nLab triangulated bifunctor

Definition

Definition

Given a triple of triangulated categories $\mathcal{C}, \mathcal{C}', \mathcal{D}$, then a triangulated bifunctor

$$F \,\colon\, \mathcal{C} \times \mathcal{C}' \longrightarrow \mathcal{D}$$

is a bifunctor which is a triangulated functor in each variable and respects translation coherently, in that it:

1. (additivity) is additive,

2. (triangulation) preserves distinguished triangles in each variable,

3. (translation) is equipped with natural isomorphisms

   $$\begin{array}{ccc} F\big(\Sigma X, Y\big) &\underoverset{\sim}{\theta_{X,Y}}{\longrightarrow}& \Sigma F\big(X,Y\big) \\ F\big(X, \Sigma Y\big) &\underoverset{\sim}{\theta'_{X,Y}}{\longrightarrow}& \Sigma F\big(X,Y\big) \end{array}$$

   such that the following diagram anti-commutes (one composite equals minus the other):

   $$\begin{array}{ccc} F(\Sigma X, \Sigma Y) &\overset{\theta_{X,\Sigma Y}}{\longrightarrow}& \Sigma F(X,\Sigma Y) \\ \mathllap{{}^{\theta'_{\Sigma X, Y}}} \big\downarrow && \big\downarrow \mathrlap{{}^{\Sigma \theta'_{X,Y}}} \\ \Sigma F(\Sigma X, Y) &\underset{\Sigma \theta_{X,Y}}{\longrightarrow}& \Sigma^2 F(X,Y) \mathrlap{\,.} \end{array}$$

(Kashiwara & Schapira 2006 Def. 10.3.6 with 10.1.1(v))

Examples

Example

An exact continuous $\infty$-functor out of a tensor product of stable $\infty$-categories to a stable $\infty$-category induces a triangulated bifunctor on triangulated homotopy categories.

Related entries

• monoidal triangulated category

References

• Masaki Kashiwara, Pierre Schapira; Def. 10.3.6, 10.1.1(v) in: Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006) [doi:10.1007/3-540-27950-4, pdf]