Category

Definition

A Category $C$ consists of 3 things:

1. A class of objects $\mathcal{X}$
2. For every ordered pair, $(X,Y)$ of objects a set $\hom(X,Y)$ of morphisms $f$
3. A function called composition of morphisms:
   • $F_{(X,Y,Z)}:\hom(X,Y)\times\hom(Y,Z)\rightarrow\hom(X,Z)$

   defined for every triple, $(X,Y,Z)$ of objects where

   • Where $F_{(X,Y,Z)}(f,g)$ is denoted $g\circ f$

and the following $2$ properties are satisfied:

1. (Associativity) if $f\in\hom(W,X)$ and $g\in\hom(X,Y)$ and $h\in\hom(Y,Z)$ then
   • $h\circ(g\circ f)=(h\circ g)\circ f$
2. (Existence of identities) if $X$ is an object then there exists a $1_X\in\hom(X,X)$ such that:
   • $1_X\circ f=f$ and $g\circ 1_X=g$

   for every $f\in\hom(W,X)$ and $g\in\hom(X,Y)$ where $W$ and $Y$ are any class of objects

Uniqueness of the identity

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TODO: Be bothered to prove

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Left & right inverses

Let $f\in\hom(X,Y)$ and $g,\ g'\in\hom(Y,X)$, if:

• $g\circ f=1_X$ we call $g$ a left inverse for $f$ and if
• $f\circ g'=1_X$ we call $g'$ a right inverse for $f$

See also

• Natural transformation
• Covariant functor
• Contravariant functor