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An object that is simultaneously initial and terminal.
A zero object in a category is an object that is both initial and terminal. Thus for every object , there is exactly one morphism and exactly one morphism .
Any two zero objects are uniquely isomorphic. In a preadditive category, the composite is the zero morphism from to . A category equipped with a zero object is often called pointed.
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.
Let be a category. An initial object is an object such that for every object , there is exactly one morphism . Any two initial objects are uniquely isomorphic.
Reversing all arrows gives a terminal object. An object that is both initial and terminal is a zero object. The empty set is initial in the category of sets, while the trivial group is both initial and terminal in the category of groups.
Let be a category. A terminal object is an object such that for every object there is exactly one morphism
Any two terminal objects are uniquely isomorphic, so notation such as refers to a choice that is immaterial up to unique isomorphism.
For example, a one-point set is terminal among sets, and a one-point space is terminal among topological spaces. In a category of objects over a fixed base , the identity is terminal. Consequently, on a site with terminal object , a covering family expresses a global object as being covered by local pieces.
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.