category with translation

(also nonabelian homological algebra)

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.

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**.

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

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”.

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

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

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.

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

Revised on April 15, 2015 23:00:32
by Anonymous Coward
(108.28.194.5)