nLab
category with translation

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

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 CC equipped with an auto-equivalence functor

T:CC T : C \to C

called the shift functor or translation functor or suspension functor.

Remark

Frequently CC is an additive category in which case TT is also required to be an additive functor.

Definition

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

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

If CC,CC' are additive and FF is additive FF 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

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

In some variants the translation endofunctor TT 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)