# Categories, abelian categories and cohomologies.

Yoshifumi Tsuchimoto

Let us denote by the set of all differentiable maps from to . A so-called de Rham cohomology'' of is computed as a cohomology of a complex

We see that:

Actually, the dimension of the 0 -th cohomology is related to a number of the connected component of . The dimension of the -st cohomology is related to a number of the hole' of . Cohomology is then a good tool to obtain numbers (invariants'') of geometric objects.

Cohomology also arises as obstructions''. Indeed, the de Rham cohomology of the tells us a hint about which functions are integrable'', etc.

In this talk we give a definition and explain some basic properties of cohomologies. But before that, we first deal with some category theory.

DEFINITION 01.1   A category is a collection of the following data
1. A collection of objects of .
2. For each pair of objects , a set

of morphisms.
3. For each triple of objects , a map(`composition (rule)'')

satisfying the following axioms
1. unless .
2. (Existence of an identity) For any , there exists an element such that

holds for any ( ).
3. (Associativity) For any objects , and for any morphisms , we have

DEFINITION 01.2   A universe is a nonempty set satisfying the following axioms:
1. If and , then .
2. If , then .
3. If , then the power set .
4. If is a family of elements of indexed by an element , then .

LEMMA 01.3   Let be an universe. Then the following statements hold.
1. If , then .
2. If is a subset of , then .
3. If , then the ordered pair is in .
4. If , then and are in .
5. If is a family of elements of indexed by an element , then we have .

In this text we always assume the following.

For any set , there always exists a universe such that .

EXERCISE 01.1   Let us put

support of $f$ is compact

Then:
1. Compute the cohomology group of the following complex.

2. Compute the cohomology group of the following complex.