_{n}. There is a morphism X′ \overline{}X′′ that restricts to the identity of X′′|Δ̄_{n}^{△, ·} on Δ̄_{n}^{△, ·}, and hence to the identity of X|Δ̇_{n} on Δ̇_{n}. Now suppose given X_{1}, X_{2} ∈ Ob F^{□}(Δ̄_{n}^{#}) such that X_{1}|Δ̇_{n} = X_{2}|Δ̇_{n}. Then there is a sequence of morphisms

X_{1} \overline{}X′_{1} \overline{}X′′_{1} = X′′_{2} \overline{}X′_{2} \overline{}X_{2}

each of which restricts to the identity of X_{1}|Δ̇_{n} = X_{2}|Δ̇_{n} on Δ̇_{n}, as required.

Lemma 4.4 The functors

[figure: F(Δ̇_{n}); N(Δ̇_{n}); \underline{F}(Δ̇_{n}); F(Δ̇_{n}); N(Δ̇_{n})N′; \underline{\underline{F}(Δ̇_{n})}]

are 1-epimorphic.

Proof. Since N′ is full and dense, it is 1-epimorphic by Corollary A.37. Therefore, it suffices to show that N(Δ̇_{n}) is 1-epimorphic.

We will apply Lemma A.35. Choosing representatives of the occurring morphisms Z_{i} \overline{}Z_{i+1} in an object Z of \underline{F}(Δ̇_{n}), where i ∈ [1, n − 1], we see that N(Δ̇_{n}) is dense.

To fulfill condition (C) of loc. cit., we will show that given X, Y ∈ Ob F(Δ̇_{n}) and a morphism (X)(N(Δ̇_{n})) ^{\overline{}^{f}} (Y)(N(Δ̇_{n})), there are morphisms X′ ^{\overline{}^{h}} X and X′ ^{\overline{}^{f′}} Y in F(Δ̇_{n}) such that (h)(N(Δ̇_{n})) is an isomorphism and such that (h)(N(Δ̇_{n}))f = (f′)(N(Δ̇_{n})).

We proceed by induction on k ∈ [1, n]. Suppose given a diagram

\xymatrix{X̃_{1} \ar[r]^{x̃} \ar[d]_{f̃_{1}} & X̃_{2} \ar[r]^{x̃} \ar[d]_{f̃_{2}} & X̃_{3} \ar[r]^{x̃} \ar[d]_{f̃_{3}} & ⋯ \ar[r]^{x̃} & X̃_{n−1} \ar[r]^{x̃} \ar[d]_{f̃_{n−1}} & X̃_{n} \ar[d]_{f̃_{n}} \\ Y_{1} \ar[r]^{y} & Y_{2} \ar[r]^{y} & Y_{3} \ar[r]^{y} & ⋯ \ar[r]^{y} & Y_{n−1} \ar[r]^{y} & Y_{n}}

in F such that x̃f̃_{i+1} = f̃_{i}y for i ∈ [1, k − 1], and such that x̃f̃_{i+1} ≡_{B} f̃_{i}y for i ∈ [k, n], and a morphism X̃ ^{\overline{}^{h̃}} X in F(Δ̇_{n}) such that (h̃)(N(Δ̇_{n})) is an isomorphism and such that (h̃)(N(Δ̇_{n}))f is the morphism in \underline{F}(Δ̇_{n}) represented by f̃.

If k < n, we shall construct a morphism X̃′ ^{\overline{}^{h̃′}} X̃ in F(Δ̇_{n}) with each h̃′_{i}N being an isomorphism, and a diagram

\xymatrix{X̃′_{1} \ar[r]^{x̃′} \ar[d]_{f̃′_{1}} & X̃′_{2} \ar[r]^{x̃′} \ar[d]_{f̃′_{2}} & X̃′_{3} \ar[r]^{x̃′} \ar[d]_{f̃′_{3}} & ⋯ \ar[r]^{x̃′} & X̃′_{n−1} \ar[r]^{x̃} \ar[d]_{f̃′_{n−1}} & X̃′_{n} \ar[d]_{f̃′_{n}} \\ Y_{1} \ar[r]^{y} & Y_{2} \ar[r]^{y} & Y_{3} \ar[r]^{y} & ⋯ \ar[r]^{y} & Y_{n−1} \ar[r]^{y} & Y_{n}}

in F such that x̃′f̃′_{i+1} = f̃′_{i}y for i ∈ [1, k], such that x̃′f̃′_{i+1} − f̃′_{i}y ≡_{B} 0 for i ∈ [k + 1, n], and such that h̃′_{i}f̃_{i} − f̃′_{i} ≡_{B} 0 for all i ∈ [1, n]. For then we obtain a commutative diagram in \underline{F}(Δ̇_{n})

\xymatrix{X̃′ \ar[d]_{h̃′} \ar[ddr]^{f̃′} \\ X̃ \ar[d]_{h̃} \ar[dr]_{f̃} \\ X \ar[r]_{f} & Y ,}

in which we denoted morphisms by their representatives.

Let X̃_{k} [figure: j] B be a pure monomorphism to an object B in B, and let x̃f̃_{k+1} − f̃_{k}y = jg. Let

X̃′ := ( X̃_{1} ^{\overline{}^{x̃}} ⋯ ^{\overline{}^{x̃}} X̃_{k} ^{\overline{}^{(^{x̃ j})}} X̃_{k+1} ⊕ B ^{\overline{}^{(x̃ ; 0)}} X̃_{k+2} ^{\overline{}^{x̃}} ⋯ ^{\overline{}^{x̃}} X̃_{n} ) ,

and let

f̃′_{i} := \{ \begin{matrix}f̃_{i} & for i ∈ [1, n] ∖ \{k + 1\} \\ (f̃_{k+1} ; −g) & for i = k + 1 \end{matrix} , h̃′_{i} := \{ \begin{matrix}1X̃_{i} & for i ∈ [1, n] ∖ \{k + 1\} \\ (\begin{matrix}1X̃_{k+1} \\ 0 \end{matrix} ) & for i = k + 1 \end{matrix} .

Proposition 4.5 The residue class functor F^{□}(Δ̄_{n}^{#}) ^{\overline{}^{M}} \underline{\underline{F}^{+}(Δ̄_{n}^{#})} is 1-epimorphic.

Proof. Consider the commutative quadrangle

\xymatrix{F^{□}(Δ̄_{n}^{#}) \ar[r]^{M} \ar[d]_{(−)|Δ̇n} & \underline{\underline{F}^{+}(Δ̄_{n}^{#})} \ar[d]^{(−)|Δ̇n} \\ F(Δ̇_{n}) \ar[r]^{N(Δ̇_{n})N′} & \underline{\underline{F}(Δ̇_{n})} .}

Therein, the functor F^{□}(Δ̄_{n}^{#}) ^{\overline{}^{(−)|Δ̇n}} F(Δ̇_{n}) is 1-epimorphic by Lemma 4.3. The functor F(Δ̇_{n}) ^{\overline{}^{N(Δ̇_{n}) N′}} \underline{\underline{F}(Δ̇_{n})} is 1-epimorphic by Lemma 4.4. The functor \underline{\underline{F}^{+}(Δ̄_{n}^{#})} ^{\overline{}^{(−)|Δ̇n}} \underline{\underline{F}(Δ̇_{n})}

is an equivalence by Proposition 2.6. Hence by Remark A.34, the functor F^{□}(Δ̄_{n}^{#}) ^{\overline{}^{M}} \underline{\underline{F}^{+}(Δ̄_{n}^{#})} is 1-epimorphic.

We do not claim that the residue class functor F^{□}(Δ̄_{n}^{#}) ^{\overline{}^{N(Δ̄_{n}^{#})}} \underline{F}^{+}(Δ̄_{n}^{#}) is 1-epimorphic.

4.4 Construction of ϑ

Let n ⩾ 0. In the notation of Lemma A.32, we let C := Δ̄_{n}^{#}; the role of the category called E there is played by F here; we let G := F^{□}(Δ̄_{n}^{#}); and finally, we let H := \underline{F}^{+}(Δ̄_{n}^{#}). Note that \underline{F}^{+}(Δ̄_{n}^{#}) is a characteristic subcategory of \underline{F}(Δ̄_{n}^{#}).

The tuples

I(n) = ((I_{X, β/α})_{β/α∈Δ̄_{n}^{#}})_{X∈Ob F^{□}(Δ̄_{n}^{#})}
:= ((X_{β/α}  [figure: ^{(^{x x})}] X_{β/β} ⊕ X_{α^{+1}/α}  ^{\overline{}^{(\begin{matrix}x \\ −x \end{matrix})}} X_{α^{+1}/β})_{β/α∈Δ̄_{n}^{#}})_{X∈Ob F^{□}(Δ̄_{n}^{#})}

J(n) = ((J_{X, β/α})_{β/α∈Δ̄_{n}^{#}})_{X∈Ob F^{□}(Δ̄_{n}^{#})}
:= ((X_{β/α}  [figure: X_{β/α} ι] X_{β/α} I = X_{β/α}^{+1} P ^{\overline{}^{X_{β/α}^{+1} π}} X_{β/α}^{+1})_{β/α∈Δ̄_{n}^{#}})_{X∈Ob F^{□}(Δ̄_{n}^{#})}

are Δ̄_{n}^{#}-resolving systems, inducing an isomorphism T_{I(n)}  \overline{}^{α_{I(n), J(n)}}_{∼}  T_{J(n)} by Lemma A.32.(2).

Recall that \underline{F}^{+}(Δ̄_{n}^{#}) ^{\overline{}^{M′′}} \underline{\underline{F}^{+}(Δ̄_{n}^{#})} denotes the residue class functor. We have

\begin{aligned}T_{I(n)} M′′ & = & M \underline{\underline{F}^{+}(T_{n})} & = & M[−]^{+1} \\ T_{J(n)} M′′ & = & M \underline{T^{+}(Δ̄_{n})} & = & M[−^{+1}] . \end{aligned}

Since M is 1-epimorphic by Proposition 4.5, we obtain

[figure: F^{□}(Δ̄_{n}^{#}); T_{I(n)}; T_{J(n)}; M; \underline{F}^{+}(Δ̄_{n}^{#}); M′′; \underline{\underline{F}^{+}(Δ̄_{n}^{#})}; [−]^{+1}; [−^{+1}]; \underline{\underline{F}^{+}(Δ̄_{n}^{#})} ,; α_{I(n), J(n)}; ϑ_{n}]

where ϑ_{n} is characterised by this commutative diagram, i.e. by

α_{I(n), J(n)} ⋆ M′′ = M ⋆ ϑ_{n} .

Since α_{I(n), J(n)} is an isomorphism, so is ϑ_{n}. Varying n, this defines ϑ = (ϑ_{n})_{n⩾0}. Theorem 4.6 The tuple ϑ = (ϑ_{n})_{n⩾0} is a Heller triangulation on \underline{F}. Proof. According to Definition 1.5.(i), we have to show that the following conditions (∗) and (∗∗) hold. (∗) For m, n ⩾ 0, for a morphism Δ̄_{n} ^{\overline{}^{p}} Δ̄_{m} and for an object Y ∈ Ob \underline{\underline{F}^{+}(Δ̄_{n}^{#})}, we have

(Y \underline{p}^{#})ϑ_{m} = (Y ϑ_{n})\underline{p}^{#} .

(∗∗) For n ⩾ 0 and for an object Y ∈ Ob \underline{\underline{F}^{+}((2Δ̄_{n})^{#})}, we have

(Y \underline{𝔣}_{n})ϑ_{n+1} = (Y ϑ_{2n+1})\underline{𝔣}_{n} .

Ad (∗). Recall that \underline{p}^{#} stands for \underline{\underline{F}^{+}(p^{#})}, and that p^{#} stands for F^{□}(p^{#}). By Proposition 4.5, we may assume Y = XM for some X ∈ Ob F^{□}(Δ̄_{n}^{#}). Then

\begin{aligned}(XM\underline{p}^{#})ϑ_{m} & = & (Xp^{#}M)ϑ_{m} & = & (Xp^{#}α_{I(m), J(m)})M′′ \\ (XMϑ_{n})\underline{p}^{#} & = & (Xα_{I(n), J(n)}M′′)\underline{p}^{#} & = & (Xα_{I(n), J(n)}p^{#})M′′ , \end{aligned}

so that it suffices to show that Xp^{#}α_{I(m), J(m)} = Xα_{I(n), J(n)}p^{#}. Starting with Xp^{#} , the object Xp^{#} T_{I(m)} is calculated by means of (I_{Xp^{#}, β/α})_{β/α∈Δ̄_{m}^{#}} ; whereas X T_{I(n)} is calculated by means of (I_{X, δ/γ})_{δ/γ∈Δ̄_{n}^{#}} , so that X T_{I(n)} p^{#} can be regarded as being calculated by means of (I_{X, βp/αp})_{β/α∈Δ̄_{m}^{#}}. But

I_{X, βp/αp} = (X_{βp/αp} [figure: ^{(^{x x})}] X_{βp/βp} ⊕ X_{(αp)^{+1}/αp}^{\overline{}^{(\begin{matrix}x \\ −x \end{matrix})}}X_{(αp)^{+1}/βp}) = I_{Xp^{#}, β/α} ,

whence Xp^{#} T_{I(m)} = X T_{I(n)} p^{#}. Next, starting with Xp^{#} , the object Xp^{#} T_{J(m)} is calculated by means of (J_{Xp^{#}, β/α})_{β/α∈Δ̄_{m}^{#}} ; whereas X T_{J(n)} p^{#} can be regarded as being calculated by means of (J_{X, βp/αp})_{β/α∈Δ̄_{m}^{#}}. But

J_{X, βp/αp} = (X_{βp/αp} [figure: X_{βp/αp}ι] X_{βp/αp} I = X_{βp/αp}^{+1} P ^{\overline{}^{X_{βp/αp}^{+1}π}} X_{βp/αp}^{+1}) = J_{Xp^{#}, β/α} , whence Xp^{#} T_{J(m)} = X T_{J(n)} p^{#}. Now

Xp^{#} T_{I(m)} \overline{}^{Xp^{#}α_{I(m), J(m)}}_{∼} Xp^{#} T_{J(m)}

is induced by (I_{Xp^{#}, β/α})_{β/α∈Δ̄_{m}^{#}} and by (J_{Xp^{#}, β/α})_{β/α∈Δ̄_{m}^{#}} ; whereas

X T_{I(n)} p^{#}  \overline{}^{Xα_{I(n), J(n)}p^{#}}_{∼}  X T_{J(n)} p^{#}

can be regarded as being induced by (I_{X, βp/αp})_{β/α∈Δ̄_{m}^{#}} and by (J_{X, βp/αp})_{β/α∈Δ̄_{m}^{#}}. We have just seen, however, that these pairs of tuples coincide. Ad (∗∗). By Proposition 4.5, we may assume Y = XM for some X ∈ Ob F^{□}((2Δ̄_{n})^{#}). By Remark 4.1, we have

\begin{aligned}(XM\underline{𝔣}_{n})ϑ_{n+1} & = & (X𝔣̃_{n}M)ϑ_{n+1} & = & (X𝔣̃_{n}α_{I(n+1), J(n+1)})M′′ \\ (XMϑ_{2n+1})\underline{𝔣}_{n} & = & (Xα_{I(2n+1), J(2n+1)}M′′)\underline{𝔣}_{n} & = & (Xα_{I(2n+1), J(2n+1)}𝔣_{n})M′′ , \end{aligned}

so that it suffices to show that X𝔣̃_{n}α_{I(n+1), J(n+1)} = Xα_{I(2n+1), J(2n+1)}𝔣_{n}. Starting with X𝔣̃_{n}, the object X𝔣̃_{n} T_{I(n+1)} is calculated by means of (I_{X𝔣̃_{n}, β/α})_{β/α ∈ \overline{ρ⊔Δ_{n}}^{#}} ; whereas X T_{I(2n+1)} 𝔣_{n} can be regarded as being calculated by means of the tuple of pure short exact sequences consisting of

0  at (ρ/ρ)^{+z}, z ∈ Z
I_{X, (α^{+1}/α)^{+z}}  at (α/ρ)^{+z}, α ∈ Δ_{n}, z ∈ Z
I_{X, (β^{+1}/α^{+1})^{+z}} ⊕ I_{X, (α^{+2}/β)^{+z}} at (β/α)^{+z}, α, β ∈ Δ_{n}, α ⩽ β, z ∈ Z .

We have I_{X𝔣̃_{n}, (ρ/ρ)^{+z}} = 0 for z ∈ Z. For α ∈ Δ_{n}, we have

I_{X𝔣̃_{n}, α/ρ} = (X_{α^{+1}/α} [figure: ^{(^{x x})}] X_{α^{+1}/α^{+1}} ⊕ X_{α^{+2}/α}^{\overline{}^{(\begin{matrix}x \\ −x \end{matrix})}}X_{α^{+2}/α^{+1}}) = I_{X, α^{+1}/α} ,

and accordingly at (α/ρ)^{+z} for z ∈ Z. Moreover, for α, β ∈ Δ_{n} with α ⩽ β, we have

\begin{matrix}I_{X𝔣̃_{n}, β/α} = \\ (X_{β^{+1}/α^{+1}} ⊕ X_{α^{+2}/β} [figure: ^{(\begin{matrix}x & 0 & x & 0 \\ 0 & x & 0 & −x \end{matrix})}] X_{β^{+1}/β^{+1}} ⊕ X_{β^{+2}/β} ⊕ X_{α^{+3}/α^{+1}} ⊕ X_{α^{+2}/α^{+2}}^{\overline{}^{(\begin{matrix}x & 0 \\ 0 & −x \\ −x & 0 \\ 0 & −x \end{matrix})}}X_{α^{+3}/β^{+1}} ⊕ X_{β^{+2}/α^{+2}}) \end{matrix}

and

\begin{matrix}I_{X, β^{+1}/α^{+1}} ⊕ I_{X, α^{+2}/β} = \\ (X_{β^{+1}/α^{+1}} ⊕ X_{α^{+2}/β} [figure: ^{(\begin{matrix}x & x & 0 & 0 \\ 0 & 0 & x & x \end{matrix})}] X_{β^{+1}/β^{+1}} ⊕ X_{α^{+3}/α^{+1}} ⊕ X_{α^{+2}/α^{+2}} ⊕ X_{β^{+2}/β}^{\overline{}^{(\begin{matrix}x & 0 \\ −x & 0 \\ 0 & x \\ 0 & −x \end{matrix})}}X_{α^{+3}/β^{+1}} ⊕ X_{β^{+2}/α^{+2}}) \end{matrix}

Accordingly at (β/α)^{+z} for z ∈ Z. Since there is an isomorphism from I_{X𝔣̃_{n}, α/ρ} to I_{X, β^{+1}/α^{+1}} ⊕ I_{X, α^{+2}/β} that has identities on the first and on the third terms of the short exact sequences, completed by

X_{β^{+1}/β^{+1}} ⊕ X_{β^{+2}/β} ⊕ X_{α^{+3}/α^{+1}} ⊕ X_{α^{+2}/α^{+2}}  \overline{}^{(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \end{matrix})}_{∼}  X_{β^{+1}/β^{+1}} ⊕ X_{α^{+3}/α^{+1}} ⊕ X_{α^{+2}/α^{+2}} ⊕ X_{β^{+2}/β}

on the second terms, the characterisation in Lemma A.32.(1) shows that we end up altogether with X𝔣̃_{n} T_{I(n+1)} = X T_{I(2n+1)} 𝔣_{n}.

Starting with X𝔣̃_{n}, the object X𝔣̃_{n} T_{J(n+1)} is calculated by means of (J_{X𝔣̃_{n}, β/α})_{β/α ∈ \overline{ρ⊔Δ_{n}}^{#}} ; whereas X T_{J(2n+1)} 𝔣_{n} can be regarded as being calculated by means of the tuple of pure short exact sequences consisting of

0  at (ρ/ρ)^{+z}, z ∈ Z
J_{X, (α^{+1}/α)^{+z}}  at (α/ρ)^{+z}, α ∈ Δ_{n}, z ∈ Z
J_{X, (β^{+1}/α^{+1})^{+z}} ⊕ J_{X, (α^{+2}/β)^{+z}} at (β/α)^{+z}, α, β ∈ Δ_{n}, α ⩽ β, z ∈ Z .

We have J_{X𝔣̃_{n}, (ρ/ρ)^{+z}} = 0 for z ∈ Z. For α ∈ Δ_{n}, we have

J_{X𝔣̃_{n}, α/ρ} = (X_{α^{+1}/α} [figure: X_{α^{+1}/α} ι] X_{α^{+1}/α} I = X_{α^{+1}/α}^{+1} P ^{\overline{}^{X_{α^{+1}/α}^{+1} π}} X_{α^{+1}/α}^{+1}) = J_{X, α^{+1}/α} ,

and accordingly at (α/ρ)^{+z} for z ∈ Z.

Moreover, for α, β ∈ Δ_{n} with α ⩽ β, we have

J_{X𝔣̃_{n}, β/α} = (X_{β^{+1}/α^{+1}} ⊕ X_{α^{+2}/β} [figure: (X_{β^{+1}/α^{+1}}⊕X_{α^{+2}/β}) ι]

(X_{β^{+1}/α^{+1}} ⊕ X_{α^{+2}/β}) I = (X_{β^{+1}/α^{+1}} ⊕ X_{α^{+2}/β})^{+1} P ^{\overline{}^{(X_{β^{+1}/α^{+1}}⊕X_{α^{+2}/β})^{+1} π}}

X_{α^{+3}/β^{+1}} ⊕ X_{β^{+2}/α^{+2}}) = J_{X, β^{+1}/α^{+1}} ⊕ J_{X,α^{+2}/β} ,

and accordingly at (β/α)^{+z} for z ∈ Z.

Hence altogether, we conclude that X𝔣̃_{n} T_{J(n+1)} = X T_{J(2n+1)} 𝔣_{n}.

Now

X𝔣̃_{n} T_{I(n+1)} \overline{}^{X𝔣̃_{n}α_{I(n+1), J(n+1)}}_{∼} X𝔣̃_{n} T_{J(n+1)}

is induced by (I_{X𝔣̃_{n}, β/α})_{β/α∈Δ̄_{n+1}^{#}} and by (J_{X𝔣̃_{n}, β/α})_{β/α∈Δ̄_{n+1}^{#}} ; whereas

X T_{I(2n+1)} 𝔣_{n}  \overline{}^{Xα_{I(2n+1), J(2n+1)}𝔣_{n}}_{∼}  X T_{J(2n+1)} 𝔣_{n}

can be regarded as being induced by the tuple consisting of

0  at (ρ/ρ)^{+z}, z ∈ Z
I_{X, (α^{+1}/α)^{+z}}  at (α/ρ)^{+z}, α ∈ Δ_{n}, z ∈ Z
I_{X, (β^{+1}/α^{+1})^{+z}} ⊕ I_{X, (α^{+2}/β)^{+z}} at (β/α)^{+z}, α, β ∈ Δ_{n}, α ⩽ β, z ∈ Z .

and by (J_{X𝔣̃_{n}, β/α})_{β/α∈Δ̄_{n+1}^{#}}.

Since the respective former tuples are isomorphic by a tuple of isomorphisms that has identities on the first and on the third term, and since the respective latter tuples are equal, the characterisation in Lemma A.32.(3) shows that in fact X𝔣̃_{n}α_{I(n+1), J(n+1)} = Xα_{I(2n+1), J(2n+1)}𝔣_{n}.

Corollary 4.7 Let E be a Frobenius category. There exists a Heller triangulation on (\underline{\underline{E}}, T).

Concerning the stable category \underline{\underline{E}}, cf. Definition A.7.

Proof. Let B ⊆ E be the full subcategory of bijectives. The category B^{ac} is functorially Frobenius by Example A.6. Hence \underline{\underline{E}} = \underline{B^{ac}}, equipped with the complex shift T, carries a Heller triangulation by virtue of Theorem 4.6.

4.5 Exact functors induce strictly exact functors

Proposition 4.8 Suppose given an exact functor

F ^{\overline{}^{F}} F̃

between functorial Frobenius categories F = (F, T, I, ι, P, π) and F̃ = (F̃, T̃, Ĩ, ι̃, P̃, π̃) that satisfies

\begin{aligned}F T̃ & = & TF \\ F Ĩ & = & I F \\ F P̃ & = & PF . \end{aligned}

Then the induced functor

\underline{F} ^{\overline{}^{\underline{F}}} \underline{F̃}

is strictly exact with respect to the Heller triangulations introduced in Theorem 4.6.

Proof. Condition (1) of Definition 1.5.(iii) is satisfied. Condition (2) of loc. cit. holds since each morphism has a weak kernel that is sent to a weak kernel of its image; and dually. In fact,

given a morphism represented by X ^{\overline{}^{f}} Y, the residue class of the kernel of X ⊕ Y P^{\overline{}^{(\begin{matrix}f \\ Y π \end{matrix} )}}Y in

F, composed with X ⊕ Y P ^{\overline{}^{(\begin{matrix}1 \\ 0 \end{matrix})}} X, is a weak kernel of the residue class of X ^{\overline{}^{f}} Y by Lemma A.31 and Remark A.27. Since pure short exact sequences and bijectives are preserved by F, this weak kernel is preserved by \underline{F}.

Consider condition (3) of loc. cit. Let ϑ resp. ϑ̃ be the Heller triangulation on \underline{F} resp. on \underline{F̃} characterised as in Theorem 4.6 by

\begin{aligned}α_{I(n), J(n)}M′′ & = & Mϑ_{n} \\ α_{Ĩ(n), J̃(n)}M̃′′ & = & M̃ϑ̃_{n} , \end{aligned}

where M̃, M̃′, M̃′′, Ĩ(n) resp. J̃(n) is defined over F̃ as M, M′, M′′, I(n) resp. J(n) is over F. To prove (3), i.e. to show that for n ⩾ 0 and Y ∈ Ob \underline{\underline{F}^{+}(Δ̄_{n}^{#})}, we have

(Y ϑ_{n})\underline{\underline{F}^{+}(Δ̄_{n}^{#})} = (Y \underline{\underline{F}^{+}(Δ̄_{n}^{#})})ϑ̃_{n} ,

we may assume by Proposition 4.5 that Y = XM for some X ∈ Ob F^{□}(Δ̄_{n}^{#}). Since

\begin{aligned}(XMϑ_{n})\underline{\underline{F}^{+}(Δ̄_{n}^{#})} & = & (Xα_{I(n), J(n)}M′′)\underline{\underline{F}^{+}(Δ̄_{n}^{#})} & = & (Xα_{I(n), J(n)}\underline{F}^{+}(Δ̄_{n}^{#}))M̃′′ \\ (XM\underline{\underline{F}^{+}(Δ̄_{n}^{#})})ϑ̃_{n} & = & (XF^{□}(Δ̄_{n}^{#})M̃)ϑ̃_{n} & = & (XF^{□}(Δ̄_{n}^{#})α_{Ĩ(n), J̃(n)})M̃′′ , \end{aligned}

it suffices to show that Xα_{I(n), J(n)}\underline{F}^{+}(Δ̄_{n}^{#}) = XF^{□}(Δ̄_{n}^{#})α_{Ĩ(n), J̃(n)}.

Starting with XF^{□}(Δ̄_{n}^{#}) , the object XF^{□}(Δ̄_{n}^{#}) T_{Ĩ(n)} is calculated by means of (Ĩ_{XF^{□}(Δ̄_{n}^{#}), β/α})_{β/α∈Δ̄_{n}^{#}} ; whereas X T_{Ĩ(n)} \underline{F}^{+}(Δ̄_{n}^{#}) can be regarded as being calculated by means of (I_{X, β/α}F)_{β/α∈Δ̄_{n}^{#}}, where I_{X, β/α}F is defined by an application of F to all three terms and both morphisms of the pure short exact sequence I_{X, β/α}. Since F is additive, we get

(Ĩ_{XF^{□}(Δ̄_{n}^{#}), β/α})_{β/α∈Δ̄_{n}^{#}} = (I_{X, β/α}F)_{β/α∈Δ̄_{n}^{#}} ,

whence XF^{□}(Δ̄_{n}^{#}) T_{Ĩ(n)} = X T_{I(n)} \underline{F}^{+}(Δ̄_{n}^{#}).

Starting with XF^{□}(Δ̄_{n}^{#}) , the object XF^{□}(Δ̄_{n}^{#}) T_{J̃(n)} is calculated by means of (J̃_{XF^{□}(Δ̄_{n}^{#}), β/α})_{β/α∈Δ̄_{n}^{#}} ; whereas X T_{J(n)} \underline{F}^{+}(Δ̄_{n}^{#}) can be regarded as being calculated by means of (J_{X, β/α}F)_{β/α∈Δ̄_{n}^{#}}, where J_{X, β/α}F is obtained by entrywise application of F. Since F commutes with P and P̃, and with I and Ĩ, we get

(J̃_{XF^{□}(Δ̄_{n}^{#}), β/α})_{β/α∈Δ̄_{n}^{#}} = (J_{X, β/α}F)_{β/α∈Δ̄_{n}^{#}} ,

whence XF^{□}(Δ̄_{n}^{#}) T_{J̃(n)} = X T_{J(n)} \underline{F}^{+}(Δ̄_{n}^{#}).

Moreover, since the defining pairs of tuples coincide, we finally get XF^{□}(Δ̄_{n}^{#})α_{Ĩ(n), J̃(n)} = Xα_{I(n), J(n)}\underline{F}^{+}(Δ̄_{n}^{#}).

Suppose given an exact functor

E ^{\overline{}^{E}} Ẽ

between Frobenius categories E and Ẽ that sends all bijective objects in E to bijective objects in Ẽ. Let B ⊆ E resp. B̃ ⊆ Ẽ be the respective subcategories of bijectives. We obtain an induced functor B^{ac} ^{\overline{}^{E^{ac}}} B̃^{ac}, inducing in turn a functor

\underline{\underline{E}} := \underline{E^{ac}} : \underline{\underline{E}} = \underline{B^{ac}} \overline{}\underline{B̃^{ac}} = ˜ \underline{\underline{E}}

modulo split acyclic complexes; cf. Example A.6.(2).

Corollary 4.9 The induced functor

\underline{\underline{E}} ^{\overline{}^{\underline{\underline{E}}}} ˜ \underline{\underline{E}}

is strictly exact with respect to the Heller triangulations on \underline{\underline{E}} and on ˜ \underline{\underline{E}} introduced in Theorem 4.6 via the functorial Frobenius categories B^{ac} and B̃^{ac}.

Proof. We may apply Proposition 4.8 to (F ^{\overline{}^{F}} F̃) := (B^{ac} ^{\overline{}^{E^{ac}}} B̃^{ac}).

5 Some quasicyclic categories

In the definition of a Heller triangulated category, the categories \underline{C^{+}(Δ̄_{n}^{#})} occur. Replacing this classical stable category by its stable counterpart, these turn out to be Heller triangulated themselves. So we can iterate. Cf. [2, Prop. 8.4].

Let C be a weakly abelian category. Let n ⩾ 0.

5.1 The category C^{+}(Δ̄_{n}^{#}) is Frobenius

5.1.1 The category A^{0}(Δ̄_{n}^{#}) is Frobenius

We proceed in a slightly more general manner than necessary. We generalise the fact that the category of complexes A^{0}(Δ̄_{2}^{#}) over an additive category A is a Frobenius category, to a category A^{0}(Δ̄_{n}^{#}) for n ⩾ 0; cf. Lemma 5.2 below. Then we will specialise to our weakly abelian category C and pass to the full subcategory C^{+}(Δ̄_{n}^{#}) ⊆ C^{0}(Δ̄_{n}^{#}); cf. Proposition 5.5 below.

5.1.1.1 Notation

Let A be an additive category. Let A^{0}(Δ̄_{n}^{#}) be the full subcategory of A(Δ̄_{n}^{#}) defined by

Ob A^{0}(Δ̄_{n}^{#}) := {X ∈ Ob A(Δ̄_{n}^{#}) : X_{α/α} = 0 and X_{α^{+1}/α} = 0 for all α ∈ Δ̄_{n}} .

A sequence X′ ^{\overline{}^{i}} X ^{\overline{}^{p}} X′′ in A^{0}(Δ̄_{n}^{#}) is called pointwise split short exact if the sequence X′_{ξ} ^{\overline{}^{i_{ξ}}} X_{ξ} ^{\overline{}^{p_{ξ}}} X′′_{ξ} is split short exact for all ξ ∈ Δ̄_{n}^{#}. A morphism is called pointwise split monomorphic (resp. epimorphic ) if it appears as a kernel (resp. cokernel) in a pointwise split short exact sequence.

The category A^{0}(Δ̄_{n}^{#}) carries an outer shift functor X \overline{}[X]^{+1}, where [X]_{β/α}^{+1} := X_{(β/α)^{+1}} = X_{α^{+1}/β} for β/α ∈ Δ̄_{n}^{#}.

Recall that A, together with the set of split short exact sequences, is an exact category; cf. Example A.3. So the additive category A^{0}(Δ̄_{n}^{#}), equipped with the set of pointwise split short exact sequences as pure short exact sequences, is an exact category; cf. Example A.4.

Given β/α, δ/γ ∈ Δ̄_{n}^{#}, we write β/α ⋖ δ/γ if α < γ and β < δ.

Given A ∈ Ob A and β/α ∈ Δ̄_{n}^{#}, we denote by A_{]α,β]} the object in A^{0}(Δ̄_{n}^{#}) consisting of identical morphisms wherever possible and having

(A_{]α,β]})_{δ/γ} := \{ \begin{matrix}A & if α/β^{−1} ⋖ δ/γ ⩽ β/α \\ 0 & else \end{matrix}

for δ/γ ∈ Δ̄_{n}^{#}. Such an object is called an extended interval.

Intuitively, it is a rectangle with upper right corner at β/α, and as large as possible in A^{0}(Δ̄_{n}^{#}).

Let A^{+, split}(Δ̄_{n}^{#}) be the full subcategory of A^{0}(Δ̄_{n}^{#}) consisting of objects isomorphic to summands of objects of the form

⨁_{β/α ∈ Δ̄_{n}^{#}}(A_{β/α})_{]α,β]} ,

where A_{β/α} ∈ Ob A for β/α ∈ Δ̄_{n}^{#}. This direct sum exists since it is a finite direct sum at each δ/γ ∈ Δ̄_{n}^{#}. Concerning the notation A^{+, split}(Δ̄_{n}^{#}), cf. also Remark 5.3 below.

5.1.1.2 The periodic case

Let A′ be an additive category, equipped with a graduation shift automorphism X \overline{}X[+1]. We write X \overline{}X[m] for its m-th iteration, where m ∈ Z.

By entrywise application, there is also a graduation shift on A^{′0}(Δ̄_{n}^{#}), likewise denoted by X \overline{}X[+1].

As in §2.5.3, we define the subcategory A^{′0,periodic}(Δ̄_{n}^{#}) ⊆ A^{′0}(Δ̄_{n}^{#}) to consist of the morphisms X ^{\overline{}^{f}} Y in A^{′0}(Δ̄_{n}^{#}) that satisfy

(X[+1]^{\overline{}^{f[+1]}} Y [+1]) = ([X]^{+1} ^{\overline{}^{[f]^{+1}}} [Y]^{+1})

So the subcategory A^{′0,periodic}(Δ̄_{n}^{#}) ⊆ A^{′0}(Δ̄_{n}^{#}) is not full in general.

Given A ∈ Ob A′ and 0 ⩽ i ⩽ j ⩽ n, we denote by A_{]i,j]} the object in A^{′0}(Δ̄_{n}^{#}) consisting only of zero and identical morphisms and having

(A_{]i,j]})_{δ/γ} := \{ \begin{matrix}A[m] & if (i/j^{−1})^{+m} ⋖ δ/γ ⩽ (j/i)^{+m} for some m ∈ Z \\ 0 & else \end{matrix}

for δ/γ ∈ Δ̄_{n}^{#}.

Intuitively, it is a rectangle with upper right corner at j/i, and as large as possible in A^{′0, periodic}(Δ̄_{n}^{#}), repeated Z-periodically up to according graduation shift.

Let A^{′+, split, periodic}(Δ̄_{n}^{#}) be the full subcategory of A^{′0, periodic}(Δ̄_{n}^{#}) consisting of objects isomorphic to summands of objects of the form

⨁_{0⩽i⩽j⩽n}(A_{j,i})_{]i,j]} ,

where A_{j,i} ∈ Ob A′ for 0 ⩽ i ⩽ j ⩽ n. Such an object is called a periodic extended interval.

Lemma 5.1 The category A^{′0, periodic}(Δ̄_{n}^{#}), equipped with the pointwise split short exact sequences, is a Frobenius category, having A^{′+, split, periodic}(Δ̄_{n}^{#}) as its subcategory of bijectives.

Proof. By duality, it suffices to show that the following assertions (1, 2) hold.

(1) The object A_{]i,j]} is injective in A^{′0, periodic}(Δ̄_{n}^{#}) for any A ∈ Ob A′ and any 0 ⩽ i ⩽ j ⩽ n.

(2) For each object of A^{′0, periodic}(Δ̄_{n}^{#}), there exists a pure monomorphism into an object of A^{′+, split, periodic}(Δ̄_{n}^{#}).

Ad (1). Note that we have an adjunction isomorphism

A^{′0, periodic}(Δ̄_{n}^{#})(X, A_{]i,j]}) ^{\overline{}^{∼}}  _{A′}(X_{j/i} , A)
f \overline{}f_{j/i} ,

where X ∈ Ob A^{′0, periodic}(Δ̄_{n}^{#}). Suppose given a pure monomorphism A_{]i,j]} [figure] X for some A ∈ Ob A′. Let (A [figure] X_{j/i} \overline{}A) = 1_{A}. Let X \overline{}A_{]i,j]} correspond to X_{j/i} \overline{}A. The composition (A_{]i,j]} [figure] X \overline{}A_{]i,j]}) restricts to 1_{A} at j/i, hence equals 1_{A_{]i,j]}}.

Ad (2). Suppose given X ∈ Ob A^{′0, periodic}(Δ̄_{n}^{#}). Given 0 ⩽ i ⩽ j ⩽ n, we let

X ^{\overline{}^{Xs_{j/i}}} (X_{j/i})_{]i,j]}

be the morphism corresponding to 1_{X_{j/i}} by adjunction, which is natural in X. Collecting these morphisms yields a morphism

X ^{\overline{}^{Xs}} ⨁_{0⩽i⩽j⩽n}(X_{j/i})_{]i,j]} ,

which is pointwise split monomorphic since at j/i, its component X_{j/i} \overline{}X_{j/i} is an identity.

5.1.1.3 The general case

Lemma 5.2 The category A^{0}(Δ̄_{n}^{#}), equipped with the pointwise split short exact sequences, is a Frobenius category, having A^{+, split}(Δ̄_{n}^{#}) as its subcategory of bijectives.

Proof. To prove that A^{0}(Δ̄_{n}^{#}) is a Frobenius category, we more precisely claim that A^{+, split}(Δ̄_{n}^{#}) is a sufficiently big category of bijective objects in the exact category A^{0}(Δ̄_{n}^{#}).

Abbreviate A^{Z} := A(Ż), where Ż denotes the discrete category with Ob Ż = Z and only identical morphisms. The category A^{Z} carries the graduation shift automorphism

\begin{matrix}A^{Z} & ^{\overline{}^{∼}} & A^{Z} \\ (X ^{\overline{}^{f}} Y) & \overline{} & (X[+1] ^{\overline{}^{f[+1]}} Y [+1]) := (X_{i+1} ^{\overline{}^{f_{i+1}}} Y_{i+1})_{i∈Z} . \end{matrix}

We have an isomorphism of categories

A^{+}(Δ̄_{n}^{#}) \overline{}^{Φ}_{∼} (A^{Z})^{+, periodic}(Δ̄_{n}^{#})
X \overline{}((X_{(β/α)^{+i}})_{i∈Z})_{β/α∈Δ̄_{n}^{#}}
((Y_{β/α})_{0})_{β/α∈Δ̄_{n}^{#}}  \overline{}Y .

Both categories are exact when equipped with pointwise split short exact sequences, and Φ and Φ^{−1} are exact functors. We have A^{+, split}(Δ̄_{n}^{#})Φ = (A^{Z})^{+, split, periodic}(Δ̄_{n}^{#}).

Putting A′ := A^{Z}, the result follows by Lemma 5.1.

If A = C is a weakly abelian category, we have two definitions of C^{+, split}(Δ̄_{n}^{#}).

The first one, given in §1.2.1.1, defines this category as a full subcategory of C^{+}(Δ̄_{n}^{#}) containing those diagrams in which all morphisms are split in Ĉ.

The second one, just given, defines this category as a full subcategory of C^{0}(Δ̄_{n}^{#}) containing, up to isomorphism, summands of direct sums of extended intervals.

Remark 5.3 If A = C is a weakly abelian category, then the two aforementioned definitions of C^{+, split}(Δ̄_{n}^{#}) coincide.

Proof. First, we notice that an extended interval lies in C^{+}(Δ̄_{n}^{#}), and that all its diagram morphisms are split in Ĉ.

It remains to be shown that an object in C^{+}(Δ̄_{n}^{#}) all of whose diagram morphisms are split in Ĉ, is, up to isomorphism, a summand of a direct sum of extended intervals.

Passing to (C^{Z})^{+, periodic}(Δ̄_{n}^{#}), we have to show that an object X ∈ Ob(C^{Z})^{+, periodic}(Δ̄_{n}^{#}) all of whose diagram morphisms are split in Ĉ^{Z}, is, up to isomorphism, a summand of a direct sum of periodic extended intervals.

By Lemma A.25, applied to the abelian Frobenius category Ĉ^{Z}, the object X|Δ̇_{n} ∈ Ob C^{Z}(Δ̇_{n}) is isomorphic to a summand of a finite direct sum of intervals. Hence, by Lemma 2.11, the object X is isomorphic to a summand of a finite direct sum of images of intervals under S̄, i.e. of periodic extended intervals, as was to be shown.

5.1.2 The subcategory C^{+}(Δ̄_{n}^{#}) ⊆ C^{0}(Δ̄_{n}^{#})

Recall that C is a weakly abelian category.

Lemma 5.4 Suppose given a pure short exact sequence

X′ [figure] X \overline{}X′′

in C^{0}(Δ̄_{n}^{#}). If two out of the three objects X′, X and X′′ are in C^{+}(Δ̄_{n}^{#}), so is the third.

Proof. For an object X ∈ Ob C^{0}(Δ̄_{n}) to lie in Ob C^{+}(Δ̄_{n}), it suffices to know that the complex

X(α, β, γ) := ( ⋯ \overline{}X_{β/γ^{−1}} \overline{}X_{β/α} \overline{}X_{γ/α} \overline{}X_{γ/β} \overline{}X_{α^{+1}/β} \overline{}⋯)

is acyclic in Ĉ for all α, β, γ ∈ Δ̄_{n} with α ⩽ β ⩽ γ ⩽ α^{+1}; which is true, as we take from Lemma A.17; cf. Remark A.27.

Now the long exact homology sequence, applied in Ĉ to the short exact sequence X′(α, β, γ) [figure] X(α, β, γ) \overline{}X′′(α, β, γ) of complexes, shows that if two of these complexes are acyclic, so is the third.

Proposition 5.5

(1) The category C^{+}(Δ̄_{n}^{#}), equipped with the pointwise split short exact sequences, is a Frobenius category, having C^{+, split}(Δ̄_{n}^{#}) as its subcategory of bijectives.

Hence its stable category \underline{\underline{C^{+}(Δ̄_{n}^{#})}} is equivalent to its classical stable category \underline{C^{+}(Δ̄_{n}^{#})} =

C^{+}(Δ̄_{n}^{#})/C^{+, split}(Δ̄_{n}^{#}). So both \underline{\underline{C^{+}(Δ̄_{n}^{#})}} and \underline{C^{+}(Δ̄_{n}^{#})} are weakly abelian.

(2) Suppose C to be equipped with an automorphism X \overline{}X^{+1}. The category C^{+, periodic}(Δ̄_{n}^{#}) (cf. §2.5.3), equipped with the pointwise split short exact sequences, is an additively functorial Frobenius category, having C^{+, split, periodic}(Δ̄_{n}^{#}) as its subcategory of bijectives.

We remark that \underline{C^{+}(Δ̄_{n}^{#})} appears in Definition 1.5.

Proof. Ad (1). To prove that C^{+}(Δ̄_{n}^{#}) is an exact category, it remains to be shown, in view of Lemma 5.2 and of §A.2.2, that a pure short exact sequence in C^{0}(Δ̄_{n}^{#}) that has the first and the third term in Ob C^{+}(Δ̄_{n}^{#}), has the second term in Ob C^{+}(Δ̄_{n}^{#}), too. This follows by Lemma 5.4.

To prove that C^{+}(Δ̄_{n}^{#}) is Frobenius, we may use that C^{0}(Δ̄_{n}^{#}) is Frobenius, with the bijective objects already lying in C^{+}(Δ̄_{n}^{#}), thus being a fortiori bijective with respect to C^{+}(Δ̄_{n}^{#}). By duality, it remains to be shown that the kernel of a pointwise split epimorphism of a bijective object to a given object in C^{+}(Δ̄_{n}^{#}) is again in C^{+}(Δ̄_{n}^{#}), thus showing that this epimorphism is actually pure in C^{+}(Δ̄_{n}^{#}). This follows by Lemma 5.4.

Ad (2). In view of Lemma 5.1, this follows as (1).

I do not know whether C^{+, ϑ=1}(Δ̄_{n}^{#}) is Frobenius. It seems doubtful, since this question is reminiscent of the example of A. Neeman that shows that the mapping cone of a morphism of distinguished triangles in the sense of Verdier need not be distinguished [26, p. 234].

5.1.3 Two examples

Suppose C to be equipped with an automorphism X \overline{}X^{+1}.

The category C^{+, periodic}(Δ̄_{n}^{#}) being a Frobenius category by Proposition 5.5.(2), its classical stable category \underline{C^{+, periodic}(Δ̄_{n}^{#})} carries a Heller operator, defined on X ∈ Ob C^{+, periodic}(Δ̄_{n}^{#}) as the kernel of B \overline{}X, where B ∈ Ob C^{+, split, periodic}(Δ̄_{n}^{#}). As examples, we calculate the Heller operator for n ∈ {2, 3} on periodic n-pretriangles.

Suppose n = 2. Let X ∈ Ob C^{+, periodic}(Δ̄_{2}^{#}) be a periodic 2-pretriangle. We obtain

[figure: X_{1/0}; x; X_{2/0}; x; X_{2/1}; x; X_{1/0}^{+1}; x; X_{2/0}^{+1}; ^{[\begin{matrix}x \\ 1 \end{matrix}]}; ^{[\begin{matrix}x \\ 1 \end{matrix}]}; ^{[\begin{matrix}x \\ 1 \end{matrix}]}; ^{[\begin{matrix}x \\ 1 \end{matrix}]}; ^{[\begin{matrix}x \\ 1 \end{matrix}]}; X_{2/1}^{−1}⊕X_{1/0}; ^{[ \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ]}; X_{1/0}⊕X_{2/0}; ^{[ \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ]}; X_{2/0}⊕X_{2/1}; ^{[ \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ]}; X_{2/1}⊕X_{1/0}^{+1}; ^{[ \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ]}; X_{1/0}^{+1}⊕X_{2/0}^{+1}; ^{[^{1 −x}]}; ^{[^{1 −x}]}; ^{[^{1 −x}]}; ^{[^{1 −x}]}; ^{[^{1 −x}]}; X_{2/1}^{−1}; −x; X_{1/0}; −x; X_{2/0}; −x; X_{2/1}; −x; X_{1/0}^{+1}]

In particular, if X is a 2-triangle, i.e. an object of C^{+, ϑ=1}(Δ̄_{2}^{#}), then this Heller shift of X is also a 2-triangle; cf. Lemma 3.5.

Suppose n = 3. Let X ∈ Ob C^{+, periodic}(Δ̄_{3}^{#}) be a periodic 3-pretriangle. We obtain

[figure: X_{1/0}; x; X_{2/0}; x; X_{3/0}; x; x; X_{2/1}; x; X_{3/1}; x; X_{1/0}^{+1}; x; x; X_{3/2}; x; X_{2/0}^{+1}; x; X_{3/0}^{+1}; ^{[\begin{matrix}x \\ x \\ 1 \end{matrix}]}; ^{[\begin{matrix}x \\ x \\ x \\ 1 \end{matrix}]}; ^{[\begin{matrix}x \\ x \\ 1 \end{matrix}]}; ^{[\begin{matrix}x \\ x \\ 1 \end{matrix}]}; ^{[\begin{matrix}x \\ x \\ x \\ 1 \end{matrix}]}; ^{[\begin{matrix}x \\ x \\ 1 \end{matrix}]}; ^{[\begin{matrix}x \\ x \\ 1 \end{matrix}]}; ^{[\begin{matrix}x \\ x \\ x \\ 1 \end{matrix}]}; ^{[\begin{matrix}x \\ x \\ 1 \end{matrix}]}; X_{2/1}^{−1}⊕X_{3/1}^{−1}⊕X_{1/0}; ^{[ \begin{matrix}0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix} ]}; X_{3/1}^{−1}⊕X_{3/2}^{−1}⊕X_{1/0}⊕X_{2/0}; ^{[ \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ]}; X_{1/0}⊕X_{2/0}⊕X_{3/0}; ^{[ \begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ]}; ^{[ \begin{matrix}0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ]}; X_{3/2}^{−1}⊕X_{2/0}⊕X_{2/1}; ^{[ \begin{matrix}0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix} ]}; X_{2/0}⊕X_{3/0}⊕X_{2/1}⊕X_{3/1}; ^{[ \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ]}; X_{2/1}⊕X_{3/1}⊕X_{1/0}^{+1}; ^{[ \begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ]}; ^{[ \begin{matrix}0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix} ]}; X_{3/0}⊕X_{3/1}⊕X_{3/2}; ^{[ \begin{matrix}0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ]}; X_{3/1}⊕X_{3/2}⊕X_{1/0}^{+1}⊕X_{2/0}^{+1}; ^{[ \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ]}; X_{1/0}^{+1}⊕X_{2/0}^{+1}⊕X_{3/0}^{+1}; ^{[ \begin{matrix}1 & 0 & −x \\ 0 & 1 & −x \end{matrix} ]}; ^{[ \begin{matrix}1 & 0 & 0 & −x \\ 0 & 1 & 0 & −x \\ 0 & 0 & 1 & −x \end{matrix} ]}; ^{[ \begin{matrix}1 & 0 & −x \\ 0 & 1 & −x \end{matrix} ]}; ^{[ \begin{matrix}1 & 0 & −x \\ 0 & 1 & −x \end{matrix} ]}; ^{[ \begin{matrix}1 & 0 & 0 & −x \\ 0 & 1 & 0 & −x \\ 0 & 0 & 1 & −x \end{matrix} ]}; ^{[ \begin{matrix}1 & 0 & −x \\ 0 & 1 & −x \end{matrix} ]}; ^{[ \begin{matrix}1 & 0 & −x \\ 0 & 1 & −x \end{matrix} ]}; ^{[ \begin{matrix}1 & 0 & 0 & −x \\ 0 & 1 & 0 & −x \\ 0 & 0 & 1 & −x \end{matrix} ]}; ^{[ \begin{matrix}1 & 0 & −x \\ 0 & 1 & −x \end{matrix} ]}; X_{2/1}^{−1}⊕X_{3/1}^{−1}; ^{[ \begin{matrix}0 & 0 & −x \\ 1 & 0 & −x \end{matrix} ]}; X_{3/1}^{−1}⊕X_{3/2}^{−1}⊕X_{1/0}; ^{[ \begin{matrix}0 & −x \\ 0 & −x \\ 1 & −x \end{matrix} ]}; X_{1/0}⊕X_{2/0}; ^{[ \begin{matrix}0 & −x \\ 1 & −x \\ 0 & −x \end{matrix} ]}; ^{[ \begin{matrix}0 & −x & 0 \\ 1 & −x & 0 \end{matrix} ]}; X_{3/2}^{−1}⊕X_{2/0}; ^{[ \begin{matrix}0 & 0 & −x \\ 1 & 0 & −x \end{matrix} ]}; X_{2/0}⊕X_{3/0}⊕X_{2/1}; ^{[ \begin{matrix}0 & −x \\ 0 & −x \\ 1 & −x \end{matrix} ]}; X_{2/1}⊕X_{3/1}; ^{[ \begin{matrix}0 & −x \\ 1 & −x \\ 0 & −x \end{matrix} ]}; ^{[ \begin{matrix}0 & 0 & −x \\ 1 & 0 & −x \end{matrix} ]}; X_{3/0}⊕X_{3/1}; ^{[ \begin{matrix}0 & −x & 0 \\ 1 & −x & 0 \end{matrix} ]}; X_{3/1}⊕X_{3/2}⊕X_{1/0}^{+1}; ^{[ \begin{matrix}0 & −x \\ 0 & −x \\ 1 & −x \end{matrix} ]}; X_{1/0}^{+1}⊕X_{2/0}^{+1}]

If X is a 3-triangle, i.e. an object C^{+, ϑ=1}(Δ̄_{3}^{#}), I do not know whether this Heller shift of X is again a 3-triangle.

5.2 A quasicyclic category

The category of quasicyclic categories is defined to be the category of contravariant functors from Δ̄^{∘} to the (1-)category (Cat) of categories. Recall that we have a functor Δ \overline{}Δ̄, Δ_{n} \overline{}Δ̄_{n} that allows to restrict a quasicyclic category to its underlying simplicial category. Given a category U, we denote by Iso U ⊆ U its subcategory consisting of isomorphisms. Given a functor U ^{\overline{}^{U}} U′, we denote by Iso F : Iso U \overline{}Iso U′ the induced functor. We define

Δ̄^{∘} ^{\overline{}^{qcyc_{•} C}} (Cat)
(Δ̄_{n} ^{\overline{}^{p}} Δ̄_{m})  \overline{}(Iso C^{+}(Δ̄_{n}^{#}) ^{\overline{}^{Iso C^{+}(p^{#})}} Iso C^{+}(Δ̄_{m}^{#})) .

More intuitively written, qcyc_{•} C := Iso C^{+}(Δ̄_{•}^{#}). Note that qcyc_{0} C consists only of zero-objects. A strictly exact functor C ^{\overline{}^{F}} C′ induces a functor C^{+}(Δ̄_{n}^{#}) ^{\overline{}^{F^{+}(Δ̄_{n}^{#})}} C^{′+}(Δ̄_{n}^{#}) for n ⩾ 0, and thus a morphism

qcyc_{•} C ^{^{qcyc_{•} F}\overline{}}qcyc_{•} C′

of quasicyclic categories. As variants, we mention

qcyc_{•}^{periodic} C := Iso C^{+, periodic}(Δ̄_{•}^{#})
qcyc_{•}^{ϑ=1} C := Iso C^{+, ϑ=1}(Δ̄_{•}^{#}) .

5.3 A biquasicyclic category

As an attempt in the direction described in [32, p. 330], we define a first step of an “iteration” of the construction C \overline{}qcyc_{•} C.

By Proposition 5.5, we may form the category \underline{\underline{C^{+}(Δ̄_{n}^{#})}}^{+}(Δ̄_{m}^{#}). Note that a morphism Δ̄_{m} ^{\overline{}^{f}} Δ̄_{m′} of periodic linearly ordered sets induces a functor \underline{\underline{C^{+}(Δ̄_{n}^{#})}}^{+}(f^{#}) in the second variable. By Lemma 4.8, a morphism Δ̄_{n} ^{\overline{}^{g}} Δ̄_{n′} of periodic linearly ordered sets induces a strictly exact functor \underline{\underline{C^{+}(g^{#})}}, and so a functor \underline{\underline{C^{+}(g^{#})}}^{+}(Δ̄_{m}^{#}) in the first variable for m ⩾ 0. The functors induced by f and by g commute. We may define

qcyc_{••} C := Iso (\underline{\underline{C^{+}(Δ̄_{•}^{#})}}^{+}(Δ̄_{•}^{#})) ,

which yields a biquasicyclic category, i.e. a functor from Δ̄^{∘} × Δ̄^{∘} to the (1-)category (Cat) of categories. By Lemma 4.8, a strictly exact functor C ^{\overline{}^{F}} C′ induces a functor \underline{\underline{C^{+}(Δ̄_{n}^{#})}} ^{\overline{}^{\underline{\underline{F^{+}(Δ̄_{n}^{#})}}}} \underline{\underline{C^{′+}(Δ̄_{n}^{#})}} for n ⩾ 0, and thus a morphism

qcyc_{••} C ^{\overline{}^{qcyc_{••} F}} qcyc_{••} C′

of quasicyclic categories.

As variants, we mention

qcyc_{••}^{periodic} C := Iso (\underline{\underline{C^{+}(Δ̄_{•}^{#})}}^{+, periodic}(Δ̄_{•}^{#}))

qcyc_{••}^{ϑ=1} C := Iso (\underline{\underline{C^{+}(Δ̄_{•}^{#})}}^{+, ϑ=1}(Δ̄_{•}^{#})) .

Cf. Remark 3.3.

This procedure can be iterated to obtain triquasicyclic categories etc.

A Some general lemmata

This appendix is a tool kit consisting of folklore lemmata (with proof) and known results (mainly without proof). We do not claim originality.

A.1 An additive lemma

Let A and B be additive categories, and let A ^{\overline{}^{F}} B be a full and dense additive functor. Let N ⊆ B be a full additive subcategory. Let M ⊆ A be the full subcategory determined by

Ob M := {A ∈ Ob A : AF is isomorphic to an object of N } .

Lemma A.1 Suppose that for each morphism A ^{\overline{}^{a_{0}}} A′ in A such that a_{0}F = 0, there exists a factorisation

(A ^{\overline{}^{a_{0}}} A′) = (A ^{\overline{}^{a′_{0}}} M_{0} ^{\overline{}^{a′′_{0}}} A′)

with M_{0} ∈ Ob M. Then the induced functor

\begin{matrix}A/M & ^{\overline{}^{\underline{F}}} & B/N \\ (A ^{\overline{}^{a}} A′) & \overline{} & (AF ^{\overline{}^{aF}} A′F) \end{matrix}

is an equivalence.

Proof. We have to show that \underline{F} is faithful. Suppose given A ^{\overline{}^{a}} A′ in A such that

(AF ^{\overline{}^{aF}} A′F) = (AF ^{\overline{}^{b′}} N ^{\overline{}^{b′′}} A′F) ,

where N ∈ Ob N. Since F is dense, we may assume N = MF for some M ∈ Ob M. Since F is full, there exist a′ and a′′ in A with a′F = b′ and a′′F = b′′. Then

(A ^{\overline{}^{a}} A′) = (A ^{\overline{}^{a′a′′}} A′) + (A ^{\overline{}^{a_{0}}} A′)

with a_{0}F = 0. Since a′a′′ factors over M ∈ Ob M, and since a_{0} factors over an object of M by assumption on F, we conclude that a factors over an object of M.

A.2 Exact categories

A.2.1 Definition

The concept of exact categories is due to Quillen [29], who uses a different, but equivalent set of axioms. In [17, App. A], Keller has cut down redundancies in this set of axioms. We give still another equivalent reformulation.

An additive category A is a category with zero object, binary products and binary coproducts such that the natural map from the coproduct to the product is an isomorphism; which allows to define a commutative and associative addition (+) on _{A}(X, Y), where X, Y ∈ Ob A; and such that there exists an endomorphism −1_{X} for each X ∈ Ob A that is characterised by 1_{X} + (−1_{X}) = 0_{X}.

A sequence X ^{\overline{}^{f}} Y ^{\overline{}^{g}} Z in A is called short exact if f is a kernel of g and g is a cokernel of f.

A short exact sequence isomorphic to a short exact sequence of the form

X ^{\overline{}^{(^{1 0} )}} X ⊕ Y ^{\overline{}^{( \begin{matrix}0 \\ 1 \end{matrix} )}} Y ,

where X, Y ∈ Ob A, is called split short exact. A morphism appearing as a kernel in a split short exact sequence is split monomorphic, a morphism appearing as a cokernel in a split short exact sequence is called split epimorphic. A split short exact sequence is isomorphic to a sequence of the form just displayed by an isomorphism having an identity on the first and on the third term.

An exact category (E, S) consists of an additive category E and an isomorphism closed set S of short exact sequences in E, called pure short exact sequences (^{7}), such that the following axioms (Ex 1, 2, 3, 1^{∘}, 2^{∘}, 3^{∘}) are satisfied.

A monomorphism fitting into a pure short exact sequence is called a pure monomorphism, denoted by [figure] ; an epimorphism fitting into a pure short exact sequence is called a pure epimorphism, denoted by \overline{}. A morphism which can be written as a composition of a pure epimorphism followed by a pure monomorphism is called pure.

(Ex 1) Split monomorphisms are pure monomorphisms.
(Ex 1^{∘}) Split epimorphisms are pure epimorphisms.
(Ex 2) The composite of two pure monomorphisms is purely monomorphic.
(Ex 2^{∘}) The composite of two pure epimorphisms is purely epimorphic.

(Ex 3) Given a commutative diagram

[figure: Y; X; •; Z ,]

we may insert it into a commutative diagram

[figure: A; •; •; B; Y; X; •; •; Z]

with (X, Y, B) and (A, Y, Z) pure short exact sequences.

7This notion is borrowed from the particular cases of pure short exact sequences of lattices over orders and of ⊗-pure short exact sequences of modules. Other frequently used names are admissible short exact sequence, consisting of an admissible monomorphism and an admissible epimorphism; and conflation, consisting of an inflation and a deflation.

(Ex 3^{∘}) Given a commutative diagram

[figure: Y; X; •; Z ,]

we may insert it into a commutative diagram

[figure: A; •; B; Y; X; •; Z]

with (X, Y, B) and (A, Y, Z) pure short exact sequences.

An exact functor from an exact category (E, S) to an exact category (F, T ) is given by an additive functor

E ^{\overline{}^{F}} F such that SF ⊆ T , where, by abuse of notation, F also denotes the functor induced by F on diagrams of shape • \overline{}• \overline{}•.

Frequently, the exact category (E, S) is simply referred to by E.

Example A.2

(1) An abelian category, equipped with the set of all short exact sequences as pure short exact sequences, is an exact category.

(2) If E is an exact category, so is E^{∘}, equipped with the pure short exact sequences of E considered as short exact sequences in E^{∘}, with the roles of kernel and cokernel interchanged.

Example A.3 An additive category A, equipped with the set of split short exact sequences as pure short exact sequences, is an exact category.

In fact, (Ex 1, 2) are fulfilled, and it remains to prove (Ex 3); then the dual axioms ensue by duality. Given

[figure: X ⊕Y ⊕Z; ( \begin{matrix}1 & 0 \\ 0 & 1 \\ 0 & 0 \end{matrix} ); X; •; _{(_{1 0} )}; X ⊕Y ,]

we get

[figure: Z; •; _{(_{0 0 1} )}; •; ^{(^{0 1} )}; Y ⊕Z; X ⊕Y ⊕Z; _{( \begin{matrix}1 & 0 \\ 0 & 1 \\ 0 & 0 \end{matrix} )}; _{( \begin{matrix}0 & −a \\ 1 & 0 \\ 0 & 1 \end{matrix} )}; X; •; _{(_{1 0 a})}; •; _{(_{1 0} )}; X ⊕Y ,]

where X ^{\overline{}^{a}} Z is the third component of the given morphism X \overline{}X ⊕ Y ⊕ Z.

Example A.4 Suppose given an exact category E and a category D. Let a short exact sequence (X, Y, Z) in E(D) be pure if the sequence (X_{d}, Y_{d}, Z_{d}) is a pure short exact sequence in E for all d ∈ Ob D. Then E(D) is an exact category.

A.2.2 Embedding exact categories

By a theorem

• stated by Quillen [29, p. 100],

• proven by Laumon [22, Th. 1.0.3],

• re-proven by Keller [17, Prop. A.2],

• where Quillen resp. Keller refer to [9] for a similar resp. an auxiliary technique,

for any exact category E, there exists an abelian category Ẽ containing E as a full subcategory closed under extensions, the pure short exact sequences in E being the short exact sequences in Ẽ with all three objects in Ob E.

Conversely, suppose given an exact category E and a full subcategory E′ ⊆ E such that whenever (X, Y, Z) is a pure short exact sequence in E with X, Z ∈ Ob E′, then also Y ∈ Ob E′. Then the subcategory E′, equipped with the pure short exact sequences in E with all three terms in Ob E′ as pure short exact sequences in E′, is an exact category.

A.2.3 Frobenius categories : definitions

Definition A.5

(1) A bijective object in an exact category E is an object B for which _{E}(B, −) and _{E}(−, B) are exact functors from E resp. from E^{∘} to Z -Mod.

(2) A Frobenius category is an exact category for which each object X allows for a diagram B \overline{}X [figure] B′ with B and B′ bijective.

(3) Suppose given an exact category F carrying a shift automorphism T : X \overline{}X T = X^{+1} and two

additive endofunctors I and P together with natural transformations 1_{F} ^{\overline{}^{ι}} I and P ^{\overline{}^{π}} 1_{F} such that T P = I and such that

X ^{\overline{}^{Xι}} X I = X^{+1} P^{\overline{}^{X^{+1}π}} X^{+1}

is a pure short exact sequence with bijective middle term for all X ∈ Ob C. Then (F, T, I, ι, P, π) is called a functorial Frobenius category. Often we write just F for (F, T, I, ι, P, π).

Example A.6

(1) Let A be an additive category. Let Ż denote the discrete category with Ob Ż = Z and only identical morphisms. The category A(Ż) carries the shift functor X^{•} \overline{}X^{•+1}, where (X^{•+1})^{i} = X^{i+1}. An object in the category C(A) of complexes with entries in A is written (X^{•}, d^{•}), where X is an object

of A(Ż) and where X^{•} ^{\overline{}^{d^{•}}} X^{•+1} with d^{•}d^{•+1} = 0. The category C(A), equipped with pointwise split short exact sequences, is an exact category; cf. Examples A.3, A.4. Given a complex (X^{•}, d^{•}), we let (X^{•}, d^{•}) T = (X^{•}, d^{•})^{+1} := (X^{•+1}, −d^{•+1}) and

\begin{aligned}& ((X^{•}, d^{•}) ^{^{(X^{•},d^{•})ι}\overline{}}(X^{•}, d^{•}) I = (X^{•}, d^{•})^{+1} P ^{\overline{}^{(X^{•},d^{•})^{+1}π}} (X^{•}, d^{•})^{+1}) \\ := & ((X^{•}, d^{•}) ^{^{(^{1 d^{•}})} \overline{}}(X^{•} ⊕ X^{•+1}, (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix})) ^{^{( \begin{matrix}−d^{•} \\ 1 \end{matrix} )} \overline{}}(X^{•+1}, −d^{•+1})) . \end{aligned}

Then (C(A), T, I, ι, P, π) is a functorial Frobenius category.

(2) Suppose E to be a Frobenius category. Let B ⊆ E be a sufficiently big full subcategory of bijective objects, i.e. each object of B is bijective in E, and each object X of E admits B \overline{}X [figure] B′ with B, B′ ∈ Ob B. In other words, each bijective object of E is isomorphic to a direct summand of an object of B. Let B^{ac} ⊆ C(B) denote the full subcategory of purely acyclic complexes, i.e. complexes (X^{•}, d^{•}) such

that all differentials X^{i} ^{\overline{}^{d}} X^{i+1} are pure, factoring in E as d = d̄ḋ with d̄ purely epi- and ḋ purely monomorphic, and such that all resulting sequences (ḋ, d̄) are purely short exact. For short, a complex is purely acyclic if it decomposes into pure short exact sequences.

Then B^{ac} is a functorial Frobenius category, equipped with the restricted functors and transformations of C(B) as defined in (1); cf. [25, Lem. 1.1]. Let B^{sp ac} ⊆ B^{ac} be the full subcategory of split acyclic complexes, i.e. of complexes isomorphic to a complex of the form (T^{•} ⊕ T^{•+1}, (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix})) for some T^{•} ∈ Ob B(Ż). Then B^{sp ac} is a sufficiently big full subcategory of bijective objects in B^{ac}.

Definition A.7 Suppose given a Frobenius category E, and a sufficiently big full subcategory B ⊆ E of bijectives. Let

\underline{E} := E/B  be the classical stable category of E ;
\underline{\underline{E}} := B^{ac}/B^{sp ac} be the stable category of E .

In other words, the stable category \underline{\underline{E}} of E is defined to be the classical stable category \underline{B^{ac}} of B^{ac}. The shift functor induced by the automorphism T of B^{ac} on \underline{\underline{E}} is also denoted by T.

Lemma A.8 The functor

\begin{matrix}B^{ac} & ^{\overline{}^{I}} & E \\ (X, d) & \overline{} & Im(X^{0} ^{\overline{}^{d}} X^{1}) \end{matrix}

induces an equivalence

\underline{\underline{E}} = \underline{B^{ac}} \overline{}^{\underline{I}}_{∼} \underline{E} .

Cf. [19, Sec. 4.3].

Proof. This is an application of Lemma A.1.

We choose an inverse equivalence R to \underline{I}. We have the residue class functor E ^{\overline{}^{N}} \underline{E}, and, by abuse of notation,

a second residue class functor (E ^{\overline{}^{N}} \underline{\underline{E}}) := (E ^{\overline{}^{N}} \underline{E} \overline{}^{R}_{∼} \underline{\underline{E}}).

A morphism X ^{\overline{}^{f}} Y is zero in \underline{E} if and only if for any monomorphism X [figure: i] X′ and any epimorphism Y′ ^{\overline{}^{p}} Y, there is a factorisation f = if′p. This defines \underline{E} without mentioning bijective objects in E. So one might speculate whether the class of Frobenius categories within the class of exact categories could be extended still without losing essential properties of Frobenius categories.

A.3 Kernel-cokernel-criteria

Let A be an abelian category. The circumference lemma states that given a commutative triangle in A, the induced sequence on kernels and cokernels, with zeroes attached to the ends, is long exact.

Definition A.9 A weak square in A is a commutative quadrangle (A, B, C, D) in A whose diagonal sequence (A, B ⊕ C, D) is exact at B ⊕ C. It is denoted by a “ + ”-sign in the commutative diagram,

[figure: C; D; A; +; B .]

A pullback is a weak square with first morphism in the diagonal sequence being monomorphic. It is denoted

[figure: C; D; A; B .]

A pushout is a weak square with second morphism in the diagonal sequence being epimorphic. It is denoted

[figure: C; D; A; B .]

A square is a commutative quadrangle that is a pullback and a pushout, i.e. that has a short exact diagonal sequence. It is denoted

[figure: C; D; A; □; B .]

Remark A.10 If a commutative quadrangle in A

\xymatrix{C \ar[r]^{c} & D \\ A \ar[r]_{b} \ar[u]^{a} & B \ar[u]_{d}}

is a square, then the induced morphism from the kernel of A ^{\overline{}^{a}} C to the kernel of B ^{\overline{}^{d}} D is an isomorphism

and the induced morphism from the cokernel of A ^{\overline{}^{a}} C to the cokernel of B ^{\overline{}^{d}} D is an isomorphism.

Proof. If (A, B, C, D) is a square, then the circumference lemma, applied to the commutative triangle

\xymatrix{C \\  & B ⊕ C \ar[ul]_{_{( \begin{matrix}0 \\ 1 \end{matrix} )}} \\ A \ar[ur]_{_{(_{b a})}} \ar[uu]^{a} & ,}

yields a long exact sequence

0 \overline{}K_{a} ^{\overline{}^{j}} B ^{\overline{}^{−d}} D ^{\overline{}^{q}} C_{a} \overline{}0 ,

where K_{a} ^{\overline{}^{i}} A is the kernel of a, and where C ^{\overline{}^{p}} C_{a} is the cokernel of a. Since ib = j and cb = p, the induced morphisms on the kernels and on the cokernels of a and d are isomorphisms.

Lemma A.11 A commutative quadrangle in A

\xymatrix{C \ar[r]^{c} & D \\ A \ar[r]_{b} \ar[u]^{a} & B \ar[u]_{d}}

is a weak square if and only if the induced morphism K_{a} \overline{}K_{d} from the kernel of A ^{\overline{}^{a}} C to the kernel of

B ^{\overline{}^{d}} D is an epimorphism and the induced morphism C_{a} \overline{}C_{d} from the cokernel of A ^{\overline{}^{a}} C to the cokernel

of B ^{\overline{}^{d}} D is a monomorphism.

It is a pullback if and only if K_{a} ^{\overline{}^{∼}} K_{d} and C_{a} [figure] C_{d}.

It is a pushout if and only if K_{a} \overline{}K_{d} and C_{a} ^{\overline{}^{∼}} C_{d}.

It is a square if and only if K_{a} ^{\overline{}^{∼}} K_{d} and C_{a} ^{\overline{}^{∼}} C_{d}.

Proof. Let A′ be the pullback of (C, B, D), and let D′ be the pushout of (A′, C, B). We obtain induced morphisms A \overline{}A′ and D′ [figure] D. The circumference lemma, applied to (B, D′, D), shows C_{B→D′} [figure] C_{B→D}.

The quadrangle (A, B, C, D) is a weak square if and only if A \overline{}A′; which in turn, by the circumference lemma applied to (A, A′, C), is equivalent to K_{A→C} \overline{}K_{A′→C} and C_{A→C} ^{\overline{}^{∼}} C_{A′→C}; which, by composition and by Remark A.10, applied to the square (A′, B, C, D′), is equivalent to K_{A→C} \overline{}K_{B→D} and C_{A→C} [figure] C_{B→D}.

The quadrangle (A, B, C, D) is a pullback if and only if A ^{\overline{}^{∼}} A′; which in turn, by the circumference lemma applied to (A, A′, C), is equivalent to K_{A→C} ^{\overline{}^{∼}} K_{A′→C} and C_{A→C} ^{\overline{}^{∼}} C_{A′→C}; which, by composition and by Remark A.10, applied to the square (A′, B, C, D′), is equivalent to K_{A→C} ^{\overline{}^{∼}} K_{B→D} and C_{A→C} [figure] C_{B→D}.

The quadrangle (A, B, C, D) is a square if and only if A ^{\overline{}^{∼}} A′ and D′ ^{\overline{}^{∼}} D; which in turn, by the circumference lemma applied to (A, A′, C), is equivalent to K_{A→C} ^{\overline{}^{∼}} K_{A′→C}, C_{A→C} ^{\overline{}^{∼}} C_{A′→C} and C_{B→D′} ^{\overline{}^{∼}} C_{B→D}; which, by composition and by Remark A.10, applied to the square (A′, B, C, D′), is equivalent to K_{A→C} ^{\overline{}^{∼}} K_{B→D} and C_{A→C} ^{\overline{}^{∼}} C_{B→D}.

A.4 An exact lemma

Let E be an exact category. A pure square in E is a commutative quadrangle (A, B, C, D) in E that has a pure short exact diagonal sequence (A, B ⊕ C, D). Just as a square in abelian categories, a pure square is denoted by a box “ □ ”.

Lemma A.12 Suppose given a composition

\xymatrix{X′ \ar[r] & Y′ \ar[r] & Z′ \\ X \ar[r] \ar[u] & Y \ar[r] \ar[u] & Z \ar[u]}

of commutative quadrangles in E. If two out of the three quadrangles (X, Y, X′, Y′), (Y, Z, Y′, Z′), (X, Z, X′, Z′) are pure squares, so is the third.

Proof. In an abelian category, this follows from Lemma A.11.

As explained in §A.2.2, we may embed E fully, faithfully and additively into an abelian category Ẽ such that the pure short exact sequences in E are precisely the short exact sequences in Ẽ with all three objects in Ob E. In particular, the pure squares in E are precisely the squares in Ẽ with all four objects in Ob E, and the assertion in E follows from the assertion in Ẽ.

A.5 Some abelian lemmata

Let A be an abelian category.

Lemma A.13 Inserting images, a weak square (A, B, C, D) in A decomposes into

[figure: C; •; D; •; □; •; •; •; A; •; □; B]

Proof. The assertion follows using the characterisation of weak squares, pullbacks and pushouts given in Lemma A.11.

Lemma A.14 If, in a commutative diagram

[figure: X′; Y′; Z′; X; +; Y; +; Z]

in A, the quadrangles (X, Y, X′, Y′) and (Y, Z, Y′, Z′) are weak squares, then the composite quadrangle (X, Z, X′, Z′) is also a weak square.

Proof. The assertion follows using the characterisation of weak squares given in Lemma A.11. Lemma A.15 If, in a commutative diagram

[figure: X′; Y′; Z′; X; Y; Z]

in A, the left hand side quadrangle (X, Y, X′, Y′) is a pushout, as indicated, and the outer quadrangle (X, Z, X′, Z′) is a weak square, then the right hand side quadrangle (Y, Z, Y′, Z′) is also a weak square. If the left hand side quadrangle (X, Y, X′, Y′) and the outer quadrangle (X, Z, X′, Z′) are pushouts, then the right hand side quadrangle (Y, Z, Y′, Z′) is also a pushout. Proof. This follows using Lemma A.11. Lemma A.16 If, in a commutative quadrangle in A

\xymatrix{X′ \ar[r] & Y′ \\ X \ar[u] \ar[r] & Y , \ar[u]}

the morphism X \overline{}Y is an epimorphism and the morphism X′ \overline{}Y′ is a monomorphism, then the quadrangle is a weak square. Proof. This follows using Lemma A.11, applied horizontally. Lemma A.17 Given a commutative diagram

\xymatrix{0 \ar[r] & Y′ \ar[r] & Z′ \ar[r] & W′ \\ X \ar[u] \ar[r] & Y \ar[u] \ar[r] & Z \ar[u] \ar[r] & 0 \ar[u]}

in A such that (X, Z, 0, Z′) and (Y, 0, Y′, W′) are weak squares, then (Y, Z, Y′, Z′) is a weak square. Proof. This follows using Lemma A.11. Lemma A.18 Given a diagram

[figure: 0; Y′′; v′′; Z′′; X′; u′; +; Y′; v′; y′; +; Z′; z′; X; x; u; +; Y; y; +; 0]

in A consisting of weak squares, as indicated by +, the sequence

X ^{\overline{}^{xu′}} Y′ ^{\overline{}^{(^{y′ v′})}} Y′′ ⊕ Z′ ^{\overline{}^{( \begin{matrix}v′′ \\ −z′ \end{matrix} )}} Z′′

is exact at Y′ and at Y′′ ⊕ Z′.

Proof. At Y′, we reduce to the case u, u′, x and y monomorphic and (X, Y, X′, Y′) being a pullback via Lemma

A.13. Suppose given T ^{\overline{}^{t}} Y′ with ty′ = 0 and tv′ = 0. First of all, there exist T ^{\overline{}^{a}} X′ and T ^{\overline{}^{b}} Y such

that au′ = t = by. Thus there exists T ^{\overline{}^{c}} X such that cx = a and cu = b. In particular, cxu′ = au′ = t. Hence a factorisation of t over xu′ exists. Uniqueness follows by monomorphy of xu′.

Lemma A.19

(1) Suppose given a weak square in A

[figure: X′; Y′; X; +; Y]

with X′ bijective. If the images of X \overline{}Y, of X \overline{}X′ and of Y \overline{}Y′ are bijective, then the images of X′ \overline{}Y′ and of X \overline{}Y′ are bijective, too.

(2) Suppose given a weak square in A

[figure: X′; Y′; X; +; Y]

with Y bijective. If the images of X′ \overline{}Y′, of X \overline{}X′ and of Y \overline{}Y′ are bijective, then the images of X \overline{}Y and of X \overline{}Y′ are bijective, too.

Proof. Ad (1). We decompose (X, Y, X′, Y′) according to Lemma A.13 and denote the image of X \overline{}Y by Im_{X,Y}, etc.

The diagonal sequence of the square (Im_{X,Y} , Y, Im_{X,Y′}, Im_{Y,Y′}) shows that Im_{X,Y′} is bijective.

The diagonal sequence of the square (Im_{X,X′}, Im_{X,Y′}, X′, Im_{X′,Y′}) shows that Im_{X′,Y′} is bijective.

Lemma A.20 Given a pullback

[figure: X; f; Y; X′; f′; •; x; Y′ ,; y]

in A with Y′ injective, the morphism (X′, Y′) ^{\overline{}^{(x,y)}} (X, Y) is split monomorphic in A(Δ_{1}). More precisely, any retraction for x may be extended to a retraction for (x, y).

Proof. Let xx′ = 1_{X′}. We form the pushout.

[figure: Y; X; f; P; •; X′; f′; •; x; Y′; •; y]

There is an induced morphism P \overline{}Y′ such that (X \overline{}P \overline{}Y′) = (X ^{\overline{}^{x′f′}} Y′) and such that

(Y′ \overline{}P \overline{}Y′) = (Y′ ^{\overline{}^{1_{Y ′}}} Y′). Since Y′ is injective, we obtain a factorisation (P \overline{}Y′) = (P [figure] Y \overline{}Y′).

Lemma A.21 Suppose given a morphism X \overline{}Y of commutative quadrangles in A, i.e. a morphism in A(Δ_{1} × Δ_{1}).

(1) If X is a pushout and Y is a weak square, then the cokernel of X \overline{}Y is a weak square.

(2) If X is a weak square and Y is a pullback, then the kernel of X \overline{}Y is a weak square.

Proof. Ad (1). A morphism of commutative quadrangles gives rise to a morphism of the diagonal sequences; namely from a sequence that is exact in the middle and has an epimorphic second morphism, stemming from X, to a sequence that is exact in the middle, stemming from Y. In order to prove that the cokernel sequence is exact in the middle, we reduce by insertion of the image of the first morphism of the diagonal sequence and by an application of the circumference lemma to the case in which the sequence stemming from Y has a monomorphic first morphism. Then the snake lemma yields the result.

Lemma A.22 Suppose given a diagram

[figure: 0; Y′′; Z′′; X′; +; Y′; +; Z′; X; +; Y; +; Z]

in A, consisting of weak squares. The induced morphisms furnish a short exact sequence

Im(X \overline{}Z′) \overline{}Im(Y \overline{}Z′) \overline{}Im(Y \overline{}Z′′) .

Proof. Abbreviate Im(X \overline{}Z′) by Im_{X,Z′} etc. The morphism Im_{X,Z′} \overline{}Im_{Y,Z′} is monomorphic by composition, and, dually, the morphism Im_{Y,Z′} \overline{}Im_{Y,Z′′} is epimorphic. Now since Im_{X,X′} \overline{}Im_{X,Z′} is epimorphic and Im_{Y,Z′′} \overline{}Im_{Y′′,Z′′} is monomorphic, it suffices to show that

[figure: Im; X,X′; Im; Y,Z′; Im; Y′′,Z′′]

is exact at Im_{Y,Z′}. This follows from the diagram obtained by Lemma A.13

[figure: 0; Y′′; Im_{Y′′,Z′′}; X′; +; Y′; _{Im_{Y′,Z′}}; _{Im_{X,X′}}; •; _{Im_{Y,Y′}}; •; □; Im_{Y,Z′} ,; •]

since by Lemma A.14, weak squares are stable under composition.

A.6 On Frobenius categories

A.6.1 Some Frobenius-abelian lemmata

Suppose given an abelian Frobenius category A; cf. Definition A.5. Let B be its full subcategory of bijective objects. Recall that the classical stable category of A is defined as \underline{A} = A/B; cf. Definition A.7. A morphism in A whose residue class in \underline{A} is an isomorphism is called a homotopism. A morphism in A whose residue class in \underline{A} is a retraction is called a retraction up to homotopy.

Lemma A.23 Given a retraction up to homotopy X ^{\overline{}^{f}} Y and an epimorphism Y′ ^{\overline{}^{y}} Y in A, in the pullback

[figure: X′; x; f′; X; f; Y′; y; Y ,]

the morphism X′ ^{\overline{}^{f′}} Y′ is a retraction up to homotopy, too. More precisely, if gf ≡_{B} 1_{Y}, then we may find a morphism g′ with g′f′ ≡_{B} 1_{Y′} as a pullback of g along x.

Proof. Let Y ^{\overline{}^{g}} X be such that gf = 1_{Y} + h, where

(Y ^{\overline{}^{h}} Y) = (Y ^{\overline{}^{h_{1}}} B ^{\overline{}^{h_{2}}} Y)

for some B ∈ Ob B and some morphisms h_{1} and h_{2} in A. Let B ^{\overline{}^{h′_{2}}} Y′ be a morphism such that

(B ^{\overline{}^{h′_{2}}} Y′ ^{\overline{}^{y}} Y) = (B ^{\overline{}^{h_{2}}} Y) ,

which exists since B is projective and y is epimorphic. The commutative quadrangle

[figure: Y′; y; 1_{Y ′} +yh_{1}h′_{2}; Y; 1_{Y} +h; Y′; y; Y]

is a pullback since the induced morphism on the horizontal kernels is an identity; cf. Lemma A.11. So we may form the diagram

[figure: Y′; y; g′; Y; g; X′; x; f′; X; f; Y′; y; Y ,]

in which g′ with g′x = yg and g′f′ = 1_{Y′} + yh_{1}h′_{2} is induced by the universal property of the lower pullback (X′, X, Y′, Y), and in which the resulting upper quadrangle (Y′, Y, X′, X) is a pullback by Lemma A.11.

Lemma A.24 Given a homotopism X ^{\overline{}^{f}} Y and an epimorphism Y′ ^{\overline{}^{y}} Y in A, in the pullback

[figure: X′; x; f′; X; f; Y′; y; Y ,]

the morphism X′ ^{\overline{}^{f′}} Y′ is a homotopism, too.

Proof. Let gf ≡_{B} 1_{Y} and fg ≡_{B} 1_{X}. By Lemma A.23, we may form the diagram

[figure: Y′; y; g′; Y; g; X′; x; f′; X; f; Y′; y; Y ,]

in which g′f′ ≡_{B} 1_{Y′}. Since g is a retraction up to homotopy, so is g′ by Lemma A.23. Therefore g′ is a homotopism. Hence also f′ is a homotopism.

A.6.2 Decomposing split diagrams in intervals

Let A be an abelian Frobenius category, and let B be its full subcategory of bijective objects. Suppose given

n ⩾ 1. Write Δ̇_{n} := Δ_{n} ∖ \{0\}. An object X in A(Δ̇_{n}) is called split if X_{k} ^{\overline{}^{x}} X_{l} is split for all k, l ∈ [1, n] with k ⩽ l.

Given C ∈ Ob A and k, l ∈ [1, n] with k ⩽ l, we denote by C_{[k,l]} the object of A(Δ̇_{n}) given by (C_{[k,l]})_{j} = 0 for

j ∈ [1, n] ∖ [k, l], by (C_{[k,l]})_{j} = C for j ∈ [k, l], and by ((C_{[k,l]})_{j} ^{\overline{}^{c}} (C_{[k,l]})_{j′}) = (C ^{\overline{}^{1_{C}}} C) for j, j′ ∈ [k, l] with j ⩽ j′. An object in A(Δ̇_{n}) of the form C_{[k,l]} for some C ∈ Ob A and some k, l ∈ [1, n] with k ⩽ l, is called an interval.

Lemma A.25 Any split object in B(Δ̇_{n}) is isomorphic to a finite direct sum of intervals.

Proof. We proceed by induction on n. Suppose given a split object X in B(Δ̇_{n}). Let X′ := X↑_{0} be defined

as a pointwise pullback at n, using 0 ^{\overline{}^{0}} X_{n} (cf. §A.7 below). We have X′ ∈ Ob B(Δ̇_{n}) with X′_{n} = 0. Hence, by induction, X′ is isomorphic to a finite direct sum of intervals. There is a pure monomorphism X′ [figure] X whose cokernel is a diagram in Ob B(Δ̇_{n}) consisting of split monomorphisms; cf. Lemma A.11. Moreover, by an iterated application of Lemma A.20, starting at position 1, this pure monomorphism X′ [figure] X is split as a morphism of A(Δ̇_{n}) (^{8}). Thus X is isomorphic to the direct sum of X′ and the cokernel of X′ [figure] X, and it remains to be shown that this cokernel is isomorphic to a finite direct sum of intervals.

Therefore, we may assume that X consists of split monomorphisms [figure: X_{k}; •; x; X_{l}] for k, l ∈ [1, n]. We have

a monomorphism (X_{1})_{[1,n]} [figure: i] X. Choosing a retraction to [figure: X_{1}; •; x; X_{n}] and composing, we obtain a coretraction to i, so that X is isomorphic to the direct sum of the interval (X_{1})_{[1,n]} and the cokernel of i. Since the cokernel of i has a zero term at position 1, we are done by induction.

A.6.3 A Freyd category reminder

The construction of the Freyd category and its properties are due to Freyd [8, Th. 3.1].

Definition A.26 Suppose given an additive category C and a morphism X ^{\overline{}^{f}} Y in C.

(1) A morphism K ^{\overline{}^{i}} X is a weak kernel of X ^{\overline{}^{f}} Y if the sequence of abelian groups

(T, K) ^{\overline{}^{(−)i}} (T, X) ^{\overline{}^{(−)f}} (T, Y)

is exact at (T, X) for every T ∈ Ob C.

(2) A morphism Y ^{\overline{}^{p}} C is a weak cokernel of X ^{\overline{}^{f}} Y if the sequence of abelian groups

(X, T) ^{\overline{}^{f(−)}} (Y, T) ^{\overline{}^{p(−)}} (C, T)

is exact at (Y, T) for every T ∈ Ob C.

(3) The category C is called weakly abelian if every morphism has a weak kernel and a weak cokernel, and if every morphism is a weak kernel (of some morphism) and a weak cokernel (of some morphism).

Let C be a weakly abelian category. Let C^{0}(Δ_{1}) be the full subcategory of C(Δ_{1}) whose objects are zero morphisms. The Freyd category Ĉ of C is defined to be the quotient category

Ĉ := C(Δ_{1})/C^{0}(Δ_{1}) .

We collect some elementary facts and constructions and mention some conventions.

8At this point, we use that Δ̇_{n} is linearly ordered.

(1) The category Ĉ is abelian. The kernel and the cokernel of a morphism X ^{\overline{}^{f}} Y represented by (f′, f′′) are constructed as

\xymatrix{K \ar[r]^{i} \ar[d]_{ix} & X′ \ar[d]_{x} \ar[r]^{f′} & Y′ \ar[d]^{y} \ar[r]^{1_{Y ′}} & Y′ \ar[d]^{yp} \\ X′′ \ar[r]^{1_{X′′}} & X′′ \ar[r]^{f′′} & Y′′ \ar[r]^{p} & C ,}

where i is a chosen weak kernel and p a chosen weak cokernel of the diagonal morphism f′y = xf′′. If

f′y = xf′′ = 0, we choose X′ ^{\overline{}^{1_{X′}}} X′ as weak kernel and Y′′ ^{\overline{}^{1_{Y ′′}}} Y′′ as weak cokernel.

Choosing a kernel and a cokernel for each object in Ĉ(Δ_{1}), we obtain a kernel and a cokernel functor Ĉ(Δ_{1}) [figure] Ĉ, as for any abelian category.

(2) We stipulate that the pullback resp. the pushout of an identity morphism along a morphism is chosen to be an identity morphism.

(3) We have a full and faithful functor C \overline{}Ĉ, X \overline{}(X ^{\overline{}^{1_{X}}} X). Its image, identified with C, consists of bijective objects.

(4) For each X = (X′ ^{\overline{}^{x}} X′′) ∈ Ob Ĉ, we may define objects and morphisms

X P ^{\overline{}^{Xπ}} X [figure: Xι] X I

by

\xymatrix{X′ \ar[r]^{1_{X′}} \ar[d]_{1_{X′}} & X′ \ar[r]^{x} \ar[d]^{x} & X′′ \ar[d]^{1_{X′′}} \\ X′ \ar[r]^{x} & X′′ \ar[r]^{1_{X′′}} & X′′ .}

As already mentioned in (3), the objects X P and X I are bijective, and thus Ĉ is Frobenius. Sometimes, we write just ι for Xι and π for Xπ. Note that Xπ = 1_{X} and Xι = 1_{X} if X ∈ Ob C.

This construction X P ^{\overline{}^{Xπ}} X [figure: Xι] X I is not meant to be functorial in (X′ ^{\overline{}^{x}} X′′), however.

Remark A.27 Suppose given morphisms X ^{\overline{}^{f}} Y ^{\overline{}^{g}} Z in C. The following assertions are equivalent.

(i) The morphism f is a weak kernel of g.

(ii) The morphism g is a weak cokernel of f.

(iii) The sequence (f, g) is exact at Y when considered in Ĉ.

Proof. Ad (i) ⟹ (iii). Suppose that f is a weak kernel of g. Let K ^{\overline{}^{i}} Y be the kernel of g in Ĉ. Factor f = f′i. Since f is a weak kernel of g in C, we may factor (Kπ)i = uf, whence Kπ = uf′. Hence f′ is epimorphic.

Ad (iii) ⟹ (i). Suppose (f, g) to be exact at Y. Let T ^{\overline{}^{t}} Y in C be such that tg = 0. Then t factors over the kernel of g, taken in Ĉ, and therefore, by projectivity of T in Ĉ, also over X.

Remark A.28 A morphism X ^{\overline{}^{f}} Y in C is monomorphic if and only if it is a coretraction. Dually, it is epimorphic if and only if it is a retraction.

Proof. Suppose f to be monomorphic in C. It suffices to show that f is monomorphic in Ĉ, for then f is a

coretraction since X is injective in Ĉ. Let K ^{\overline{}^{i}} X be the kernel of f in Ĉ. From (Kπ)if = 0, we conclude (Kπ)i = 0 since f is monomorphic in C, and thus K ≃ 0 since Kπ is epimorphic and i is monomorphic in Ĉ.

In particular, an abelian category is weakly abelian if and only if it is semisimple, i.e. if and only if every morphism in A splits. Hence the notion “weakly abelian” is slightly abusive.

Let E be a Frobenius category; cf. §A.2.3.

Lemma A.29 Suppose given a pure short exact sequence X′ [figure: i] X ^{\overline{}^{p}} X′′ in E. In \underline{E}, the residue class iN is a weak kernel of pN, and the residue class pN is a weak cokernel of iN.

Proof. By duality, it suffices to show that iN is a weak kernel of pN. So suppose given T ^{\overline{}^{t}} X in E with

tp ≡_{B} 0. We have to show that there exists a morphism T ^{\overline{}^{t′}} X′ such that t′i ≡_{B} t. Let (T ^{\overline{}^{t}} X ^{\overline{}^{p}} X′′) =

(T ^{\overline{}^{u}} B ^{\overline{}^{q}} X′′), where B is bijective. Let P ^{\overline{}^{p̃}} B be the pullback of p along q. We have a factorisation

(T ^{\overline{}^{t}} X) = (T ^{\overline{}^{v}} P ^{\overline{}^{w}} X). We have a factorisation (X′ [figure: i] X) = (X′ [figure: ĩ] P ^{\overline{}^{w}} X); moreover, (ĩ, p̃) is a pure short exact sequence, hence split by projectivity of B; cf. Lemma A.11 and §A.2.2. Let ĩr = 1. Then rĩ − 1 ≡_{B} 0, since it factors over B. We obtain (vr)i = vrĩw ≡_{B} vw = t.

Remark A.30 The stable category \underline{\underline{E}} and the classical stable category \underline{E} of the Frobenius category E are weakly abelian. The stable category \underline{\underline{E}} carries an automorphism T, induced by shifting an acyclic complex to the left by one position and negating the differentials.

Cf. Definition A.7.

Proof. By Lemma A.8, it remains to prove that \underline{E} is weakly abelian. Suppose given a morphism X ^{\overline{}^{f}} Y in E.

By duality, it suffices to show that the residue class of X ^{\overline{}^{f}} Y in \underline{E} is a weak cokernel and has a weak kernel. Substituting isomorphically in \underline{E} by adding a bijective object to X, we may assume f to be a pure epimorphism in E. So we may complete to a pure short exact sequence and apply Lemma A.29.

Lemma A.31 A pure short exact sequence X′ [figure: i] X ^{\overline{}^{p}} X′′ in E is mapped via the residue class functor N to a sequence in \underline{E} that is exact at X when considered in the Freyd category \underline{Ê} of \underline{E}. In particular, a pure square in E is mapped to a weak square in \underline{E}.

Proof. By Remark A.27, we may apply Lemma A.29.

A.6.4 Heller operators for diagrams

In Definition 1.5, the central role is attributed to the tuple ϑ = (ϑ_{n})_{n⩾0} of isomorphisms. In the case of C being the stable category of a Frobenius category, such an isomorphism ϑ_{n} arises from different choices of pure monomorphisms into bijective objects. To that end, we provide a comparison lemma, which suitably organises wellknown facts.

Let C be a category.

Given a category D and a full subcategory U ⊆ D(C), we say that U is characteristic in D(C) if the image of

U under A(C) is contained in U for any autoequivalence D \overline{}^{A}_{∼} D, and if U is closed under isomorphy in D(C), i.e. X ≃ X′ in D(C) and X′ ∈ Ob U implies X ∈ Ob U.

Let E be a Frobenius category. Denote by \underline{E} its classical stable category, and denote by E ^{\overline{}^{N}} \underline{E} the residue class functor. Let G ⊆ E(C) be a full additive subcategory. Let H ⊆ \underline{E}(C) be a full additive characteristic subcategory such that (G)(N(C)) ⊆ H.

\xymatrix{G \ar@{^{(}->}[r] \ar[d] & E(C) \ar[d]^{N(C)} \\ H \ar@{^{(}->}[r] & \underline{E}(C)}

A C-resolving system I consists of pure short exact sequences

I = ((X_{c} [figure: i_{X,c}] I_{X,c} ^{\overline{}^{p_{X,c}}} X̃_{c})_{c∈Ob C})_{X∈Ob G} ,

with bijective objects I_{X,c} in E as middle terms.

Lemma A.32 (1) Given a C-resolving system

I = ((X_{c} [figure: i_{X,c}] I_{X,c} ^{\overline{}^{p_{X,c}}} X̃_{c})_{c∈Ob C})_{X∈Ob G} ,

there exists a functor

G ^{\overline{}^{T^{I}}} H

that is uniquely characterised by the following properties. On objects X ∈ Ob G ⊆ Ob E(C), the image X T_{I} ∈ Ob H ⊆ Ob \underline{E}(C) is characterised as follows.

(∗) For any (c ^{\overline{}^{γ}} d) ∈ C, there exist

• a representative (X T_{I})_{γ}^{∼} in E of the evaluation (X T_{I})_{c}^{\overline{}^{(X T_{I})_{γ}}} (X T_{I})_{d} in \underline{E} at

c ^{\overline{}^{γ}} d of the diagram X T_{I} ∈ Ob H ⊆ Ob \underline{E}(C), and

• a morphism I_{X,c} \overline{}I_{X,d} in E such that

[figure: X_{c}; •; i_{X,c}; X_{γ}; I_{X,c}; p_{X,c}; (X T_{I})_{c}; (X T_{I})_{γ}^{∼}; X_{d}; •; i_{X,d}; I_{X,d}; p_{X,d}; (X T_{I})_{d}]

is a morphism of pure short exact sequences.

On morphisms (X ^{\overline{}^{f}} Y) ∈ G ⊆ E(C), the image (X T_{I} ^{\overline{}^{f T_{I}}} Y T_{I}) ∈ H ⊆ \underline{E}(C) is characterised as follows.

(∗∗) For any c ∈ Ob C, there exist

• a representative (f T_{I})_{c}^{∼} in E of the evaluation (X T_{I})_{c} ^{\overline{}^{(f T_{I})_{c}}} (Y T_{I})_{c} in \underline{E} at

c of the diagram morphism (X T_{I} ^{\overline{}^{f T_{I}}} Y T_{I}) ∈ H ⊆ \underline{E}(C), and

• a morphism I_{X,c} \overline{}I_{Y,c} in E such that

[figure: X_{c}; •; i_{X,c}; f_{c}; I_{X,c}; p_{X,c}; (X T_{I})_{c}; (f T_{I})_{c}^{∼}; Y_{c}; •; i_{Y,c}; I_{Y,c}; p_{Y,c}; (Y T_{I})_{c}]

is a morphism of pure short exact sequences.

(2) Given C-resolving systems

I = ((X_{c} [figure: i_{X,c}] I_{X,c} ^{\overline{}^{p_{X,c}}} X̃_{c})_{c∈Ob C})_{X∈Ob G} ,
I′ = ((X_{c} [figure: i′_{X,c}] I′_{X,c} ^{\overline{}^{p′_{X,c}}} X̃′_{c})_{c∈Ob C})_{X∈Ob G} ,

there exists an isomorphism

T_{I}  \overline{}^{α_{I,I′}}_{∼} T_{I′}

that is uniquely characterised by the following property.

(∗∗∗) For any X ∈ Ob G ⊆ Ob E(C) and for any c ∈ Ob C, there exist

• a representative (Xα_{I,I′})^{∼} in E of the evaluation (X T_{I})_{c}^{\overline{}^{(Xα_{I,I′})_{c}}} (X T_{I′})_{c} in

\underline{E} at c of the evaluation X T_{I} ^{\overline{}^{Xα_{I,I′}}} X T_{I′} in H ⊆ \underline{E}(C) of α_{I,I′} at X, and

• a morphism I_{X,c} \overline{}I′_{X,c} in E

such that

[figure: X_{c}; •; i_{X,c}; I_{X,c}; p_{X,c}; (X T_{I})_{c}; (Xα_{I,I′})^{∼}; X_{c}; •; i′_{X,c}; I′_{X,c}; p′_{X,c}; (X T_{I′})_{c}]

is a morphism of pure short exact sequences.

Proof. Let us first assume that H = \underline{E}(C). Having proven all assertions in this case, it then finally will remain to be shown that given H ⊆ \underline{E}(C) and a C-resolving system I, we have X T_{I} ∈ Ob H ⊆ Ob \underline{E}(C) for X ∈ Ob G.

We remark that starting from a morphism U ^{\overline{}^{u}} U′ in E and from chosen pure short exact sequences (U, B, V)

and (U′, B′, V′) with bijective middle terms B resp. B′, we may define a morphism V ^{\overline{}^{v}} V′ in \underline{E} by the existence of a morphism

[figure: U; •; i; u; B; p; V; v^{∼}; U′; •; i′; B′; p′; V′]

of pure short exact sequences in E, where V ^{\overline{}^{v}} V′ is the image in \underline{E} of the morphism V ^{\overline{}^{v^{∼}}} V′ in E.

Ad (1). Given X ∈ Ob G, we define X T_{I} ∈ \underline{E}(C) at the morphism c ^{\overline{}^{γ}} d of C by the diagram in (∗). The characterisation (∗) shows that X T_{I} is in fact in Ob \underline{E}(C).

Given a morphism X ^{\overline{}^{f}} Y in G, we define the morphism X T_{I} ^{\overline{}^{f T_{I}}} Y T_{I} in \underline{E}(C) at c ∈ Ob C by the diagram in (∗∗). Combining (∗) and (∗∗), we see that f T_{I} is in fact in \underline{E}(C). From (∗∗) we conclude that T_{I} is indeed a functor.

Ad (2). Given X ∈ Ob G, we define X T_{I} ^{\overline{}^{Xα_{I,I′}}} X T_{I′} at c ∈ Ob C by the diagram in (∗∗∗).

Combining (∗∗∗) and (∗), we see that X T_{I} ^{\overline{}^{Xα_{I,I′}}} X T_{I′} is indeed in \underline{E}(C). Combining (∗∗∗) and (∗∗), we see that α_{I,I′} is indeed a transformation.

Suppose given resolving systems I, I′ and I′′. The characterisation of α_{I,I′} etc. implies that α_{I,I′}α_{I′,I′′} = α_{I,I′′} and that α_{I,I} = 1_{T_{I}}. Hence in particular, α_{I,I′}α_{I′,I} = 1_{T_{I}} and α_{I′,I}α_{I,I′} = 1_{T_{I′}}, and so α_{I,I′} is an isomorphism from T_{I} to T_{I′}.

Consider the case C = Δ_{0}, i.e. the terminal category, let G = E(Δ_{0}) = E and let H = \underline{E}(Δ_{0}) = \underline{E}. For a

Δ_{0}-resolving system J, we obtain a functor E ^{\overline{}^{T^{J}}} \underline{E} that factors as

(E ^{\overline{}^{T^{J}}} \underline{E}) = (E ^{\overline{}^{N}} \underline{E} \overline{}^{T̄^{J}}_{∼} \underline{E}) .

In fact, for a morphism b that factors over a bijective object B, we can choose 0 as a representative of b T_{J}, inserting the pure short exact sequence (B, B, 0). Moreover, T̄_{J} is an equivalence, for it is full; faithful, using the dual of the argument just given; and dense, since given a morphism of short exact sequences in E with bijective middle terms and an identity on the kernels, the morphism on the cokernels is a homotopism.

Now return to the general case H ⊆ \underline{E}(C). Let J′ be a C-resolving system consisting of pure short exact sequences with bijective middle term that already occur in the chosen Δ_{0}-resolving system J. Then, for X ∈ Ob G, we have X T_{J′} = X(N(C))(T̄_{J}(C)). Since X(N(C)) ∈ Ob H by assumption, and since, moreover, H is assumed to be a characteristic subcategory of \underline{E}(C), we conclude that X(N(C))(T̄_{J}(C)) = X T_{J′} is in Ob H. Finally,

let I be an arbitrary C-resolving system. We have X T_{I} \overline{}^{α_{I,J′}}_{∼} X T_{J′} in \underline{E}(C), and thus X T_{J′} ∈ Ob H implies X T_{I} ∈ Ob H, since a characteristic subcategory of \underline{E}(C) is, by definition, closed under isomorphy.

A.7 Pointwise pullback and pushout

Suppose given an abelian category A, a poset E and an element ε ∈ E. Let E^{ε} := E ⊔ {ε′} be the poset defined by requiring that ε ⩽ ε′, that α ⩽̸ ε′ whenever α ⩽̸ ε and that ε′ ⩽̸ α for all α ∈ E; and the remaining relations within E ⊆ E^{ε} inherited from E. We define the pushout at ε

[figure: A(E^{ε}); (−)↑^{(=)}; A(E); X′; X↑^{x′} ,]

where X := X′|_{E}, and (X′_{ε} ^{\overline{}^{x′}} X′_{ε′}) = (X′_{ε} ^{\overline{}^{X′_{ε′/ε}}} X′_{ε′}); and a transformation

X′|_{E} = X ^{^{i = iX′} \overline{}}X↑^{x′} ,

natural in X′, by the following construction. Abbreviating X↑^{x′} by X̃, we let

[figure: X′_{ε′} = X̃_{ε}; X̃_{α/ε}; X̃_{α}; X_{ε}; X_{α/ε}; x′; X_{α}; i_{α}]

for α ∈ E with ε ⩽ α. If ε ⩽̸ α, we let X̃_{α} = X_{α} and i_{α} = 1_{X_{α}}.

Given α ⩽ β in E, we let

(X̃_{α} ^{\overline{}^{X̃_{β/α}}} X̃_{β})  be induced by pushout if ε ⩽ α ⩽ β ,
(X̃_{α} ^{\overline{}^{X̃_{β/α}}} X̃_{β}) := (X_{α} ^{\overline{}^{X_{β/α}}} X_{β} ^{\overline{}^{i_{β}}} X̃_{β}) if ε ⩽̸ α, but ε ⩽ β ,
(X̃_{α} ^{\overline{}^{X̃_{β/α}}} X̃_{β}) := (X_{α} ^{\overline{}^{X_{β/α}}} X_{β})  if ε ⩽̸ β .

The morphism X ^{\overline{}^{i}} X ↑^{x′} is the solution to the following universal problem. Suppose given a morphism

X ^{\overline{}^{f}} Y in A(E) such that at ε ∈ E we have a factorisation

(X_{ε} ^{\overline{}^{f_{ε}}} Y_{ε}) = (X_{ε} ^{\overline{}^{x′}} X′_{ε′} \overline{}Y_{ε}) .

Then there is a unique morphism X↑^{x′} ^{\overline{}^{g}} Y such that

(X ^{\overline{}^{f}} Y) = (X ^{\overline{}^{i}} X↑^{x′} ^{\overline{}^{g}} Y) .

Dually, let E_{ε} := E ⊔ {ε′} be the poset defined by requiring that ε ⩾ ε′, that α ⩾̸ ε′ whenever α ⩾̸ ε and that ε′ ⩾̸ α for all α ∈ E; and the remaining relations within E ⊆ E_{ε} inherited from E. We define the pullback at ε

[figure: A(E_{ε}); (−)↑_{(=)}; A(E); X′; X↑_{x′} ,]

where X := X′|_{E}, and (X′_{ε′} ^{\overline{}^{x′}} X′_{ε}) = (X′_{ε′} ^{\overline{}^{X′_{ε/ε′}}} X′_{ε}); and a transformation

X′|_{E} = X ^{\overline{}^{p = pX′}} X↑_{x′} ,

natural in X′, being the solution to the universal problem dual to the one described above.

A.8 1-epimorphic functors

Let C ^{\overline{}^{F}} D be a functor between categories C and D. Definition A.33 The functor C ^{\overline{}^{F}} D is 1-epimorphic if the induced functor “restriction along F ”

C, E ^{\overline{}^{F (−)}} D, E

is full and faithful for any category E. In particular, given functors D [figure: G; H] E with F G ≃ F H, we can conclude that G ≃ H; whence the notion of 1-epimorphy. Remark A.34 Suppose given a diagram of categories and functors

\xymatrix{C \ar[r]^{F} \ar[d]_{S}^{≀} & D \ar[d]^{T}_{≀} \\ C′ \ar[r]^{F′} & D′}

with equivalences S and T, and with F T ≃ SF′. Then F is 1-epimorphic if and only if F′ is 1-epimorphic. Let C, C′ ∈ Ob C. An F -epizigzag (resp. an F -monozigzag) C ⇝^{u} C′ is a finite sequence of morphisms

C = C_{0} ^{\overline{}^{u_{0}}} Z_{0} ^{\overline{}^{u′_{0}}} C_{1} ^{\overline{}^{u_{1}}} Z_{1} ^{\overline{}^{u′_{1}}} C_{1} ^{\overline{}^{u_{2}}} ⋯ ^{\overline{}^{u′_{k−2}}} C_{k−1} ^{\overline{}^{u_{k−1}}} Z_{k−1} ^{\overline{}^{u′_{k−1}}}C_{k} = C′ in C of length k ⩾ 0 such that u′_{i}F is an isomorphism for all i ∈ [0, k], and such that

uF := (u_{0}F)(u′_{0}F)^{−}(u_{1}F)(u′_{1}F)^{−} ⋯ (u_{k−1}F)(u′_{k−1}F)^{−} : CF \overline{}C′F

is a retraction (resp. a coretraction) in D. Lemma A.35 Suppose the functor

C ^{\overline{}^{F}} D

to be dense, and to satisfy the following condition (C).

(C)  \{ Given objects C, C′ ∈ Ob C and a morphism CF ^{\overline{}^{d}} C′F in D, there exists an F -epizigzag C_{s} ⇝^{c_{s}} C, an F -monozigzag C′ ⇝^{c′_{t}} C′_{t} and a morphism C_{s} ^{\overline{}^{c}} C′_{t} such that (C_{s}F ^{\overline{}^{c_{s}F}} CF ^{\overline{}^{d}} C′F ^{\overline{}^{c′_{t}F}} C′_{t}F) = (C_{s}F ^{\overline{}^{cF}} C′_{t}F) .

Then F is 1-epimorphic. Proof. Since F is dense, Remark A.34 allows to assume that F is surjective on objects, i.e. (Ob C)F = Ob D. Let us prove that E(C) ^{\overline{}^{F (−)}} E(D) is faithful. Suppose given functors C ^{\overline{}^{F}} D [figure: G; H] E and morphisms G ^{\overline{}^{γ}} H and G ^{\overline{}^{γ′}} H such that F γ = F γ′. Given D ∈ Ob D, we have to show that Dγ = Dγ′. Writing D = CF for some C ∈ Ob C, this follows from Dγ = CF γ = CF γ′ = Dγ′. Let us prove that E(C) ^{\overline{}^{F (−)}} E(D) is full. Suppose given functors C ^{\overline{}^{F}} D [figure: G; H] E and a morphism F G ^{\overline{}^{δ}} F H. Define G ^{\overline{}^{δ̂}} H by (CF)δ̂:= Cδ for C ∈ Ob C. We have to prove that Dδ̂ is a welldefined morphism for D ∈ Ob D. So suppose that D = CF = C′F. We have to show that Cδ = C′δ. By assumption (C), applied to d = 1_{D} = 1_{CF} = 1_{C′F}, there exist an F -epizigzag C_{s} ⇝^{c_{s}} C, an F -monozigzag C′ ⇝^{c′_{t}} C′_{t} and a morphism C_{s} ^{\overline{}^{c}} C′_{t} such that (c_{s}F)(c′_{t}F) = cF. We obtain

(c_{s}F G)(Cδ)(c′_{t}F H) = (C_{s}δ)(c_{s}F H)(c′_{t}F H)
= (C_{s}δ)(cF H)
= (cF G)(C′_{t}δ)
= (c_{s}F G)(c′_{t}F G)(C′_{t}δ)
= (c_{s}F G)(C′δ)(c′_{t}F H) ,

whence Cδ = C′δ by epimorphy of c_{s}F G and by monomorphy of c′_{t}F H.

We have to prove that δ̂ is natural. Suppose given CF ^{\overline{}^{d}} C′F in D for some C, C′ ∈ Ob C. We have to show that (dG)((C′F)δ̂) = ((CF)δ̂)(dH), i.e. that (dG)(C′δ) = (Cδ)(dH). By assumption (C), there exist an

F -epizigzag C_{s} ⇝^{c_{s}} C, an F -monozigzag C′ ⇝^{c′_{t}} C′_{t} and a morphism C_{s} ^{\overline{}^{c}} C′_{t} such that (c_{s}F)d(c′_{t}F) = cF. We obtain

(c_{s}F G)(dG)(C′δ)(c′_{t}F H) = (c_{s}F G)(dG)(c′_{t}F G)(C′_{t}δ)
= (cF G)(C′_{t}δ)
= (C_{s}δ)(cF H)
= (C_{s}δ)(c_{s}F H)(dH)(c′_{t}F H)
= (c_{s}F G)(Cδ)(dH)(c′_{t}F H) ,

whence (dG)(C′δ) = (Cδ)(dH) by epimorphy of c_{s}F G and by monomorphy of c′_{t}F H.

Corollary A.36 If C ^{\overline{}^{F}} D is a functor such that (i, ii) hold, then F is 1-epimorphic.

(i) For all morphisms D ^{\overline{}^{d}} D′ in D, there is a morphism C ^{\overline{}^{c}} C′ in C such that

(C ^{\overline{}^{c}} C′)F = (D ^{\overline{}^{d}} D′) .

(ii) For any C, C′ ∈ Ob C such that CF = C′F, there exists a finite sequence of morphisms

C = C_{0} ^{\overline{}^{u_{0}}} Z_{0} ^{\overline{}^{u′_{0}}} C_{1} ^{\overline{}^{u_{1}}} Z_{1} ^{\overline{}^{u′_{1}}} C_{2} ^{\overline{}^{u_{2}}} ⋯ ^{\overline{}^{u′_{k−2}}} C_{k−1} ^{\overline{}^{u_{k−1}}} Z_{k−1} ^{\overline{}^{u′_{k−1}}}C_{k} = C′

from C to C′ such that u_{i}F = u′_{i}F = 1_{CF} = 1_{C′F} for all i ∈ [0, k].

Proof. The functor F is dense, even surjective on objects, because identities have inverse images under F. To

fulfill condition (C) of Lemma A.35, given objects C, C′ ∈ Ob C and a morphism CF ^{\overline{}^{d}} C′F in D, we may

take some morphism C_{s} ^{\overline{}^{c}} C′_{t} in C such that (C_{s} ^{\overline{}^{c}} C′_{t})F = (CF ^{\overline{}^{d}} C′F), we may take for c_{s} a sequence as given by assumption because of C_{s}F = CF, and we may take for c_{t} a sequence as given by assumption because of C′_{t}F = C′F.

Corollary A.37 If C ^{\overline{}^{F}} D is a full and dense functor, then F is 1-epimorphic.

Proof. In fact, in condition (C) of Lemma A.35, we may take an F -monozigzag and an F -epizigzag of length 0.

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Matthias Künzer

Lehrstuhl D für Mathematik

RWTH Aachen Templergraben 64 D-52062 Aachen

kuenzer@math.rwth-aachen.de www.math.rwth-aachen.de/∼kuenzer