nLab category with translation

Idea

A category with translations is a category equipped with a rudimentary notion of suspension objects. Categories with translation underlie triangulated categories where the “translation” becomes a genuine suspension as in homotopy fiber sequences.

Definition

Definition

A category with translation is a category $C$ equipped with an auto-equivalence functor

$$T : C \to C$$

called the shift functor or translation functor or suspension functor.

Remark

Frequently $C$ is an additive category in which case $T$ is also required to be an additive functor.

Definition

A morphism of categories with translation $F:(C,T)\to (C',T')$ is a functor $F:C\to C'$ equipped with an isomorphism $F\circ T\cong T'\circ F$:

$$\array{ C &\stackrel{F}{\to}& C' \\ \downarrow^T &\swArrow^{\simeq}& \downarrow^{T'} \\ C &\stackrel{F}{\to}& C' } \,.$$

If $C$,$C'$ are additive and $F$ is additive $F$ is a “morphism of additive categories with translation”.

Definition

In any additive category with translation a triangle is a sequence of morphisms of the form

$$a\stackrel{f}\to b\stackrel{g}\to c\stackrel{h}\to T a \,.$$

Remark

In some variants the translation endofunctor $T$ is not required to be an equivalence. This is the case for instance for the presuspended categories of Keller and Vossieck.

Examples

• The “translation” functor models the shift operation in a triangulated category, where one chooses a distinguished collection of triangles satisfying some axioms.

Related concepts

• suspension object

• triangulated category

  • exact triangle, distinguished triangle