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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeSep 22nd 2017

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.

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeSep 22nd 2017

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?

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeSep 22nd 2017

I also added that this constitutes of one of the two symmetric monoidal closed structures on $Cat$.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeSep 22nd 2017

I added to David’s addition, commenting that the other was of course the cartesian closed structure, and that both products are semicartesian.

• CommentRowNumber5.
• CommentAuthormaxsnew
• CommentTimeSep 22nd 2017

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

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeSep 22nd 2017

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.

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeSep 22nd 2017

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.

• CommentRowNumber8.
• CommentAuthormaxsnew
• CommentTimeMay 4th 2018

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”.

• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeMay 4th 2018

I think I’d prefer “separately functorial map”.

• CommentRowNumber10.
• CommentAuthorMike Shulman
• CommentTimeMay 4th 2018

Or “separately functorial operation”.

• CommentRowNumber11.
• CommentAuthorJohn Baez
• CommentTimeMar 31st 2021

Added reference to the paper showing Cat has just two symmetric monoidal closed structures.

• CommentRowNumber12.
• CommentAuthorJohn Baez
• CommentTimeMar 31st 2021

Pointed out that premonoidal categories are monoids in Cat with its funny tensor product.

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