Map (Category Theory)

Idea

The map of a category gives a way of talking uniformly about morphisms in a category without fixing a domain and codomain.

Defintion

Let $\mathcal{C}$ be a category. $\mathrm{map}\mathcal{C}:=\Sigma x,y:\mathcal{C}.\mathcal{C}[x,y]$

Map Category

The map category of $\mathcal{C}$ consists of:

• Objects are $\mathrm{map}\mathcal{C}$.
• Morphisms are pairs of morphisms such that commuting squares:

$$\begin{array}{rl}& \mathrm{record}((\mathrm{map}\mathcal{C})[(x_1,y_1,f_1),(x_2,y_2,f_2)]):=\\ & (g:\mathcal{C}[x_1,x_2])\times \\ & (h:\mathcal{C}[y_1,y_2])\times \\ & (h\circ f_1=f_2\circ g)\end{array}$$