Categories, abelian categories and cohomologies.

Yoshifumi Tsuchimoto

Let us denote by $C^\infty(M,N)$ the set of all differentiable maps from $M$ to $N$ . A so-called ``de Rham cohomology'' of $S^1$ is computed as a cohomology of a complex

$$C^\infty(S^1;\mathbb{R}) \overset{d/dt}{\to} C^\infty(S^1;\mathbb{R}) .$$

We see that:

$$H^0_{\text{de Rham}}(S^1;\mathbb{R})=\mathbb{R},\quad H^1_{\text{de Rham}}(S^1;\mathbb{R})=\mathbb{R}$$

Actually, the dimension of the 0 -th cohomology is related to a number of the connected component of $S^1$ . The dimension of the $1$ -st cohomology is related to a number of the `hole' of $S^1$ . 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 $S^1$ 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 $\mathcal{C}$ is a collection of the following data

1. A collection $\operatorname{Ob}(\mathcal{C})$ of objects of $\mathcal{C}$ .
2. For each pair of objects $X,Y \in \operatorname{Ob}(\mathcal{C})$ , a set
   $$\operatorname{Hom}_{\mathcal{C}}(X,Y)$$
   of morphisms.
3. For each triple of objects $X,Y,Z \in \operatorname{Ob}(\mathcal{C})$ , a map(``composition (rule)'')
   $$\operatorname{Hom}_{\mathcal{C}}(X,Y)\times \operatorname{Hom}_{\mathcal{C}}(Y,Z)\to \operatorname{Hom}_{\mathcal{C}}(X,Z)$$

satisfying the following axioms

1. $\operatorname{Hom}(X,Y)\cap \operatorname{Hom}(Z,W) =\emptyset$ unless $(X,Y)=(Z,W)$ .
2. (Existence of an identity) For any $X\in \operatorname{Ob}(\mathcal{C})$ , there exists an element $\operatorname{id}_X\in \operatorname{Hom}(X,X)$ such that
   $$\operatorname{id}_X \circ f=f,\quad g \circ \operatorname{id}_X=g$$
   holds for any $f \in \operatorname{Hom}(S,X), g \in \operatorname{Hom}(X,T)$ ( $\forall S,T \in \operatorname{Ob}(\mathcal{C})$ ).
3. (Associativity) For any objects $X,Y,Z,W \in \operatorname{Ob}(\mathcal{C})$ , and for any morphisms $f\in \operatorname{Hom}(X,Y), g\in \operatorname{Hom}(Y,Z), h \in \operatorname{Hom}(Z,W)$ , we have
   $$(f\circ g)\circ h =f\circ (g\circ h).$$

DEFINITION 01.2   A universe $U$ is a nonempty set satisfying the following axioms:

1. If $x\in U$ and $y\in x$ , then $y\in U$ .
2. If $x,y\in U$ , then $\{x,y\}\in U$ .
3. If $x\in U$ , then the power set $2^x\in U$ .
4. If $\{x_i \vert i\in I\}$ is a family of elements of $U$ indexed by an element $I\in U$ , then $\cup_{i\in I} x_i\in U$ .

LEMMA 01.3   Let $U$ be an universe. Then the following statements hold.

1. If $x\in U$ , then $\{x\}\in U$ .
2. If $x$ is a subset of $y\in U$ , then $x\in U$ .
3. If $x,y\in U$ , then the ordered pair $(x,y) = \{\{x,y\},x\}$ is in $U$ .
4. If $x,y\in U$ , then $x\cup y$ and $x\times y$ are in $U$ .
5. If $\{x_i \vert i\in I\}$ is a family of elements of $U$ indexed by an element $I\in U$ , then we have $\prod_{i\in I} x_i \in U$ .

In this text we always assume the following.

For any set $S$ , there always exists a universe $U$ such that $S\in U$ .

EXERCISE 01.1   Let us put

$$C^\infty(\mathbb{R};\mathbb{R})=\{f: \in C^\infty(\mathbb{R};\mathbb{R}),$$   support of $f$ is compact$$\}$$

Then:

1. Compute the cohomology group of the following complex.
   $$C^\infty(\mathbb{R};\mathbb{R}) \to C^\infty(\mathbb{R}; \mathbb{R})$$

2. Compute the cohomology group of the following complex.
   $$C_{\operatorname{cpt}}^\infty(\mathbb{R};\mathbb{R}) \to C_{\operatorname{cpt}}^\infty(\mathbb{R}; \mathbb{R})$$