Azimuth Math Book
Category

A category CC consists of the following data:

Subject to the following requirements:

Associativity. Suppose we are given a composable sequence of arrows f,g,hf,g,h:

W f X gY h Z\begin{matrix} W & \xrightarrow{f} & X & \xrightarrow{g} Y & \xrightarrow{h} & Z \end{matrix}

Then it must be that:

(fg)h=f(gh)(f \triangleright g) \triangleright h = f \triangleright (g \triangleright h)

So parentheses don’t matter, which means that we can simply write fghf \triangleright g \triangleright h.

Identities. For each object XX, there exists a designated identity arrow Id X:XXId_X: X \rightarrow X, such that for all arrows f:WXf: W \rightarrow X and g:XYg: X \rightarrow Y:

fId X=ff \triangleright Id_X = f

and

gId Y=gg \triangleright Id_Y = g