Sorry to ask such a trivial question, but I can't find the answer anywhere.
Question. What are the monomorphisms/epimorphisms in Rel?
Furthermore, what's the standard terminology for describing these ideas?
Here's my attempt at an answer, although I had to invent my own terminology. Let us write $f^{-1}$ for the converse of $f$, and $xy$ for the ordered pair $(x,y)$. Then we have:
Definition. For all relations $f : X \rightarrow Y$, define the following.
- $f$ is total iff for all $x \in X$ there exists $y \in Y$ such that $xy \in f$.
- $f$ is a tube (aka partial function) iff for all $x \in X$ and all $y,\bar{y} \in Y$ such that $xy \in f$ and $x\bar{y} \in f$, it holds that $y=\bar{y}$.
- $f$ is cototal iff $f^{-1}$ is total.
- $f$ is a cotube iff $f^{-1}$ is a tube.
Note that under these definitions, a function is a total tube. (In other words, a function is a total partial function. However, this is a very awkward wording.)
Anyway, under these definitions, I would expect that:
- The monomorphisms in Rel are precisely the total cotubes.
- The epimorphism in Rel are precisely the cototal tubes.
Once again, sorry if this is a very basic question, but I can't find the answer anywhere.