# Categories, abelian categories and cohomologies.

Yoshifumi Tsuchimoto

1. (Sets)=(Category of all sets.) For Sets , the set of morphism is defined to be the set of all maps from to .
2. (Groups)=(Category of all groups.) For Groups , the set of morphism is defined to be the set of all group homomorphisms from to .
3. (Abelian groups), (Commutative Rings), are defined in a similar manner.
4. For a given ring , we define ( -modules) to be the category of all -modules. Morphisms are -module homomorphisms.
5. (Top)=(Category of all topological spaces.) For Top , the set of morphism is defined to be the set of all continuous maps from to .
6. (Hausdorff Sp.)=(the category of all Hausdorff spaces), (Compact Sp)=(the category of all Compact spaces) are defined in a similar manner.

DEFINITION 02.1   A (covariant) functor from a category to a category consists of the following data:
1. An function which assigns to each object of an object of .
2. An function which assigns to each morphism of an morphism of .
The data must satisfy the following axioms:
functor-1.
for any object of .
functor-2.
for any composable morphisms of .

By employing an axiom

(functor- ) for any composable morphisms

instead of axiom (functor-2) above, we obtain a definition of a contravariant functor.