CATEGORIES, ABELIAN CATEGORIES AND COHOMOLOGIES.

YOSHIFUMI TSUCHIMOTO

Examples of categories

(1) (Sets)=(Category of all sets.) For X, Y ∈ (Sets), the set of morphism Hom_{(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 ∈ (Groups), the set of morphism Hom_{(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 ∈ (Top), the set of morphism Hom_{(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 2.1. A (covariant) functor F from a category 𝒞 to a category 𝒟 consists of the following data:

(1) An function which assigns to each object C of 𝒞 an object F (C) of 𝒟.

(2) An function which assigns to each morphism f of 𝒞 an mor-

phism F (f) of 𝒟. The data must satisfy the following axioms:

(functor-1) F (1_{C}) = 1_{F (C)} for any object C of 𝒞. (functor-2) F (f ∘ g) = F (f) ∘ F (g) for any composable morphisms f, g of

𝒞.

By employing an axiom (functor-2′) F (f ∘ g) = F (g) ∘ F (f) for any composable morphisms

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