Reference request: Categories enriched over $\textbf{FinLat}$ Let $\textbf{FinLat}$ be the category of finite lattices with $0$, regarded as a monoidal category by the tensor product of semilattices. It is known that the tensor product of two finite lattices considered as $(\vee,0)-$semilattices is again a finite lattice.
I am looking for some sources that discuss categories enriched over the category $\textbf{FinLat}$. Actually, more specifically, the category I am studying is enriched over the category of finite distributive lattices, but I highly doubt much research has been done in such a specialized setting.
I would also be happy with sources discussing categories enriched over the category of (commutative and/or idempotent) semirings, since every distributive lattice is also an semiring. However, I am actually not aware of any characterization of tensor products in the category of (commutative and/or idempotent) semirings, so I am not sure that this can be made into a monoidal category.
EDIT: Optimally I would like research papers that study these topics and give some theorems about their specific properties. However, if there has not been much research done on these specialized topics, I will still appreciate answers that give any interesting properties that may be applicable in this context (e.g., properties of categories enriched over semi-lattices or even just categories enriched over monoids). Alternatively, if anyone is aware of some interesting properties but cannot find any mention of them in any research papers, please feel free to simply post an answer discussing those properties without linking a reference.
 A: It may surprise you that research into what you want has been performed!
─rather variations on your objects of interest.
A pair of cool guys named Freyd and Scedrov wrote a book called
“Categories, Allegories” ─lovingly refereed to as “cats and alligators”.
In this second half of the text, they consider “allegories”:
categories whose homset is a meet-semilattice that are endowed
with an identity-on-objects involutive functor that satisfy
a law called the modal rule which documents the relation of
the involution, the order, and the meet operation.
For your needs, you can use the results that do not depend on an
involution ─or just take it to be the identity, or complementation
if you decide to to use complemented lattices─ and ignore the
results that use the modal rule.
Later on, they consider greater order structure on the homsets:
a lattice, a complete lattice, and other variations that I do not
remember.
Later on ─and hold onto your hats as this may surprise you─, they
show that if the allegory we're interested in satisfies some other
properties (power and tabular, I think) then it's equivalent to a topos!
This is exciting since it gives us another way to look at types
and set theory, for example.
If you dig around, you'll be giddy to find that if an allegory
has other properties instead (unitary and tabular, I think) then
its equivalent to a certain type of monoidal category!
With these bridges in hand, you may be able to use the know-how
of monoidal categories or the powerhouse of topoi to assist you!
Enjoy!
