nLab
suspended category

Contents

Definition

A suspended category is an additive category CC equipped with an additive functor S:CCS:C\to C called suspension and a class of SS-sequences called triangles satisfying axioms below. Here one calls an SS-sequence a sequence of morphisms of the form

XfYgZhSX, X\stackrel{f}\to Y\stackrel{g}\to Z\stackrel{h}\to S X,

and morphisms are ladders of the type

X f Y g Z h SX a b c Sa X f Y g Z h SX, \array{ X&\stackrel{f}\to &Y&\stackrel{g}\to &Z&\stackrel{h}\to& S X\\ a\downarrow&&b\downarrow&&c\downarrow&&\downarrow S a\\ X'&\stackrel{f'}\to &Y'&\stackrel{g'}\to &Z'&\stackrel{h'}\to& S X',\\ }

where all the squares commute. Axioms:

(SP0) Each sequence isomorphic to a triangle is a triangle.

(SP1) Each sequence of the form 0XidXS00\to X\stackrel{id}\to X\to S0 is a triangle.

(SP2) If XfYgZhSXX\stackrel{f}\to Y\stackrel{g}\to Z\stackrel{h}\to S X is a triangle, then YgZhSXSfSYY\stackrel{g}\to Z\stackrel{h}\to S X\stackrel{-S f}\to S Y is also a triangle.

(SP3) Every diagram of the form

X f Y g Z h SX a b Sa X f Y g Z h SX \array{ X&\stackrel{f}\to &Y&\stackrel{g}\to &Z&\stackrel{h}\to& S X\\ a\downarrow&&b\downarrow&&&&\downarrow S a\\ X'&\stackrel{f'}\to &Y'&\stackrel{g'}\to &Z'&\stackrel{h'}\to& S X'\\ }

can be completed to a morphism of SS-sequences.

(SP4) For any two morphisms XfYX\stackrel{f}\to Y and YgZY\stackrel{g}\to Z there is a commuting diagram

X f Y g Z SX = f = X Z Y SX Sf X id X j SY j SY Si SZ \array{ X&\stackrel{f}\to &Y&\stackrel{g}\to &Z'&\to& S X\\ =\downarrow&&f\downarrow&&\downarrow&&=\downarrow \\ X&\to &Z&\to &Y'&\to& S X\\ &&\downarrow&&\downarrow&&\downarrow S f\\ & &X'&\stackrel{id}\to &X'&\stackrel{j}\to& S Y\\ &&j\downarrow&&\downarrow&&\\ &&S Y&\stackrel{S i}\to &S Z'&&\\ }

where the first two rows and the middle two columns are triangles.

Examples

References

Suspended categories were introduced in

  • Bernhard Keller, Dieter Vossieck, Sous les catégories dérivées. [Beneath the derived categories] C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 6, 225–228.

See also

  • B. Keller, Chain complexes and stable categories, Manus. Math. 67 (1990), 379-417.
Revised on February 15, 2010 19:33:48 by Urs Schreiber (131.211.234.10)