Let $X$ and $Y$ denote sets. Then a relation $X \rightarrow Y$ is, by definition, a subset of $X \times Y$. Dually, we can define that a "corelation" $X \rightarrow Y$ is a partitioning of $X \uplus Y.$
General Question. Where can I learn more about corelations?
Here's a list of specific questions that I am interested in.
Can corelations can be composed in a sensible way? If so:
Do they constitute the arrows of a poset-enriched dagger category $\mathrm{cRel}$, in the same way that relations constitute the arrows of $\mathrm{Rel}$? If so, is this an allegory?
What are the mappings in $\mathrm{cRel},$ by which I mean the arrows $f : X \rightarrow Y$ such that $\mathrm{id}_X \leq f^\dagger \circ f$ and $f \circ f^\dagger \leq \mathrm{id}_Y.$
Can $\mathrm{Set}$ be viewed as a wide subcategory of $\mathrm{cRel}$?
What are the pre-orders in $\mathrm{cRel}$? By which I mean the endomorphisms $f : X \rightarrow X$ such that $\mathrm{id}_X \leq f$ and $f \circ f \leq f.$
What are the equivalence relations? By which I mean the pre-orders $f$ such that $f^\dagger = f.$
To reiterate: what I'd really like to know is, where can I learn more?