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Finally created funny tensor product. This is not really a very good name for a serious mathematical concept, but I don’t know of a better one.
Seems like some useful characterisation in section 2 of Mark Weber’s Free Products of Higher Operad Algebras, arXiv:0909.4722, so I’ll add that in.
How common is the use of “white” rather than “funny” as mentioned at Gray tensor product?
I also added that this constitutes of one of the two symmetric monoidal closed structures on $Cat$.
I added to David’s addition, commenting that the other was of course the cartesian closed structure, and that both products are semicartesian.
I added an explicit description as a pushout which I found in the Weber paper and I added it as an example to semicartesian category
I thought I saw somewhere once that the cartesian product, if it exists in a category, is terminal among all semicartesian monoidal structures? Probably easy if true, but I’ve not sat down to work it out.
Re #6 that sounds plausible, since if we have projections $A\otimes B \to A$ and $A\otimes B\to B$ they induce a map $A\otimes B \to A\times B$…
Re #2 I’ve never heard “white” in the context of 1-categories, only in the context of 2-categories where it’s being compared to the “gray” as well as the “black” one, and even that I think I’ve only heard as a joke.
Add a section about “separate functoriality”. Terminology is a little awkward here: should we say “separately functorial bifunctor” or “separately functorial functor of many arguments”? I stuck with the awkward but at least brief “separately functorial functor” vs “jointly functorial functor”.
I think I’d prefer “separately functorial map”.
Or “separately functorial operation”.
In this page it mentions that a general premonoidal category ought to be a pseudomonoid in the monoidal 2-category $(Cat, \otimes)$, but I strongly suspect that multiplying natural transformations isn’t possible. I worked it out in some detail and quickly ran into problems. I’m curious if anyone has reason to think it should work?
Edit: Apologies for not checking there first, but I see in the discussion for premonoidal category that Power and Robinson already said this wouldn’t work.
cross-linked the section “separate functoriality” with the corresponding new section at multifunctor.
Added reference to sesquicategories.
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