CATEGORIES, ABELIAN CATEGORIES AND COHOMOLOGIES.

YOSHIFUMI TSUCHIMOTO

Generalities on categories and definition of abelian categories Our treatment here is a (rather strange) mixture of [2],[1]

Definition 3.1. Let F, G be two functors from a category 𝒞 to a category 𝒟. A morphism of functros from F to G is a family of morphisms in 𝒟:

f(X) : F (X) → G(X)

one for each X ∈ Ob(𝒞), satisfying the following condition: for any morphism φ : X → Y in 𝒞, the diagram

F (X) ⟶^{f(X)} G(X)
F (φ)↓  ^{G(φ)}↓
F (X) ⟶^{f(X)} G(Y)

is commutative.

Definition 3.2. Let 𝒞 be a category, X, Y be objects of 𝒞. Then an morphism a ∈ Hom_{𝒞}(X, Y) is an isomorphism in 𝒞 if there exists b ∈ Hom_{𝒞}(Y, X) such that the relations

ab = 1_{Y} ba = 1_{X}

hold. Objects X, Y in a category 𝒞 are said to be isomorphic if there exists at least one isomorphism between them.

Note that by combining the above two definitions, we obtain a definition of a notion of isomorphisms of functors.

Definition 3.3. A functior F : 𝒞 → 𝒟 is said to be an equivalene of category if there exists a functor G : 𝒟 → 𝒞 such that the functor GF is isomorphic to Id_{𝒞}, and the functor F G is isomorphic to Id_{𝒟} . If such a thing exists, we say that the two categories are equivalent.

Definition 3.4. Let 𝒞 be a category. Then:

(1) s: initial ⟺^{def} (∀a ∈ Ob(𝒞) (# Hom_{𝒞}(s, a) = 1)).

(2) t: terminal ⟺^{def} (∀a ∈ Ob(𝒞) (# Hom_{𝒞}(a, t) = 1)).

(3) n: null ⟺^{def} (n: initial and n:terminal)

Definition 3.5. An category 𝒞 is an additive category if it satisfies the following axioms: (A1) Any set Hom_{𝒞}(X, Y) is an additive group. The composition of

morphisms is bi-additive. (A2) There exists a null object 0 ∈ Ob(𝒞). (A3) For any objects X, Y ∈ Ob(𝒞), there exists a biproduct of

X, Y. Namely, there exists a diagram

X ⟶_{i_{1}}^{⟵^{p_{1}}} Z ⟵_{i_{2}}^{⟶^{p_{2}}} Y

YOSHIFUMI TSUCHIMOTO

in 𝒞 such that

p_{1}i_{1} = 1_{X}, p_{2}i_{2} = 1_{Y} , i_{1}p_{1} + i_{2}p_{2} = 1_{Z}

holds.

Definition 3.6. Let 𝒞 be a category, X, Y ∈ Ob(𝒞), adn f, g ∈ Hom_{𝒞}(X, Y). An equalizer k of f, g is an arrow K → X in 𝒞 which satisfies the following properties:

(1) f ∘ k = g ∘ k.

(2) k is “universal” amoung morphisms which satisfies (1). In other words, if m : M → X is a morphism in 𝒞 such that f ∘m = g∘m, then there exists a unique arrow h : M → K in 𝒞 which satisfy

m = k ∘ h.

By reversing the directions of arrows above, one may define the notion of coequalizers

Definition 3.7. Let 𝒞 be an additive category. Then the equalizer (respectively, coequalizer) of an arrow f : X → Y and 0 : X → Y is called the kernel (respectively, cokernel) of f.

Definition 3.8. An additive category 𝒞 is said to be abelian if it satisfies the following axioms. (A4-1) Every morphism f : X → Y in 𝒞 has a kernel ker(f) : Ker(f) →

X. (A4-2) Every morphism f : X → Y in 𝒞 has a cokernel coker(f) : Y →

Coker(f). (A4-3) For any given morphism f : X → Y, we have a suitably defined

isomorphism

l : Coker(ker(f)) ≅ Ker(coker(f))

in 𝒞. More precisely, l is a morphism which is defined by the following relations:

ker(coker(f)) ∘ \overline{f} = f (∃\overline{f}), \overline{f} = l ∘ coker(ker(f)).

References

[1] S. I. Gelfand and Y. Manin, Methods of homological algebra, Springer-Verlag, 1997.

[2] S. S. Mac Lane, Categories for the working mathematicians, Springer Verlag,