next up previous
Next: universe Up: Topics in Non commutative Previous: Ultra filter

Elementary category theory

We need to develop a fine theory of (n-)category theory in this lecture. But that will take time. Meanwhile, we use an easy part of elementary category theory as a convenient language.

DEFINITION 5.1   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

    $\displaystyle \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)'')

    $\displaystyle \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

    $\displaystyle \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

    $\displaystyle (f\circ g)\circ h
=f\circ (g\circ h).
$



Subsections

2007-12-11