Category Theory
Overview
Category theory is a branch of mathematics that casts mathematical structures into a framework consisting of objects
and arrows between those objects.
Definition
A category A is a collection (class) of objects, denoted Obj(A) and a collection of morphisms Mor(A) defined as follows:
- Every morphism has a domain object associated with it
{% dom_A : Mor(A) \rightarrow Obj(A) %}
- Every morphism has a co-domain object associated with it
{% cod_A : Mor(A) \rightarrow Obj(A) %}
-
For any two morphisms, {% f %} and {% g %} such that {% cod_A(f) = dom_A(g) %},
there is a third morphism {% g \circ f : dom_A(f) \rightarrow cod_A(g) %}
called the composition of {% f %} and {% g %}.
-
For any object a in A, there is a morphism {% id_a %} with {% dom_A(id_a) = a %}
and {% cod_A(id_a) = a %}, such that {% id_a \circ g = g %} and {% g \circ id_a = g %}.
-
Composition is associative, that is
{% h \circ (g \circ f) = (h \circ g) \circ f %}
Topics
