nLab
suspended category

Definition

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

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

and morphisms are ladders of the type

$$\begin{array}{ccccccc}X& \stackrel{f}{\to }& Y& \stackrel{g}{\to }& Z& \stackrel{h}{\to }& SX\\ a\downarrow & & b\downarrow & & c\downarrow & & \downarrow Sa\\ X\prime & \stackrel{f\prime }{\to }& Y\prime & \stackrel{g\prime }{\to }& Z\prime & \stackrel{h\prime }{\to }& SX\prime ,\\ \end{array}$$\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 $0\to X\stackrel{\mathrm{id}}{\to }X\to S0$ is a triangle.

(SP2) If $X\stackrel{f}{\to }Y\stackrel{g}{\to }Z\stackrel{h}{\to }SX$ is a triangle, then $Y\stackrel{g}{\to }Z\stackrel{h}{\to }SX\stackrel{-Sf}{\to }SY$ is also a triangle.

(SP3) Every diagram of the form

$$\begin{array}{ccccccc}X& \stackrel{f}{\to }& Y& \stackrel{g}{\to }& Z& \stackrel{h}{\to }& SX\\ a\downarrow & & b\downarrow & & & & \downarrow Sa\\ X\prime & \stackrel{f\prime }{\to }& Y\prime & \stackrel{g\prime }{\to }& Z\prime & \stackrel{h\prime }{\to }& SX\prime \\ \end{array}$$\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 $S$-sequences.

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

$$\begin{array}{ccccccc}X& \stackrel{f}{\to }& Y& \stackrel{g}{\to }& Z\prime & \to & SX\\ =\downarrow & & f\downarrow & & \downarrow & & =\downarrow \\ X& \to & Z& \to & Y\prime & \to & SX\\ & & \downarrow & & \downarrow & & \downarrow Sf\\ & & X\prime & \stackrel{\mathrm{id}}{\to }& X\prime & \stackrel{j}{\to }& SY\\ & & j\downarrow & & \downarrow & & \\ & & SY& \stackrel{Si}{\to }& SZ\prime & & \\ \end{array}$$\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

• Every triangulated category is suspended.

• Every suspended category in which $S$ is an equivalence is triangulated.

• Under mild assumptions, the stable category of Quillen exact category $(A,E)$ is a suspended category. If $A$ is Frobenius category then $A$ is triangulated category.

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.