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# Categories, abelian categories and cohomologies.

Yoshifumi Tsuchimoto

Our treatment here is a (rather strange) mixture of [2],[1]

DEFINITION 03.1   Let be two functors from a category to a category . A morphism of functros from to is a family of morphisms in :

one for each , satisfying the following condition: for any morphism in , the diagram

is commutative.

DEFINITION 03.2   Let be a category, be objects of . Then an morphism is an isomorphism in if there exists such that the relations

hold. Objects in a category are said to be isomorphic if there exists at least one isomorphism between them.

Note that by combining the above two definitions, we obtain a definition of a notion of isomorphisms of functors.

DEFINITION 03.3   A functior is said to be an equivalene of category if there exists a functor such that the functor is isomorphic to , and the functor is isomorphic to . If such a thing exists, we say that the two categories are equivalent.

DEFINITION 03.4   Let be a category. Then:
1. : initial .
2. : terminal .
3. : null ( : initial and :terminal)

DEFINITION 03.5   An category is an additive category if it satisfies the following axioms:
A1.
Any set is an additive group. The composition of morphisms is bi-additive.
A2.
There exists a null object .
A3.
For any objects , there exists a biproduct of . Namely, there exists a diagram

in such that

holds.

DEFINITION 03.6   Let be a category, , adn . An equalizer of is an arrow in which satisfies the following properties:
1. .
2. is universal'' amoung morphisms which satisfies (1). In other words, if is a morphism in such that , then there exists a unique arrow in which satisfy

By reversing the directions of arrows above, one may define the notion of coequalizers

DEFINITION 03.7   Let be an additive category. Then the equalizer (respectively, coequalizer) of an arrow and is called the kernel (respectively, cokernel) of .

DEFINITION 03.8   An additive category is said to be abelian if it satisfies the following axioms.
A4-1.
Every morphism in has a kernel .
A4-2.
Every morphism in has a cokernel .
A4-3.
For any given morphism , we have a suitably defined isomorphism

in . More precisely, is a morphism which is defined by the following relations:

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2009-05-15