1 What is a Heller triangulated category?

A Heller triangulated category is a triple C = (C, T, ϑ), where C is a weakly abelian category, T a shift functor and ϑ a tuple of isotransformations.

1.1 What is a weakly abelian category?

A weakly abelian category is an additive category, with split idempotents to simplify, in which each morphism is and has a weak kernel and a weak cokernel. It is the category of bijective objects of an abelian category, viz. its Freyd category.

1.2 What is theta?

1.2.1 It lives on n-pretriangles…

A 2-pretriangle is a diagram of the form

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

All quadrangles marked + are weak squares, i.e. their respective diagonal sequence is exact in the middle.

In other words, a 2-pretriangle is an acyclic complex, indexed in a convenient manner.

Modulo split 2-pretriangles, we obtain the homotopy category of acyclic complexes. Analogously n-pretriangles and split n-pretriangles.

1.2.2 … as follows

Given an n-pretriangle X. We can apply T pointwise to obtain the n-pretriangle [X^{+1}] :

[X^{+1}]_{β/α} = X_{β/α} T .

This operation is called the inner shift. We can apply a diagram shift to obtain the n-pretriangle [X]^{+1} :

([X]^{+1})_{α/β} = X_{β^{+1}/α} .

This operation is called the outer shift.

Now, ϑ = ( [figure: \frac{n-pretriangles}{split n-pretriangles}; [−]^{+1}; [−^{+1}]; \frac{n-pretriangles}{split n-pretriangles}; ϑ_{n}; ≀] )_{n ⩾ 0}.

1.3 Any axioms? The tuple ϑ should be compatible with

• generalised simplicial operations, and with

• folding.

1.3.1 What is folding?

Roughly the following. Given a (2n + 1)-pretriangle X. Consider a sequence of morphisms lying diagonally in X. Embed this sequence canonically into an (n + 1)-pretriangle X𝔣_{n}, made out of sums of entries of X. The operation 𝔣_{n} is called folding. More details in §A.

1.4 Where are the distinguished triangles of Verdier?

An n-pretriangle X is an n-triangle if Xϑ_{n} = id. Then a 2-triangle is a distinguished triangle in the sense of Verdier. Every 3-triangle is a Verdier octahedron, but not conversely. Now, n-triangles are stable under generalised simplicial operations and under folding.

2 What is an exact functor?

Let F : C \overline{}C′ be an additive functor between Heller triangulated categories that respects weak kernels and weak cokernels, and for which T F = F T′ holds in this strict manner, to simplify. We call F strictly exact if for an n-pretriangle X we have Xϑ_{n}F = XF ϑ′_{n}, where F is to be read as applied pointwise.

3 Why Heller triangulated categories?

As S. Thomas has recently shown, one can start with “n-triangles plus axioms” and recover ϑ. So why not work with the conventional approach “n-triangles plus axioms”?

To have n-triangles at one’s disposal is surely useful.

Having to check compatibility with n-triangles can be clumsy, though.

So ϑ simplifies.

3.1 Where does theta simplify something?

• Suppose given a strictly exact functor F. Suppose G is right adjoint to F in a shift-compatible manner. Then G is strictly exact. Proof of compatibility with ϑ, modulo introduction of the obvious notation :

[figure: [X′]^{+1}G; X′ϑ′_{n}G; [X^{′+1}]G; [X′GF]^{+1}G; X′GF ϑ′_{n}G; pointwise; ηG; ⟲ : ϑ′_{n} natural; [(X′GF)^{+1}]G; pointwise; ηG; [X′G]^{+1}F G; X′Gϑ_{n}F G; ⟲ : assumption on F; [(X′G)^{+1}]F G; [X′G]^{+1}; X′Gϑ_{n}; pointwise; Gε; ⟲ : ε natural; id; [(X′G)^{+1}]; pointwise; Gε; id]

• Dropping our assumption that idempotents split in C, one can use ϑ for a simple proof that the Karoubi hull of a Heller triangulated category is Heller triangulated.

• Not only linear bases are allowed to build n-triangles on and to prolong morphisms, but also zigzag bases.

4 Is there a connection to derivators?

4.1 From triangulated derivators to n-triangles

As G. Maltsiniotis has shown, the base category of a triangulated derivator has

n-triangles [people.math.jussieu.fr/~maltsin/ps/triansup.ps].

Folding remains to be discussed. First step : does [BBD, Astérisque 100, 1.1.13] hold in such a base category?

4.2 Difference

There is a fundamental difference between triangulated derivators and Heller triangulated categories, though :

A “morphism of derivators” is a compatible family of functors.

A “morphism of Heller triangulated categories” is a single additive functor that respects shift and ϑ.

A Heller triangulation is a step into the direction of a wished-for “maximal exactness structure” on C. Its purpose is to help to clarify the question how much information the single category C can carry.

Imagine that recently we would have discovered the passage from Z to Z/pZ. Now I want to know all properties of Z/pZ. Whether \underleftarrow{lim}_{n} Z/p^{n}Z is a complete discrete valuation ring is a related, useful, but different question.

A More on the folding operation

Let X be a 5-pretriangle : [figure: 0; X_{5/4}; x; +; X_{0^{+1}/4}; x; +; X_{1^{+1}/4}; x; +; X_{2^{+1}/4}; x; +; X_{3^{+1}/4}; +; 0; X_{4/3}; x; +; X_{5/3}; x; +; X_{0^{+1}/3}; x; +; X_{1^{+1}/3}; x; +; X_{2^{+1}/3}; +; 0; 0; X_{3/2}; x; +; X_{4/2}; x; x; +; X_{5/2}; x; x; +; X_{0^{+1}/2}; x; x; +; X_{1^{+1}/2}; x; +; 0; 0; X_{2/1}; x; +; X_{3/1}; x; x; +; X_{4/1}; x; x; +; X_{5/1}; x; x; +; X_{0^{+1}/1}; x; +; 0; 0; X_{1/0}; x; +; X_{2/0}; x; x; +; X_{3/0}; x; x; +; X_{4/0}; x; x; +; X_{5/0}; x; +; 0; 0; X_{0/5^{−1}}; x; +; X_{1/5^{−1}}; x; x; +; X_{2/5^{−1}}; x; x; +; X_{3/5^{−1}}; x; x; +; X_{4/5^{−1}}; x; +; 0; +; +; +; +; +]

Folding turns X into the 3-pretriangle X𝔣_{2} :

[figure: 0; 0; X_{2^{+1}/5}; +; 0; X_{5/4}⊕X_{1^{+1}/2}; _{(\begin{matrix}x \\ −x \end{matrix})}; +; X_{1^{+1}/4}; x; +; 0; X_{4/3}⊕X_{0^{+1}/1}; _{(\begin{matrix}x & 0 \\ 0 & x \end{matrix})}; +; X_{5/3}⊕X_{0^{+1}/2}; _{(\begin{matrix}x \\ −x \end{matrix})}; _{(\begin{matrix}x & 0 \\ 0 & x \end{matrix})}; +; X_{0^{+1}/3}; x; +; 0; X_{3/0}; x; +; X_{4/1}; x; _{(_{x x})}; +; X_{5/2}; _{(_{x x})}; +; 0; 0; X_{2/5^{−1}}; _{(_{x x})}; +; X_{2/0}⊕X_{3/5^{−1}}; _{(\begin{matrix}x & 0 \\ 0 & x \end{matrix})}; _{(\begin{matrix}x \\ −x \end{matrix})}; +; X_{2/1}⊕X_{4/5^{−1}}; _{(\begin{matrix}x \\ −x \end{matrix})}; +; 0; +; +; +]