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A category whose morphism sets are abelian groups and whose composition is bilinear.
A category is preadditive if every morphism set is an abelian group and composition is bilinear:
In particular, there is a zero morphism between every ordered pair of objects.
An additive category is a preadditive category with finite biproducts. Preadditivity supplies the sums and signs used in complexes and distinguished triangles.
A category consists of:
These data satisfy the category axioms (associativity and identity/unit laws); see category axioms.
This abstracts the behavior of functions and their composition.
An additive category is a category in which morphisms can be added and finite direct sums exist.
A category is preadditive if:
A preadditive category is additive if, in addition:
A distinguished triangle is a triangle
belonging to a specified class that is required to satisfy the triangle axioms. It abstracts the mapping-cone sequence of a chain map and plays the role that a short exact sequence plays in an abelian category.
The class is closed under isomorphism and rotation, and every morphism must occur as the first map of some distinguished triangle. The precise coherence requirements distinguish a pretriangulated category from an arbitrary shifted preadditive category.