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Stream: theory: category theory

Topic: Lax interchange in an almost-2-category.

Chad Nester (Mar 19 2025 at 12:42):

I've encountered something that is almost a strict 2-category. However, instead of the horizontal composition operator on 2-cells satisfying $(\alpha ; \beta)*(\gamma ; \delta) = (\alpha * \gamma);(\beta * \delta)$, it only satisfies $(\alpha ; \beta)*(\gamma ; \delta) \leq (\alpha * \gamma);(\beta * \delta)$.

If it matters, the 0-cells are all poset enriched, and the ordering on my 2-cells is defined componentwise.

What sort of thing is an almost-2-category like this? Is this a good reason to abandon my notion of 2-cell?

Chad Nester (Mar 19 2025 at 12:45):

I suspect that one possible answer is that I have an intercategory of some sort, but I'm rather hoping someone knows a simpler answer.

Amar Hadzihasanovic (Mar 19 2025 at 13:25):

I'd guess that you have yourself a category enriched in the category of 2-categories with the lax (or maybe oplax, haven't checked) Gray tensor product.

Amar Hadzihasanovic (Mar 19 2025 at 13:28):

There's recently been a preprint by Dimitri Ara and @Léonard Guetta developing some of the theory of “Gray-$\omega$-categories” which are the version of this where you replace “2-category” with “$\omega$-category”.

Amar Hadzihasanovic (Mar 19 2025 at 13:30):

They make a quite strong declaration:

This theorem is an instance of an idea we would like to put forward: every functorial construction on ω-categories should be promoted to a Gray ω-functor.

So if they are correct, things like the one that you are dealing with should be quite naturally occurring.

Chad Nester (Mar 19 2025 at 15:07):

Interesting. Thanks!

David Kern (Mar 19 2025 at 15:16):

And having a Gray-category does mean that you have—as you suspected—an intercategory, as Grandis–Paré showed (in section 5) that Gray-categories can be seen as intercategories which are trivial in the transverse and one of the other two directions.
The strict Gray-$\omega$-categories should also be "degenerate" examples of what they call chiral multiple categories, but I don't think this has been done anywhere.