# Category

## Definition

A Category [ilmath]C[/ilmath] consists of 3 things[1]:

1. A class of objects [ilmath]\mathcal{X} [/ilmath][Note 1]
2. For every ordered pair, [ilmath](X,Y)[/ilmath] of objects a set [ilmath]\hom(X,Y)[/ilmath] of morphisms [ilmath]f[/ilmath]
3. A function called composition of morphisms:
• [ilmath]F_{(X,Y,Z)}:\hom(X,Y)\times\hom(Y,Z)\rightarrow\hom(X,Z)[/ilmath]
defined for every triple, [ilmath](X,Y,Z)[/ilmath] of objects where
• Where [ilmath]F_{(X,Y,Z)}(f,g)[/ilmath] is denoted [ilmath]g\circ f[/ilmath]

and the following [ilmath]2[/ilmath] properties are satisfied:

1. (Associativity) if [ilmath]f\in\hom(W,X)[/ilmath] and [ilmath]g\in\hom(X,Y)[/ilmath] and [ilmath]h\in\hom(Y,Z)[/ilmath] then
• [ilmath]h\circ(g\circ f)=(h\circ g)\circ f[/ilmath]
2. (Existence of identities) if [ilmath]X[/ilmath] is an object then there exists a [ilmath]1_X\in\hom(X,X)[/ilmath] such that[Note 2]:
• [ilmath]1_X\circ f=f[/ilmath] and [ilmath]g\circ 1_X=g[/ilmath]
for every [ilmath]f\in\hom(W,X)[/ilmath] and [ilmath]g\in\hom(X,Y)[/ilmath] where [ilmath]W[/ilmath] and [ilmath]Y[/ilmath] are any class of objects

## Uniqueness of the identity

TODO: Be bothered to prove

## Left & right inverses

Let [ilmath]f\in\hom(X,Y)[/ilmath] and [ilmath]g,\ g'\in\hom(Y,X)[/ilmath], if[1]:

• [ilmath]g\circ f=1_X[/ilmath] we call [ilmath]g[/ilmath] a left inverse for [ilmath]f[/ilmath] and if
• [ilmath]f\circ g'=1_X[/ilmath] we call [ilmath]g'[/ilmath] a right inverse for [ilmath]f[/ilmath]

## Notes

1. Munkres calls the class of objects [ilmath]X[/ilmath] and uses [ilmath]X[/ilmath] for specific objects. Not sure why, so checked definition with [Wikipedia]
2. We denote this as [ilmath]1_X[/ilmath] because it is easy to prove that it is unique, but at this point we do not know it is unique

## References

1. Elements of Algebraic Topology - James R. Munkres