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Bruno Gavranović (Apr 11 2023 at 16:03):Is there a good reference for cartesian 2-categories?
I'm trying to understand the correct analogue of the universal property of the product, which I imagine would still be
where this would be now an isomorphism of categories. I imagine one might want to potentially weaken this to an equivalence, though it's not clear whether one has to.
I'm interested in the strictest sensible version.
Mike Shulman (Apr 11 2023 at 16:13):I don't know of a reference that deals specifically and abstractly with cartesian 2-categories; I don't know if there's enough to say about them to fill a paper. As with any 2-categorical limit, finite products have both a strict version with a universal property given by an isomorphism and a weak version ("bicategorical products") with a universal property given by an equivalence. Many naturally-arising 2-categories, such as , have strict finite products; but others, such as and , only have weak bicategorical ones. However, because finite products are a [[flexible limit]], any bicategory with bicategorical finite products is equivalent to a strict 2-category having strict finite products, by a version of the [[coherence theorem for bicategories with finite bilimits]].
Niels van der Weide (Apr 11 2023 at 16:45):Relevant references on this topic:
Note: both papers are about the strict version.
Bruno Gavranović (Apr 11 2023 at 17:38):I see.
If I'm only interested in the strict version, it sounds like should be enough to prove the isomorphism between the categories holds, and nothing else.
Mike Shulman (Apr 11 2023 at 18:53):Holds naturally, of course.