nLab category with translation

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

Theorems

Stable homotopy theory

stable homotopy theory

Category with translation

Idea

A category with translations is a category equipped with a rudimentary notion of suspension objects. Categories with translation underly 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

$T:C\to C$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 $F:\left(C,T\right)\to \left(C\prime ,T\prime \right)$ is a functor $F:C\to C\prime$ equipped with an isomorphism $F\circ T\cong T\prime \circ F$:

$\begin{array}{ccc}C& \stackrel{F}{\to }& C\prime \\ {↓}^{T}& {⇙}^{\simeq }& {↓}^{T\prime }\\ C& \stackrel{F}{\to }& C\prime \end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ C &\stackrel{F}{\to}& C' \\ \downarrow^T &\swArrow^{\simeq}& \downarrow^{T'} \\ C &\stackrel{F}{\to}& C' } \,.

If $C$,$C\prime$ 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

$a\stackrel{f}{\to }b\stackrel{g}{\to }c\stackrel{h}{\to }Ta\phantom{\rule{thinmathspace}{0ex}}.$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.

Revised on September 24, 2012 14:30:03 by Urs Schreiber (82.169.65.155)