A Function as a collection of Arrows

Question

Normally you define a function to be a map on a set. But how about defining a function, in Category Theory, as a collection of arrows?

Take this cateogry

Objects: true, false.

Arrows:

true -> true
false -> false
true -> false
false -> true

Here I had to specify FOUR arrows, but I could summarize this collection of arrows as TWO functions:

id(x) = x    // replaces the first 2
not(x) = !x  //replaces the second 2 

If you want to maintain "arrowhood", which is more categoric, you could write a meta-arrow

idArrow(x)  = x -> x
notArrow(x) = x -> not(x)

Either way you compress information.

Question

Is there a strandard way of expressing these meta-arrows? Is this even part of the theory?

Answer 1

You have a set $S$ being acted on by a group $G$, and you are constructing the corresponding action groupoid. This is a standard construction. By the way, you haven't fully specified a category yet, since you haven't specified how your arrows compose.

Answer 2

Given any functor $F : X \to Y$, you can construct the category $Z$ defined by:

Start with the disjoint union of $X$ and $Y$.

For each object $x \in X$ and arrow $F(x) \xrightarrow{f} y$, add an arrow $x \xrightarrow{(x,f)} y$.

Composition is

• $ g (x,f) = (X,fg)$ if $g : y \to y'$
• $ (x, f) g = (x', f F(g))$ if $g : x'\to x$

This construction has a name, but I forget what it's called. (bridge, maybe?)

Anyways, in the special case that $X$ and $Y$ are sets (i.e. all morphisms are identity morphisms), then $F$ is a function, and this gives a way to view $F$ as a bunch of arrows.

The motivation behind the choice of new arrows and composition law above is that we want to formally add arrows $x \to F(x)$, but have a commutative diagram

$$ \begin{matrix} x &\xrightarrow{f}& x' \\ \downarrow & & \downarrow \\ F(x) &\xrightarrow{F(f)}& F(x') \end{matrix}$$

All new arrows are thus an arrow in $X$ followed by one from $X$ to $Y$ followed by an arrow from $Y$. But using the diagram, we can rewrite every such arrow as simply one from $X$ to $Y$ followed by one in $Y$, and I've used that normalization in choosing the representation. We could normalize the other way as well (make the thing in $Y$ the identity arrow)