Categories, abelian categories and cohomologies.

Yoshifumi Tsuchimoto

Our treatment here is a (rather strange) mixture of [2],[1]

DEFINITION 03.1   Let $F,G$ be two functors from a category $\mathcal{C}$ to a category $\mathcal D$ . A morphism of functros from $F$ to $G$ is a family of morphisms in $\mathcal D$ :

$$f(X):F(X)\to G(X)$$

one for each $X \in \operatorname{Ob}(\mathcal{C})$ , satisfying the following condition: for any morphism $\varphi:X\to Y$ in $\mathcal{C}$ , the diagram

$$\begin{CD} F(X) @>f(X) >> G(X) \\ @V F(\varphi) VV @V G(\varphi) VV \\ F(X) @> f(X) >> G(Y) \end{CD}$$

is commutative.

DEFINITION 03.2   Let $\mathcal{C}$ be a category, $X,Y$ be objects of $\mathcal{C}$ . Then an morphism $a\in \operatorname{Hom}_{\mathcal{C}}(X,Y)$ is an isomorphism in $\mathcal{C}$ if there exists $b \in \operatorname{Hom}_{\mathcal{C}}(Y,X)$ such that the relations

$$a b= 1_Y \qquad b a= 1_X$$

hold. Objects $X,Y$ in a category $\mathcal{C}$ 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 03.3   A functior $F:\mathcal{C}\to \mathcal D$ is said to be an equivalene of category if there exists a functor $G: \mathcal D \to \mathcal{C}$ such that the functor $G F$ is isomorphic to $\operatorname{Id}_{\mathcal{C}}$ , and the functor $F G$ is isomorphic to $\operatorname{Id}_{\mathcal D}$ . If such a thing exists, we say that the two categories are equivalent.

DEFINITION 03.4   Let $\mathcal{C}$ be a category. Then:

1. $s$ : initial $\overset{\operatorname{def}}{\iff}$ $(\forall a\in \operatorname{Ob}(\mathcal{C}) \quad(\char93 \operatorname{Hom}_{\mathcal{C}}(s,a)=1))$ .
2. $t$ : terminal $\overset{\operatorname{def}}{\iff}$ $(\forall a\in \operatorname{Ob}(\mathcal{C}) \quad(\char93 \operatorname{Hom}_{\mathcal{C}}(a,t)=1))$ .
3. $n$ : null $\overset{\operatorname{def}}{\iff}$ ($n$ : initial and $n$ :terminal)

DEFINITION 03.5   An category $\mathcal{C}$ is an additive category if it satisfies the following axioms:

A1.

Any set $\operatorname{Hom}_{\mathcal{C}}(X,Y)$ is an additive group. The composition of morphisms is bi-additive.

A2.

There exists a null object $0\in \operatorname{Ob}(\mathcal{C})$ .

A3.

For any objects $X,Y\in \operatorname{Ob}(\mathcal{C})$ , there exists a biproduct of $X,Y$ . Namely, there exists a diagram

$$X \stackrel { \overset{p_1}{\longleftarrow} } {\underset{i_1}{\l... ...tackrel { \overset{p_2}{\longrightarrow} } {\underset{i_2}{\longleftarrow}} Y$$

in $\mathcal{C}$ such that

$$p_1 i_1=1_X , \quad p_2 i_2=1_Y ,\quad i_1 p_1 +i_2 p_2=1_Z$$

holds.

DEFINITION 03.6   Let $\mathcal{C}$ be a category, $X,Y\in \operatorname{Ob}(\mathcal{C})$ , adn $f,g \in \operatorname{Hom}_{\mathcal{C}}(X,Y)$ . An equalizer $k$ of $f,g$ is an arrow $K\to X$ in $\mathcal{C}$ which satisfies the following properties:

1. $f\circ k=g\circ k$ .
2. $k$ is ``universal'' amoung morphisms which satisfies (1). In other words, if $m: M\to X$ is a morphism in $\mathcal{C}$ such that $f \circ m=g \circ m$ , then there exists a unique arrow $h: M\to K$ in $\mathcal{C}$ which satisfy
   $$m=k\circ h.$$

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

DEFINITION 03.7   Let $\mathcal{C}$ be an additive category. Then the equalizer (respectively, coequalizer) of an arrow $f:X\to Y$ and $0: X\to Y$ is called the kernel (respectively, cokernel) of $f$ .

DEFINITION 03.8   An additive category $\mathcal{C}$ is said to be abelian if it satisfies the following axioms.

A4-1.

Every morphism $f:X\to Y$ in $\mathcal{C}$ has a kernel $\ker(f):\operatorname{Ker}(f)\to X$ .

A4-2.

Every morphism $f:X\to Y$ in $\mathcal{C}$ has a cokernel $\operatorname{coker}(f):Y\to \operatorname{Coker}(f)$ .

A4-3.

For any given morphism $f:X\to Y$ , we have a suitably defined isomorphism

$$l: \operatorname{Coker}(\ker(f))\cong \operatorname{Ker}(\operatorname{coker}(f))$$

in $\mathcal{C}$ . More precisely, $l$ is a morphism which is defined by the following relations:

$$\ker(\operatorname{coker}(f))\circ \... ...exists \overline{f}),\quad \overline{f}=l \circ \operatorname{coker}(\ker(f)).$$