Dweak squares in Heller triangulated categories

Matthias Künzer

November 5, 2007

Abstract

We mention briefly an elementary compatibility of distinguished weak (dweak) squares with the shift functor, which holds in a closed Heller triangulated category.

In a Verdier triangulated category, one can still formulate this compatibility assertion, but I do not know whether it can be proven there.

To obtain a counterexample to this compatibility assertion in a Verdier triangulated category, however, one would need a Verdier triangulated category that is not a closed Heller triangulated category. I do not know an example of such a category.

Let (C, T, ϑ) be a closed Heller triangulated category; cf. [2, Def. 1.5.i], [3, Def. A.6]. Recall that closedness just means that each morphism can be completed to a 2-triangle. Heller triangulated categories in which idempotents split, are closed; cf. [2, Lem. 3.1]. The stable category of a Frobenius category is closed.

We use the conventions and notations of [2].

A closed Heller triangulated category is in particular Verdier triangulated, taking the set (^{1}) of 2-triangles as set of distinguished triangles; cf. [5, Def. 1-1]; this is proven just as [2, Prop. 3.6].

Suppose given n ⩾ 2. Each chain of morphisms X_{1} ^{\overline{}^{f_{1}}} ⋯ ^{\overline{}^{f_{n−1}}} X_{n} in C can be prolonged to an n-triangle, and this prolongation is unique up to isomorphism; cf. [2, Def. 1.5.ii.2, Lem. 3.1, Lem. 3.4.6].

A distinguished weak square, for short dweak square (^{2}), is a weak square whose diagonal sequence fits into a 2-triangle. Dweak squares are indicated by the symbol ⫀. Note that to define dweak squares, a Verdier triangulated category suffices.

Since a completion of a morphism to a 2-triangle is unique up to isomorphism, so is completion of an angle [figure] to a dweak square [figure: ⫀]; and dually.

Using a 4-triangle, one sees that dweak squares compose. There is an elegant method, due to Neeman, to show this fact already in Verdier triangulated categories; cf. [4, Lem. 2.1].

1In a sufficiently big universe.

2Also known as a homotopy cartesian square.

Completing iteratively to dweak squares, we may therefore form

[figure: 0; X′_{n}; _{...}; ⋮; ⫀; ⋮; f′_{n−1}; 0; ⋯; ⫀; ⫀; X′_{2}; f′_{2}; 0; ⫀; ⋯; ⫀; ⫀; X′_{1}; f′_{1}; X_{1}; f_{1}; ⫀; X_{2}; f_{2}; ⫀; ⋯; f_{n−1}; ⫀; X_{n}; ⫀; 0 ,]

and the resulting chain X′_{1} ^{\overline{}^{f′_{1}}} ⋯ ^{\overline{}^{f′_{n−1}}} X′_{n} is unique up to isomorphism.

Remark 1. The following compatibility of dweak squares and shift holds in our Heller triangulated category (C, T, ϑ).

(X′_{1} ^{\overline{}^{f′_{1}}} ⋯ ^{\overline{}^{f′_{n−1}}} X′_{n}) ≃ (X_{1}^{+1} ^{\overline{}^{f_{1}^{+1}}} ⋯ ^{\overline{}^{f_{n−1}^{+1}}} X_{n}^{+1}) .

Proof. Complete the chain X_{1} ^{\overline{}^{f_{1}}} ⋯ ^{\overline{}^{f_{n−1}}} X_{n} to an n-triangle. By [2, Lem. 3.4.1, Lem. 3.4.2], an n-triangle consists of dweak squares. The result follows from the uniqueness up to isomorphism stated just before Remark 1.

In a Verdier triangulated category, in which the axiom [1, 1.1.13] holds, the assertion of Remark 1 holds if

n = 2

and, by the octahedral axiom in the form of loc. cit., if

n = 3 .

But we may ask for this assertion for

n ⩾ 4

as well. In fact, both sides of the isomorphism in question are still welldefined up to isomorphism resp. welldefined. If n = 4, the valid assertion in the cases n = 2 and n = 3, together with [2, Lem. 3.4.1], yields isomorphisms

\xymatrix{X′_{1} \ar[r]^{f′_{1}} \ar[d]_{u_{1}^{1,2}}^{≀} & X′_{2} \ar[d]^{u_{2}^{1,2}}_{≀} & X′_{2} \ar[r]^{f′_{2}} \ar[d]_{u_{2}^{2,3}}^{≀} & X′_{3} \ar[d]^{u_{3}^{2,3}}_{≀} & X′_{1} \ar[r]^{f′_{1}f′_{2}} \ar[d]_{u_{1}^{1,3}}^{≀} & X′_{3} \ar[d]^{u_{3}^{1,3}}_{≀} \\ X_{1}^{+1} \ar[r]^{f_{1}^{+1}} & X_{2}^{+1} & X_{2}^{+1} \ar[r]^{f_{2}^{+1}} & X_{3}^{+1} & X_{1}^{+1} \ar[r]^{f_{1}^{+1}f_{2}^{+1}} & X_{3}^{+1} \\ X′_{1} \ar[r]^{f′_{1}f′_{2}f′_{3}} \ar[d]_{u_{1}^{1,4}}^{≀} & X′_{4} \ar[d]^{u_{4}^{1,4}}_{≀} & X′_{2} \ar[r]^{f′_{2}f′_{3}} \ar[d]_{u_{2}^{2,4}}^{≀} & X′_{4} \ar[d]^{u_{4}^{2,4}}_{≀} & X′_{3} \ar[r]^{f′_{3}} \ar[d]_{u_{3}^{3,4}}^{≀} & X′_{4} \ar[d]^{u_{4}^{3,4}}_{≀} \\ X_{1}^{+1} \ar[r]^{f_{1}^{+1}f_{2}^{+1}f_{3}^{+1}} & X_{4}^{+1} & X_{2}^{+1} \ar[r]^{f_{2}^{+1}f_{3}^{+1}} & X_{4}^{+1} & X_{3}^{+1} \ar[r]^{f_{3}^{+1}} & X_{4}^{+1}}

\xymatrix{X′_{1} \ar[r]^{f′_{1}} \ar[d]_{u_{1}^{1,2,3}}^{≀} & X′_{2} \ar[r]^{f′_{2}} \ar[d]^{u_{2}^{1,2,3}}_{≀} & X′_{3} \ar[d]^{u_{3}^{1,2,3}}_{≀} & X′_{1} \ar[r]^{f′_{1}} \ar[d]_{u_{1}^{1,2,4}}^{≀} & X′_{2} \ar[r]^{f′_{2}f′_{3}} \ar[d]^{u_{2}^{1,2,4}}_{≀} & X′_{4} \ar[d]^{u_{4}^{1,2,4}}_{≀} \\ X_{1}^{+1} \ar[r]^{f_{1}^{+1}} & X_{2}^{+1} \ar[r]^{f_{2}^{+1}} & X_{3}^{+1} & X_{1}^{+1} \ar[r]^{f_{1}^{+1}} & X_{2}^{+1} \ar[r]^{f_{2}^{+1}f_{3}^{+1}} & X_{4}^{+1} \\ X′_{1} \ar[r]^{f′_{1}f′_{2}} \ar[d]_{u_{1}^{1,3,4}}^{≀} & X′_{3} \ar[r]^{f′_{3}} \ar[d]^{u_{3}^{1,3,4}}_{≀} & X′_{4} \ar[d]^{u_{4}^{1,3,4}}_{≀} & X′_{2} \ar[r]^{f′_{2}} \ar[d]_{u_{2}^{2,3,4}}^{≀} & X′_{3} \ar[r]^{f′_{3}} \ar[d]^{u_{3}^{2,3,4}}_{≀} & X′_{4} \ar[d]^{u_{4}^{2,3,4}}_{≀} \\ X_{1}^{+1} \ar[r]^{f_{1}^{+1}f_{2}^{+1}} & X_{3}^{+1} \ar[r]^{f_{3}^{+1}} & X_{4}^{+1} & X_{2}^{+1} \ar[r]^{f_{2}^{+1}} & X_{3}^{+1} \ar[r]^{f_{3}^{+1}} & X_{4}^{+1}}

But in a Verdier triangulated category with [1, 1.1.13] added, I do not know how to prove that, say, u_{3}^{1,2,3} may be chosen equal to u_{3}^{3,4}. Neither do I know a Verdier triangulated category in which the assertion of Remark 1 fails, say, for n = 4. Worse still, I do not know an example of a Verdier triangulated category that is not closedly Heller triangulated.

References

[1] Bernstein, J.; Beilinson, A.A.; Deligne, P., Faisceaux pervers, Astérisque 100, 1982.

[2] Künzer, M., Heller triangulated categories, Homol., Homot. Appl. 9 (2), p. 233–320, 2007.

[3] Künzer, M., Nonisomorphic Verdier octahedra on the same base, preprint, arxiv, 0708.0151, 2007.

[4] Neeman, A., K-theory for triangulated categories I (A), Asian J. Math. 1 (2), p. 330–417, 1997.

[5] Verdier, J.L., Catégories Derivées, published in SGA 4 1/2, SLN 569, p. 262–311, 1977 (written 1963).

MSC2000: 18E30.