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Dylan Braithwaite (Feb 23 2023 at 13:10):Is there a name for bicategories whose hom-categories have monoidal structures but where this does not extend to the objects? My motivating example is (actually the subcategory of it spanned by monoidal categories) where the hom categories have monoidal products given by Day convolution, but I don’t think it makes much sense to extend this to profunctors defined on different pairs of categories.
Dylan Braithwaite (Feb 23 2023 at 14:29):I suppose it’s incorrect to talk about restricting to monoidal categories when that’s extra structure. Really I mean a version of whose objects are monoidal categories, but this starts to feel a bit evil because we have to distinguish objects by extra structure on them but this structure doesn’t appear in the 1-cells. This is my reason for this question: is there a good way to study this local monoidal structure more globally?
Nathanael Arkor (Feb 23 2023 at 15:01):Explicitly, given monoidal categories A and B, you are interested in the fact that Prof(A, B) ≃ Psh(B x A°) is monoidal, using Day convolution on the monoidal structure of the product category?
Dylan Braithwaite (Feb 23 2023 at 15:17):Yeah exactly. I think I’ve kind of rubber-ducked my way into thinking the original structure I was looking for doesn’t make much sense. But I’m still interesting in whether there’s a way to study this local monoidal structure more globally
Nathanael Arkor (Feb 23 2023 at 15:29):This situation seems to be quite specific to Prof. It makes use of the fact that presheaves are given by exponentiation, i.e. that Cat(A, PB) ≃ Cat(1, P(B x A°)), which isn't a relationship one would expect to hold very often. Unless you have some other similar examples, I don't see any phenomenon of which this is a particular example. Maybe others do, though.
Mike Shulman (Feb 23 2023 at 20:31):The obvious answer is "a bicategory enriched over monoidal categories". But the 2-category of monoidal categories isn't, I think, monoidal in a very useful way to enrich over. The 2-category of symmetric monoidal categories is (it has a tensor product that categorifies the tensor product of abelian monoids). But I haven't thought about whether composition of profunctors is a map in this category with respect to the Day convolution monoidal structures for profunctors between symmetric monoidal categories.
Mike Shulman (Feb 23 2023 at 20:32):If you just wanted a way to incorporate monoidal structures on objects into Prof, you could look at Prof as a double category, and take the tight morphisms to be monoidal functors.
Mike Shulman (Feb 23 2023 at 20:33):Or you could note that a monoidal category is both a monoid and a comonoid in Prof, and restrict to considering only profunctors that are monoid and/or comonoid homomorphisms.