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
and morphisms are ladders of the type
where all the squares commute. Axioms:
(SP0) Each sequence isomorphic to a triangle is a triangle.
(SP1) Each sequence of the form $0\to X\stackrel{id}\to X\to S0$ is a triangle.
(SP2) If $X\stackrel{f}\to Y\stackrel{g}\to Z\stackrel{h}\to S X$ is a triangle, then $Y\stackrel{g}\to Z\stackrel{h}\to S X\stackrel{-S f}\to S Y$ is also a triangle.
(SP3) Every diagram of the form
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
where the first two rows and the middle two columns are triangles.
Every triangulated category is suspended.
Every suspended category in which $S$ is an equivalence is triangulated.
Under mild assumptions, the stable category of a Quillen exact category $(A,E)$ is a suspended category. If $A$ is a Frobenius category, then $A$ is a triangulated category.
Suspended categories were introduced in
Sous les catégories dérivées. [Beneath the derived categories] C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 6, 225–228.
See also
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