I am presently looking at a structure that I am trying to pin down- my strategy being to pull the thing up into the greatest possible generality (based on the bits I'm sure about) and narrow it down from there.
The situation I have is somewhat similar to the dual space structure in vector spaces, though almost certainly less well-behaved and vector spaces alone will not cut it.
Consider a vector space $V$ over a field $k$- its dual space $V^*$ appears naturally as the set linear of maps
$$w^* : V \to k$$
which, coincidentally, form a vector space in themselves. We can generalise this to arbitrary categories $A$, $B$, $C$ by setting $B=hom(A,C)$. Then, at least in some sense, $B=A^*$. So far so standard, but I want more: a nice property of dual spaces is that an element of $V \otimes V^*$ can be canonically seen as an element of $hom(V,V)$- this is because of the way that $k$ acts on $V$ by multiplication. We can mimic this by letting $C$ be a monoid acting on $A$.
In summary: A category $A$ acted on by a monoid $C$ and a dual A $A^*:=hom(A,C)$
I am particularly interested in when $A$ is also a monoid, especially so when $A$ is a space of stochastic matrices.
So this isn't too unlikely a construction, in fact it's probably forehead-slappingly well known, so:
- What is it called, if anything?
- In what cases can we have $A=B=C$? Is that necessarily a permutation group for example?
- Are there any useful canonical examples, besides vector spaces?
- Better yet, theorems??? Papers???
As you can probably tell, I am no category theorist, so any help would be awesome.