A construction principle for Frobenius categories

Manuscript

Matthias Künzer

June 15, 2007

Abstract

This note is a supplement to [3]. Let C be a weakly abelian category. Let n ≥ 0. Let C(Δ̇_{n}) be the category of diagrams of shape Δ̇_{n} = [1, n] with values in C. Let \underline{C(Δ̇_{n})} be its quotient modulo split such diagrams. We know by [3, Prop. 5.5.(1),

Prop. 2.6] that there is a Frobenius category C(Δ̄_{n}^{#}) whose classical stable category \underline{C(Δ̄_{n}^{#})} is equivalent to \underline{C(Δ̇_{n})}. In particular, \underline{C(Δ̇_{n})} is weakly abelian. We give a direct proof of this fact, exhibiting a structure of a Frobenius category on C(Δ̇_{n}) such that \underline{C(Δ̇_{n})} is its classical stable category.

Contents

0 Introduction  1
0.1 A construction principle for Frobenius categories . . . . . . . . . . . . . . . . . . . . . . . 1
0.2 Application to C(Δ̇_{n}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.3 Notation and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1 Construction of exact categories  3
2 The construction principle  5
3 Application to C(Δ̇_{n})  6

0 Introduction

0.1 A construction principle for Frobenius categories

Given an exact category E and a full subcategory N ⊆ E, we ask for a modification of the exact structure on E in such a way that the result is a Frobenius category with N as a sufficiently big subcategory of bijective objects.

Declaring a pure short exact sequence in E to be N-pure if each object of N is bijective with respect to it, we verify that E, equipped with the set of N-pure short exact sequences, actually is an exact category. For it to be Frobenius, N only has to be big enough; see Remark 4.

MSC2000: 18E30.

0.2 Application to C(Δ̇_{n})

Let C be a weakly abelian category; cf. e.g. [3, Def. A.26]. Let n ≥ 0.

Let C(Δ̇_{n}) be the category of diagrams of shape Δ̇_{n} = [1, n] with values in C. Let \underline{C(Δ̇_{n})} be its quotient modulo split such diagrams.

For the definition of the poset Δ̄_{n}^{#}, see [3, §1.1]. For the definition of the category C^{+}(Δ̄_{n}^{#}), see [3, §1.2.1.1]. Roughly, it is the category of diagrams on Δ̄_{n}^{#} that have zeroes on the boundaries and weak squares wherever possible. The category C^{+}(Δ̄_{n}^{#}) is Frobenius by [3, Prop. 5.5.(1)].

Its classical stable category \underline{C^{+}(Δ̄_{n}^{#})} is equivalent to \underline{C(Δ̇_{n})} by [3, Prop. 2.6]. In particular, since \underline{C^{+}(Δ̄_{n}^{#})} is weakly abelian, so is \underline{C(Δ̇_{n})}. Cf. also [1, Prop. 8.4].

We find a structure of an exact category on C(Δ̇_{n}) such that it is a Frobenius category with \underline{C(Δ̇_{n})} as its classical stable category. This reproves the fact that \underline{C(Δ̇_{n})} is weakly abelian.

Whereas the category C(Δ̇_{n}) looks smaller and simpler than C^{+}(Δ̄_{n}^{#}), it behaves worse. Firstly, while \underline{C^{+}(Δ̄_{n}^{#})} carries a shift functor by diagram shift, the category \underline{C(Δ̇_{n})} does not allow such a diagram shift, and can only artificially be given a shift functor via the equivalence \underline{C(Δ̇_{n})} ≃ \underline{C^{+}(Δ̄_{n}^{#})}. Therefore, in the definition of a Heller triangulated category [3, Def. 1.5.(i)], we rather use \underline{C^{+}(Δ̄_{n}^{#})}. Secondly, and of relevance here, the exact structure on C^{+}(Δ̄_{n}^{#}) is the obvious one that declares pointwise split short exact sequences to be pure. The exact structure on C(Δ̇_{n}) has to be constructed; see Proposition 6 below.

0.3 Notation and conventions

(i) Given elements x, y of some set X, we let ∂_{x,y} = 1 in case x = y and ∂_{x,y} = 0 in case x ≠ y.

(ii) For an assertion X, which might be true or not, we let {X} equal 1 if X is true, and equal 0 if X is false. So for instance, {x = y} = ∂_{x,y}.

(iii) For a, b ∈ Z, we denote by [a, b] := {z ∈ Z : a ≤ z ≤ b} the integral interval.

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

(v) 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}} .

(vi) 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.

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

(viii) 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.

(ix) Pure monomorphy in an exact category is indicated by X [figure] Y, pure epimorphy by X \overline{}Y. Concerning exact categories in the sense of Quillen, cf. [3, §A.2].

(x) 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] .

(xi) A sequence X′ \overline{}X \overline{}X′′ in an additive category A is split short exact if it is isomorphic, in

A(Δ_{2}), to the sequence X′ ^{\overline{}^{(^{1 0} )}} X′ ⊕ X′′ ^{\overline{}^{( \begin{matrix}0 \\ 1 \end{matrix} )}} X′′.

(xii) For the definition of a weakly abelian category, see e.g. [3, Def. A.26]; cf. [2, §3, l. 1–2], [1, Def. 8.6].

(xiii) Given a weakly abelian category C and n ≥ 1, the category \underline{C(Δ̇_{n})} is defined as C(Δ̇_{n}) modulo the subcategory of split diagrams; cf. [3, §2.4].

(xiv) Concerning the Freyd category Ĉ of a weakly abelian category C, we refer to [3, §A.6.3]. The Freyd category Ĉ is an abelian Frobenius category that contains C as a sufficiently big subcategory of bijectives.

1 Construction of exact categories

Remark 1 If (E, S_{i}) are exact categories for i in some index set I, where S_{i} denotes the respective set of pure short exact sequences, then also (E, ⋂_{i∈I} S_{i}) is an exact category.

A sequence X′ \overline{}X \overline{}X′′ in an exact category (E, S) is called left exact if X′ \overline{}X is purely monomorphic and a kernel of X \overline{}X′′.

A sequence X′ \overline{}X \overline{}X′′ in an exact category (E, S) is called right exact if X \overline{}X′′ is purely epimorphic and a cokernel of X′ \overline{}X.

Let (E, S) and (E′, S′) be exact categories, and let E ^{\overline{}^{F}} E′ be an additive functor. Let S_{F} denote the set of short exact sequences in S whose image under F, applied pointwise, is in S′.

The short exact sequences in S will also be called S-pure; etc. The short exact sequences in S_{F} will also be called S_{F} -pure; etc. We will continue to denote an S-pure monomorphism in E by [figure] , and an S-pure epimorphism by \overline{}.

The functor F is called left exact if for any pure short exact sequence (X, Y, Z) in E, the sequence (XF, Y F, ZF) is left exact.

The functor F is called right exact if for any pure short exact sequence (X, Y, Z) in E, the sequence (XF, Y F, ZF) is right exact.

Lemma 2

(1) If E ^{\overline{}^{F}} E′ is left exact, then (E, S_{F}) is an exact category.

(2) If E ^{\overline{}^{F}} E′ is right exact, then (E, S_{F}) is an exact category.

Proof. Ad (1). Consider a left exact functor E ^{\overline{}^{F}} E′.

We use the axioms from [3, §A.2.1]. The axiom (Ex 2) is redundant; cf. [4].

Verification of (Ex 3). Suppose given a commutative triangle

[figure: M; p; X; •; ip; i; Y]

in E in which, moreover, ip is S_{F} -purely monomorphic and p is S_{F} -purely epimorphic.

By exactness of (E, S), we can complete it to a diagram

[figure: X′; •; i′; •; i′p′; Y′; s′; M; p; p′; Z ,; X; •; i; •; ip; Y; s]

in E with S-pure short exact sequences (X, M, Y′), (X′, M, Y), (X, Y, Z) and (X′, Y′, Z). Moreover, (X′, M, Y) and (X, Y, Z) are S_{F} -purely short exact, i.e. (X′F, MF, Y F) and (XF, Y F, ZF) are pure short exact sequences in E′. Hence, application of the left exact functor F yields a diagram

[figure: X′F; •; i′F; •; (i′p′)F; Y′F; s′F; MF; pF; p′F; ZF; XF; •; iF; •; (ip)F; Y F; sF]

in E′ with (XF, MF, Y′F) and (X′F, Y′F, ZF) left exact. By composition, s′F is purely epimorphic, and hence (X′F, Y′F, ZF) is a pure short exact sequence. The quadrangle (MF, Y F, Y′F, ZF) is a pure square, for on the kernels, we have the identity on X′F as induced morphism, and the cokernels are zero; cf. [3, §A.4; §A.2.2; Lem. A.11]. In particular, it is a pullback, and so p′F is purely epimorphic. We conclude that (XF, MF, Y′F) is a pure short exact sequence.

Verification of (Ex 3^{∘}). Suppose given a commutative triangle

[figure: M; p; X; ip; •; i; Y]

in which, moreover, ip is S_{F} -purely epimorphic and i is S_{F} -purely monomorphic.

By exactness of (E, S), we can complete it to a diagram

[figure: X′; •; i′; i′p′; Y′; W; •; r′; •; r; M; p; p′; X; •; i; ip; Y]

in E with pure short exact sequences (X, M, Y′), (X′, M, Y), (W, X, Y) and (W, X′, Y′). Moreover, (X, M, Y′) and (W, X, Y) are S_{F} -purely short exact, i.e. (XF, MF, Y′F) and (W F, XF, Y F) are pure short exact sequences in E′. Hence, application of F yields a diagram

[figure: X′F; •; i′F; (i′p′)F; Y′F; W F; •; r′F; •; rF; MF; pF; p′F; XF; •; iF; (ip)F; Y F]

in E′ with (W F, X′F, Y′F) and (X′F, MF, Y F) left exact. By composition, the morphism pF is purely epimorphic, and thus (X′F, MF, Y F) is a pure short exact sequence. By (Ex 3^{∘}) in E′, the morphism (i′p′)F is purely epimorphic, and thus (W F, X′F, Y′F) is a pure short exact sequence.

Remark 3 A possible source of mistakes. Given an S-pure monomorphism X [figure] Y in E such that its image F X [figure] F Y is purely monomorphic, we cannot conclude that X [figure] Y is S_{F} -purely monomorphic. In fact, the image of every S-pure monomorphism under F is purely monomorphic.

2 The construction principle

Let (E, S) be an exact category, where S denotes the set of pure short exact sequences, and let N ⊆ E be a full additive subcategory.

Consider the following set of pure short exact sequences.

S_{N} := ( ⋂_{N∈Ob N} S_{_{E}(N,−)}) ∩ ( ⋂_{N∈Ob N} S_{_{E}(−,N)}) .

Then (E, S_{N}) is an exact category by Lemma 2 and Remark 1. The short exact sequences in S_{N} are called N-pure short exact sequences. The pure monomorphisms in this exact category are called N-pure monomorphisms, and the pure epimorphisms therein are called N-pure epimorphisms.

By construction, the subcategory N ⊆ E consists of bijective objects in (E, S_{N}); that is, each N ∈ Ob N is bijective with respect to the N-pure short exact sequences.

Written out, an N-pure short exact sequence in E is a pure short exact sequence X′ [figure] X \overline{}X′′ such that for any N ∈ Ob N and any morphism N \overline{}X′′, there exists a factorisation (N \overline{}X′′) = (N \overline{}X \overline{}X′′); and, dually, such that for any N ∈ Ob N and any morphism X′ \overline{}N, there exists a factorisation (X′ \overline{}N) = (X′ [figure] X \overline{}N).

An N-pure short exact sequence X′ [figure] N \overline{}X′′ in E is called N-resolving if N ∈ Ob N.

Remark 4 The category (E, S_{N}), i.e. the given exact category E together with the set of N-pure short exact sequences S_{N}, is a Frobenius category if the following conditions (1) and (2) are fulfilled. In this case, N is a sufficiently big subcategory of bijectives.

(1) For all X′′ ∈ Ob E, there exists a N-resolving pure short exact sequence with cokernel term X′′.

(2) For all X′ ∈ Ob E, there exists a N-resolving pure short exact sequence with kernel term X′.

3 Application to C(Δ̇_{n})

Suppose given n ≥ 1. Recall that Δ̇_{n} = Δ_{n} ∖ {0} = [1, n].

Let C be a weakly abelian category. We shall consider the category C(Δ̇_{n}). For ease of notation, we formally put X_{n+1} := 0 for X ∈ Ob C(Δ̇_{n}).

A sequence X′ ^{\overline{}^{i}} X ^{\overline{}^{p}} X′′ in C(Δ̇_{n}) is called pointwise split short exact, if the sequence X′_{k} ^{\overline{}^{i_{k}}} X_{k} ^{\overline{}^{p_{k}}} X′′_{k} is split short exact for all k ∈ [1, n]. The kernel in a pointwise split short exact sequence is pointwise split monomorphic, the cokernel pointwise split epimorphic. The additive category C(Δ̇_{n}), equipped with the set of pointwise split short exact sequences as pure short exact sequences, is an exact category; cf. e.g. [3, Ex. A.3, Ex. A.4].

Consider the full subcategory C^{split}(Δ̇_{n}) ⊆ C(Δ̇_{n}) whose objects are diagrams X ∈ Ob C(Δ̇_{n}) such that X_{k} ^{\overline{}^{x}} X_{l} is split for all k, l ∈ [1, n] with k ≤ l.

Let S denote the set of pointwise split short exact sequences in C(Δ̇_{n}).

Lemma 5 Suppose given a pointwise split short exact sequence X′ ^{\overline{}^{f}} X ^{\overline{}^{g}} X′′ in C(Δ̇_{n}) such that, for all l, m ∈ [1, n] with l ≤ m, the quadrangle (X_{l}, X_{m}, X′′_{l}, X′′_{m}) has the following property (∗).

(∗) The morphism induced from the kernel of X_{l} ^{\overline{}^{x}} X_{m} in Ĉ to the kernel of X′′_{l} ^{\overline{}^{x}} X′′_{m} in Ĉ is epimorphic.

Suppose given Z ∈ Ob C^{split}(Δ̇_{n}). Then application of the functors (Z, −) = _{C(Δ̇_{n})}(Z, −) and (−, Z) = _{C(Δ̇_{n})}(−, Z) yields short exact sequences

[figure: (Z, X′); (Z,f); (Z, X); (Z,g); (Z, X′′); (X′, Z); (f,Z); (X, Z); (g,Z); (X′′, Z)]

of abelian groups. In other words, the sequence X′ ^{\overline{}^{f}} X ^{\overline{}^{g}} X′′ is C^{split}(Δ̇_{n})-purely short exact; still in other words, it is contained in S_{C^{split}(Δ̇_{n})}.

Proof. We claim that (Z, X) ^{\overline{}^{(Z,g)}} (Z, X′′) is surjective. By Lemma [3, A.25], applied to the abelian Frobenius category Ĉ, we may assume that Z is an interval, say Z = C_{[l,m]} with C ∈ Ob C and l, m ∈ [1, n] with l ≤ m; cf. [3, §A.6.2].

A morphism C_{[l,m]} \overline{}X′′ is determined by a morphism C ^{\overline{}^{t}} X′′_{l} such that the composite (C ^{\overline{}^{t}} X′′_{l} ^{\overline{}^{x′′}} X′′_{m+1}) vanishes. To prove the asserted surjectivity, we have to find a

morphism C ^{\overline{}^{t′}} X_{l} such that the composite (C ^{\overline{}^{t′}} X_{l} ^{\overline{}^{x}} X_{m+1}) vanishes and such that (C ^{\overline{}^{t′}} X_{l} ^{\overline{}^{g_{l}}} X′′_{l}) = (C ^{\overline{}^{t}} X′′_{l}).

To do so, we may assume that m < n. Let K_{x} ^{\overline{}^{s}} X_{l} denote the kernel of X_{l} ^{\overline{}^{x}} X_{m+1} in Ĉ, and let K_{x′′} ^{\overline{}^{s′′}} X′′_{l} denote the kernel of X′′_{l} ^{\overline{}^{x′′}} X′′_{m+1} in Ĉ. By (∗), we obtain an induced epimorphism K_{x} ^{\overline{}^{g̃}} K_{x′′}, characterized by g̃s′′ = sg_{l}. We factor (C ^{\overline{}^{t}} X′′_{l}) = (C ^{\overline{}^{t_{1}}} K_{x′′} ^{\overline{}^{s′′}} X′′_{l}) by the universal property of s′′. Then we factor (C ^{\overline{}^{t_{1}}} K_{x′′}) = (C ^{\overline{}^{t_{2}}} K_{x} ^{\overline{}^{g̃}} Kx̃) by epimorphy of g̃ and by bijectivity of C in Ĉ. We may use t′ := t_{2}s. This proves the claim.

We claim that (X′, Z) ^{\overline{}^{(f,Z)}} (X, Z) is surjective. By duality, it suffices to show that, given l, m ∈ [1, n] with l ≤ m, the morphism induced from the cokernel of X′_{l} \overline{}X′_{m} in Ĉ to the cokernel of X_{l} \overline{}X_{m} in Ĉ, is monomorphic. This in turn follows by an application of the snake lemma in Ĉ to the morphism (X′_{l}, X_{l}, X′′_{l}) ^{\overline{}^{(x′, x, x′′)}} (X′_{m}, X_{m}, X′′_{m}) of short exact sequences.

[figure: •; •; X′_{l}; •; f_{l}; x′; X_{l}; g_{l}; x; X′′_{l}; x′′; X′_{m}; •; f_{m}; X_{m}; g_{m}; X′′_{m}; •; 0]

This proves the second claim.

Proposition 6 The category C(Δ̇_{n}), equipped with the set S_{C^{split}(Δ̇_{n})} of C^{split}(Δ̇_{n})-pure short exact sequences, is a Frobenius category with C^{split}(Δ̇_{n}) as a sufficiently big subcategory of bijectives.

Proof. By Remark 4 and by duality, it suffices to show that for each object X ∈ Ob C(Δ̇_{n}) there exists a C^{split}(Δ̇_{n})-resolving pure exact sequence with cokernel term X.

Write K_{i,n+1} := X_{i} for i ∈ [1, n]. For the notion of a weak square, we refer to [3, Def. A.9].

Choose a diagram

[figure: K_{1,2}; k; +; 0; K_{1,3}; k; k; +; K_{2,3}; k; +; 0; K_{1,4}; k; k; +; K_{2,4}; k; k; +; K_{3,4}; k; +; 0; ⋮; k; +; ⋮; k; +; ⋮; k; +; ⋮; k; +; ⋯; +; ⋮; k; K_{1,n}; k; k; +; K_{2,n}; k; k; +; K_{3,n}; k; k; +; K_{4,n}; k; k; +; ⋯; k; +; K_{n−1,n}; k; +; 0; K_{1,n+1}; k; K_{2,n+1}; k; K_{3,n+1}; k; K_{4,n+1}; k; ⋯; k; K_{n−1,n+1}; k; K_{n,n+1}; 0]

in C, and where (K_{l,n+1} ^{\overline{}^{k}} K_{m,n+1}) = (X_{l} ^{\overline{}^{x}} X_{m}) for l, m ∈ [1, n] with l ≤ m. This is possible since [1, n] is linearly ordered, proceeding from right to left and from the bottom to the top.

We write also K_{i,j} ^{\overline{}^{k}} K_{i′,j′} whenever i, j, i′, j′ ∈ [1, n + 1] with i < j, with i′ < j′, with i ≤ i′ and j ≤ j′. In particular, for i = i′ and j = j′, the morphism K_{i,j} ^{\overline{}^{k}} K_{i,j} is an identity. Note that K_{i,j} ^{\overline{}^{k}} K_{i′,j′} is zero unless i′ < j.

The morphism (K_{i,j} ^{\overline{}^{k}} K_{i,n+1}) is a weak kernel of (K_{i,n+1} ^{\overline{}^{k}} K_{j,n+1}) = (X_{i} ^{\overline{}^{x}} X_{j}) for i, j ∈ [1, n] with i ≤ j; cf. [3, Lem. A.14, Rem. A.27].

We shall define an object P ∈ Ob C^{split}(Δ̇_{n}). Given l ∈ [1, n], we let

P_{l} := ⨁_{i∈[1,l]} ⨁_{j∈[l+1,n+1]} K_{i,j} .

Given l, m ∈ [1, n] with l < m, we let the morphism P_{l} ^{\overline{}^{p}} P_{m} be defined by the matrix p = (p_{(i,j),(i′,j′)})_{(i,j),(i′,j′)}, where

p_{(i,j),(i′,j′)} := ∂_{j,j′}(∂_{i,i′} + k∂_{i,l}{i′ ∈ [l + 1, m]}) .

First, let us verify that (P_{l} ^{\overline{}^{p}} P_{m} ^{\overline{}^{p}} P_{r}) = (P_{l} ^{\overline{}^{p}} P_{r}) for l < m < r in [1, n]. In fact, at i ∈ [1, l], j ∈ [l + 1, n + 1], i′′ ∈ [1, r], j′′ ∈ [r + 1, n + 1], we obtain

∑_{i′∈[1,m]} ∑_{j′∈[m+1,n+1]} ∂_{j,j′} (∂_{i,i′} + k∂_{i,l}{i′ ∈ [l + 1, m]}) ∂_{j′,j′′} (∂_{i′,i′′} + k∂_{i′,m}{i′′ ∈ [m + 1, r]})

= ∂_{j,j′′}(∂_{i,i′′} + k∂_{i,m}{i′′ ∈ [m + 1, r]} + k∂_{i,l}{i′′ ∈ [l + 1, m]} + k∂_{i,l}{i′′ ∈ [m + 1, r]})
= ∂_{j,j′′}(∂_{i,i′′} + k∂_{i,l}{i′′ ∈ [l + 1, r]}) .

Given l, m ∈ [1, n] with l < m, we let P_{l,m} := ⨁_{i∈[1,l]} ⨁_{j∈[m+1,n+1]} K_{i,j}. The projection P_{l} \overline{}P_{l,m} is split epimorphic. The morphism P_{l,m} \overline{}P_{m} given by the matrix p|_{P_{l,m}} = (p_{(i,j),(i′,j′)})_{(i,j),(i′,j′)} is split monomorphic, for it has the projection P_{m} \overline{}P_{l,m} as a retraction. Now since our morphism factors as (P_{l} ^{\overline{}^{p}} P_{m}) = (P_{l} \overline{}P_{l,m} \overline{}P_{m}), it is split. We conclude that P ∈ Ob C^{split}(Δ̇_{n}).

Given l ∈ [1, n], we let P_{l} ^{\overline{}^{π}} K_{l,n+1} = X_{l} be the morphism given by the column vector π = (π_{(i,j)})_{(i,j)} with

π_{(i,j)} = ∂_{i,l}k .

So P_{l} ^{\overline{}^{π}} K_{l,n+1} = X_{l} is split epimorphic, for it has the inclusion of K_{l,n+1} into P_{l} as a coretraction.

We claim that these morphisms furnish a pointwise split epimorphism P ^{\overline{}^{π}} X. Suppose given l, m ∈ [1, n] with l < m. We have to show that

(P_{l} ^{\overline{}^{π}} K_{l,n+1} ^{\overline{}^{k}} K_{m,n+1}) =^{!} (P_{l} ^{\overline{}^{p}} P_{m} ^{\overline{}^{π}} K_{m,n+1}) .

Suppose given i ∈ [1, l] and j ∈ [l+1, n]. At position (i, j), the right hand side composition has the entry

∑_{i′∈[1,m]} ∑_{j′∈[m+1,n+1]} ∂_{j,j′} (∂_{i,i′} + k∂_{i,l}{i′ ∈ [l + 1, m]}) ∂_{i′,m}k
= {j ∈ [m + 1, n + 1]}∂_{i,l}k
= π_{(i,j)}k ,

being the entry and so does the left hand side composition. We conclude that πk = pπ.

We claim that P ^{\overline{}^{π}} X is C^{split}(Δ̇_{n})-purely epimorphic. By Lemma 5, it suffices to show that for l, m ∈ [1, n] with l < m, for the quadrangle (P_{l}, P_{m}, X_{l}, X_{m}), the induced morphism from the kernel of P_{l} ^{\overline{}^{p}} P_{m} in Ĉ to the kernel of X_{l} ^{\overline{}^{x}} X_{m} in Ĉ is epimorphic. Since by [3, Rem. A.27], the induced map from the weak kernel K_{l,m} to the kernel of X_{l} ^{\overline{}^{x}} X_{m} is epimorphic, it suffices to find an epimorphic induced morphism from the kernel of P_{l} ^{\overline{}^{p}} P_{m} to K_{l,m}.

The kernel of P_{l} ^{\overline{}^{p}} P_{m} is given by ⨁_{i∈[1,l]} ⨁_{j∈[l+1,m]} K_{i,j}, together with the inclusion into P_{l}.

As induced morphism ⨁_{i∈[1,l]} ⨁_{j∈[l+1,m]} K_{i,j} \overline{}K_{l,m}, we take the column vector (∂_{i,l}k)_{(i,j)}.

This induced morphism is split epimorphic, for it has the inclusion of K_{l,m} into that kernel as a coretraction. This proves the claim on P ^{\overline{}^{π}} X.

Example 7 We display the matrix of the morphism P_{3} ^{\overline{}^{p}} P_{5} in the case n = 7 (in the notation of the proof of Proposition 6). We have

\begin{aligned}P_{3} & = & (K_{1,4}⊕K_{1,5}⊕K_{1,6}⊕K_{1,7}⊕K_{1,8})⊕(K_{2,4}⊕K_{2,5}⊕K_{2,6}⊕K_{2,7}⊕K_{2,8})⊕(K_{3,4}⊕K_{3,5}⊕K_{3,6}⊕K_{3,7}⊕K_{3,8}) \\ P_{5} & = & (K_{1,6}⊕K_{1,7}⊕K_{1,8})⊕(K_{2,6}⊕K_{2,7}⊕K_{2,8})⊕(K_{3,6}⊕K_{3,7}⊕K_{3,8})⊕(K_{4,6}⊕K_{4,7}⊕K_{4,8})⊕(K_{5,6}⊕K_{5,7}⊕K_{5,8}) , \end{aligned}

and the morphism P_{3} ^{\overline{}^{p}} P_{5} is given by the matrix

(\begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & k & 0 & 0 & k & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & k & 0 & 0 & k & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & k & 0 & 0 & k \end{matrix} ) .

References

[1] Beligiannis, A., On the Freyd categories of an additive category, Homol. Homot. Appl., 2 (11), p. 147-185, 2000.

[2] Freyd, P., Stable Homotopy, Proc. Conf. Categorical Algebra, La Jolla, p. 121–172, Springer, 1965.

[3] Künzer, M., Heller triangulated categories, preprint, math.CT/0508565, 2005.

[4] Künzer, M., A remark on the axioms for exact categories: (Ex 2) is redundant, www.math.rwth-aachen.de/∼kuenzer/manuscripts.html.

Matthias Künzer

Lehrstuhl D für Mathematik

RWTH Aachen Templergraben 64 D-52062 Aachen

kuenzer@math.rwth-aachen.de www.math.rwth-aachen.de/∼kuenzer