A suspended category is an additive category equipped with an additive functor called suspension and a class of -sequences called triangles satisfying axioms below. Here one calls an -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 is a triangle.
(SP2) If is a triangle, then is also a triangle.
(SP3) Every diagram of the form
can be completed to a morphism of -sequences.
(SP4) For any two morphisms and 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 is an equivalence is triangulated.
Under mild assumptions, the stable category of a Quillen exact category is a suspended category. If is a Frobenius category, then 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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