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 \( B = D \).