Azimuth Math Book
Category

A category CC consists of the following data:

  • A set Ob(C)Ob(C) called the objects.

  • For all objects X,YX,Y, a set of arrows C(X,Y)C(X,Y). For an arrow aa in C(X,Y)C(X,Y), we write a:XYa: X \rightarrow Y, and say that aa is an arrow from XX to YY. We may think of the arrow as an abstract process or function. XX is the domain of aa, and YY is the codomain of aa. Another name for arrow is morphism.

  • Given arrows f:XYf: X \rightarrow Y and g:YZg: Y \rightarrow Z, the category provides a composition rule that defines a composite morphism fg:XZf \triangleright g: X \rightarrow Z. Think of fgf \triangleright g as the process which first “goes through” ff from XX to YY and then goes through gg from YY to ZZ. Note: the standard notations for fgf \triangleright g are f;gf;g and gfg \circ f.

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