Multi-allegories

[Toby Bartels <toby+categories@ugcs.caltech.edu>]

Has anybody looked at a combination of allegories and multicategories
that would capture (some of) the features of the algebra
of relations of finite arity (not just binary) on sets?
So we would have Mor(A_1,...,A_n) for any n = 0,1,2,...,
each a meet semilattice with an involution for each pair of indices,
and composition Mor(A_1,...,C,...,A_n) x Mor(B_1,...,C,...,B_m)
to Mor (A_1,...,...,A_n,B_1,...,...,B_m) across any repeated object,
satisfying some (hopefully) obvious conditions.

Of course, for functions of even arbitrary finite arity,
we only need Set as a good old cartesian monoidal category;
I really am looking for something weak analogous to a multicategory.
But if there's only material on monoidal or cartesian allegories,
or something like that, then that would still be helpful.
My motivations are entirely theoretical, so don't feel limited.

There's also the matter of relations of infinite arity,
but I'm not so much concerned about that right now.

I would appreciate any pointers to literature, useful folk theorems,
or even just the correct word to search on ("multiallegory" is no good).


--Toby


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