 Math and science::Algebra::Aluffi

# Category. Definition.

### Category

A category $$\cat{C}$$ consists of:

• a class of objects, denoted as $$\catobj{C}$$
• a set, denoted as $$\cathom{C}(A, B)$$, for any objects $$A$$ and $$B$$ of $$\cat{C}$$. The elements are called morphisms.

The set of morphisms must have the following properties:

Identity
For each object $$A \in \catobj{C}$$, there exists (at least) one morphism [$$? \in \; \cathom{C}(?, ?)$$], called the identity on $$A$$.
Composition
Morphisms can be composed: any two morphisms $$f \in \cathom{C}(A, B)$$ and $$g \in \cathom{C}(B, C)$$ [determine/imply what??].
Associativity of composition
For any $$f \in \cathom{C}(A, B)$$, $$g \in \cathom{C}(B, C)$$ and $$h \in \cathom{C}(C, D)$$, we have:
[$? \quad = \quad ?$]
Identity law
The identity morphisms are identities with respect to composition. For any $$f \in \cathom{C}(A, B)$$, we have:
[$f \, ? = f, \quad ? f = f$]
Morphism sets are disjoint
For any $$A, B, C, D \in \catobj{C}$$, then $$\cathom{C}(A, B)$$ and $$\cathom{C}(C, D)$$ are disjoint unless $$A = C$$ and $$B = D$$.