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David Michael Roberts (Jun 10 2025 at 13:04):I had a vague memory it was a theorem that any global choice of cartesian products in material sets couldn't be a strict monoidal product. I was recalling/thinking of Isnell's result that one cannot find a strictly cartesian monoidal skeleton of Set. But is the imagined result about products on the category of all material sets also true?
Nathanael Arkor (Jun 10 2025 at 15:24):The cartesian monoidal structure of Set can be strictified. See, for instance, Paul Levy's talk Strictifying monoidal structure, revisited.
John Baez (Jun 10 2025 at 16:14):How is this result stronger than what we can deduce from the general strictification result that every symmetric monoidal category is equivalent, as symmetric monoidal categories, to a strict one?
John Baez (Jun 10 2025 at 16:14):Maybe the idea here is that we're using the same underlying category?
Nathanael Arkor (Jun 10 2025 at 17:00):John Baez said:
Maybe the idea here is that we're using the same underlying category?
Yes, exactly.
David Michael Roberts (Jun 10 2025 at 22:06):@Nathanael Arkor is there anything more than the slides? They are sadly much lacking in detail. Or is the labelling supposed to sort that out for us? For this labelling (item 2 in the definition of label-bearing category) is supposed to be , but what is here? The Kuratowski construction of product? And what is "thing" in the definition?
It would have been easy to give the definition of the strictification of categorical product on , it's disappointing he didn't. @Paul Blain Levy can I summon you and ask you to give more details?
David Michael Roberts (Jun 10 2025 at 22:08):@John Baez where the slack gets taken up seems to be that this hinted construction doesn't respect the M-category structure on material sets: literal inclusions aren't preserved by the product, according to the slides.
Nathanael Arkor (Jun 10 2025 at 22:23):I think Paul may not be active on Zulip, in which case the best bet would be to email him.
John Baez (Jun 10 2025 at 22:27):@David Michael Roberts - is an 'M-category' some sort of 'material' category?
Nothing to do with M-theory, I assume. :upside_down:
David Michael Roberts (Jun 11 2025 at 02:23):@John Baez [[M-category]] (you have functions as the loose arrows, and then literal subset inclusions as tight arrows)
David Michael Roberts (Jun 11 2025 at 02:24):@Nathanael Arkor yes, he hasn't been here for at least a year, but he might still have had notifications of direct mentions turned on. I will email if he doesn't appear.
David Michael Roberts (Jun 11 2025 at 04:27):I think this might be the paper Paul was talking about:
Peter Schauenburg Turning Monoidal Categories into Strict Ones
It is well-known that every monoidal category is equivalent to a
strict one. We show that for categories of sets with additional
structure (which we define quite formally below) it is not even
necessary to change the category: The same category has a different
(but isomorphic) tensor product, with which it is a strict
monoidal category. The result applies to ordinary (bi)modules,
where it shows that one can choose a realization of the tensor
product for each pair of modules in such a way that tensor
products are strictly associative. Perhaps more surprisingly,
the result also applies to such nontrivially nonstrict
categories as the category of modules over a quasibialgebra.
Nathanael Arkor (Jun 11 2025 at 06:34):It is.