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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 CatCat.

    • 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 ABAA\otimes B \to A and ABBA\otimes B\to B they induce a map ABA×BA\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”.

    diff, v5, current

    • 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.

    diff, v7, current

    • CommentRowNumber12.
    • CommentAuthorJohn Baez
    • CommentTimeMar 31st 2021

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

    diff, v7, current

    • CommentRowNumber13.
    • CommentAuthorJoe Moeller
    • CommentTimeMar 8th 2022
    • (edited Mar 8th 2022)

    In this page it mentions that a general premonoidal category ought to be a pseudomonoid in the monoidal 2-category (Cat,)(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.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2023

    cross-linked the section “separate functoriality” with the corresponding new section at multifunctor.

    diff, v8, current

    • CommentRowNumber15.
    • CommentAuthorvarkor
    • CommentTimeSep 1st 2023

    Added reference to sesquicategories.

    diff, v9, current

    • CommentRowNumber16.
    • CommentAuthorMorgan Rogers
    • CommentTimeMay 23rd 2024

    Added example illustrating failure of funny tensor product to be invariant under equivalence of categories.

    diff, v12, current

    • CommentRowNumber17.
    • CommentAuthorMorgan Rogers
    • CommentTimeMay 23rd 2024

    Added example illustrating failure of funny tensor product to be invariant under equivalence of categories. EDIT: changed \box to \BOX

    diff, v12, current