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Bruno Gavranović (Sep 05 2023 at 08:00):A lot is known about existence of pushouts in some notable categories like and . Is anything known about whether all pushouts exist in the category of of 2-categories and lax functors?
I've had a cursory look online and on nLab but couldn't find any references.
Mike Shulman (Sep 05 2023 at 13:45):Almost certainly they don't, but it would take some work to find a counterexample.
Bruno Gavranović (Sep 06 2023 at 05:47):What makes you say that? I know lax functors often behave badly, but I can't tell what goes wrong with pushouts (or say, colimits here in general), but doesn't go wrong in .
Bruno Gavranović (Sep 06 2023 at 05:49):I'm aware that this category hosts 2-dimensional things (2-cats, lax functors), and I imagine it might be too strict to ask of it to have 1-pushouts.
On the other hand, I've been surprised before, for instance, when learning about what the right notion of a 2-cell here is.
Mike Shulman (Sep 06 2023 at 06:17):In general, categories of algebras for a 2-monad and lax morphisms don't have all limits and colimits, even in a 2-dimensional sense. Intuitively, I tend to think of this as being "because" the morphisms in these categories are not just those that "preserve" some structure, but are themselves structure.