Categories, abelian categories and cohomologies.

Yoshifumi Tsuchimoto

1. (Sets)=(Category of all sets.) For $X,Y\in ($Sets$)$ , the set of morphism $\operatorname{Hom}_{\text{(set)}}(X,Y)$ is defined to be the set of all maps from $X$ to $Y$ .
2. (Groups)=(Category of all groups.) For $X,Y\in ($Groups$)$ , the set of morphism $\operatorname{Hom}_{\text{(group)}}(X,Y)$ is defined to be the set of all group homomorphisms from $X$ to $Y$ .
3. (Abelian groups), (Commutative Rings), are defined in a similar manner.
4. For a given ring $R$ , we define ($R$ -modules) to be the category of all $R$ -modules. Morphisms are $R$ -module homomorphisms.
5. (Top)=(Category of all topological spaces.) For $X,Y\in ($Top$)$ , the set of morphism $\operatorname{Hom}_{\text{(top)}}(X,Y)$ is defined to be the set of all continuous maps from $X$ to $Y$ .
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 $F$ from a category $\mathcal C$ to a category $\mathcal D$ consists of the following data:

1. An function which assigns to each object $C$ of $\mathcal C$ an object $F(C)$ of $\mathcal D$ .
2. An function which assigns to each morphism $f$ of $\mathcal C$ an morphism $F(f)$ of $\mathcal D$ .

The data must satisfy the following axioms:

functor-1.

$F(1_C)=1_{F(C)}$ for any object $C$ of $\mathcal C$ .

functor-2.

$F(f\circ g)=F(f)\circ F(g)$ for any composable morphisms $f,g$ of $\mathcal C$ .

By employing an axiom

(functor-$2'$ ) $F(f\circ g)=F(g)\circ F(f)$ for any composable morphisms

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