# Category (mathematics)

A cat­e­gory con­sists of a col­lec­tion of ob­jects with mor­phisms be­tween them. A mor­phism $$f$$ goes from one ob­ject, say $$X$$, to an­other, say $$Y$$, and is drawn as an ar­row from $$X$$ to $$Y$$. Note that $$X$$ may equal $$Y$$ (in which case $$f$$ is referred to as an en­do­mor­phism). The ob­ject $$X$$ is called the source or do­main of $$f$$ and $$Y$$ is called the tar­get or codomain of $$f$$, though note that $$f$$ it­self need not be a func­tion and $$X$$ and $$Y$$ need not be sets. This is writ­ten as $$f: X \rightarrow Y$$.

Th­ese mor­phisms must satisfy three con­di­tions:

1. Com­po­si­tion: For any two mor­phisms $$f: X \rightarrow Y$$ and $$g: Y \rightarrow Z$$, there ex­ists a mor­phism $$X \rightarrow Z$$, writ­ten as $$g \circ f$$ or sim­ply $$gf$$.

2. As­so­ci­a­tivity: For any mor­phisms $$f: X \rightarrow Y$$, $$g: Y \rightarrow Z$$ and $$h:Z \rightarrow W$$ com­po­si­tion is as­so­ci­a­tive, i.e., $$h(gf) = (hg)f$$.

3. Iden­tity: For any ob­ject $$X$$, there is a (unique) mor­phism, $$1_X : X \rightarrow X$$ which, when com­posed with an­other mor­phism, leaves it un­changed. I.e., given $$f:W \rightarrow X$$ and $$g:X \rightarrow Y$$ it holds that: $$1_X f = f$$ and $$g 1_X = g$$.

Parents:

• Category theory

How math­e­mat­i­cal ob­jects are re­lated to oth­ers in the same cat­e­gory.