CATEGORIES, ABELIAN CATEGORIES AND COHOMOLOGIES.

YOSHIFUMI TSUCHIMOTO

Let us denote by C^{∞}(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^{∞}(S^{1}; ℝ) →^{d/dt} C^{∞}(S^{1}; ℝ).

We see that:

H_{de Rham}^{0}(S^{1}; ℝ) = ℝ, H_{de Rham}^{1}(S^{1}; ℝ) = 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 1.1. A category 𝒞 is a collection of the following data (1) A collection Ob(𝒞) of objects of 𝒞.

(2) For each pair of objects X, Y ∈ Ob(𝒞), a set

Hom_{𝒞}(X, Y)

of morphisms.

(3) For each triple of objects X, Y, Z ∈ Ob(𝒞), a map(“composition (rule)”)

Hom_{𝒞}(X, Y) × Hom_{𝒞}(Y, Z) → Hom_{𝒞}(X, Z)

satisfying the following axioms (1) Hom(X, Y) ∩ Hom(Z, W) = ∅ unless (X, Y) = (Z, W).

(2) (Existence of an identity) For any X ∈ Ob(𝒞), there exists an element id_{X} ∈ Hom(X, X) such that

id_{X} ∘f = f, g ∘ id_{X} = g

holds for any f ∈ Hom(S, X), g ∈ Hom(X, T) (∀S, T ∈ Ob(𝒞)).

(3) (Associativity) For any objects X, Y, Z, W ∈ Ob(𝒞), and for any morphisms f ∈ Hom(X, Y), g ∈ Hom(Y, Z), h ∈ Hom(Z, W), we have

(f ∘ g) ∘ h = f ∘ (g ∘ h).

Definition 1.2. A universe U is a nonempty set satisfying the following axioms:

(1) If x ∈ U and y ∈ x , then y ∈ U .

(2) If x, y ∈ U , then {x, y} ∈ U .

(3) If x ∈ U , then the power set 2^{x} ∈ U.

(4) If {x_{i}|i ∈ I} is a family of elements of U indexed by an element I ∈ U, then ∪_{i∈I}x_{i} ∈ U .

YOSHIFUMI TSUCHIMOTO

Lemma 1.3. Let U be an universe. Then the following statements hold.

(1) If x ∈ U, then {x} ∈ U.

(2) If x is a subset of y ∈ U, then x ∈ U.

(3) If x, y ∈ U, then the ordered pair (x, y) = {{x, y}, x} is in U .

(4) If x, y ∈ U, then x ∪ y and x × y are in U.

(5) If {x_{i}|i ∈ I} is a family of elements of U indexed by an element I ∈ U, then we have ∏_{i∈I} x_{i} ∈ U.

In this text we always assume the following.

[figure]

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

Exercise 1.1. Let us put

C^{∞}(ℝ; ℝ) = {f :∈ C^{∞}(ℝ; ℝ), support of f is compact}

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

C^{∞}(ℝ; ℝ) → C^{∞}(ℝ; ℝ)

(2) Compute the cohomology group of the following complex.

C_{cpt}^{∞}(ℝ; ℝ) → C_{cpt}^{∞}(ℝ; ℝ)