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Daniel Teixeira (Nov 22 2022 at 19:59):On A folk model structure on omega-cat we find the following argument:
The forgetful functor U : ωCat → Glob is finitary monadic [2] and Glob is a topos of presheaves on a small category: therefore ωCat is complete and cocomplete.
What's the result being used here? I don't know what a finitary monadic functor is btw, so it might be the case that is a triviality from the definition.
fosco (Nov 22 2022 at 23:10):A finitary functor is a functor that commutes with every (finitely) [[filtered colimit]].
There is indeed a theorem that is being used here and says more or less exactly what you state -usually, it is stated for monadic functors with codomain the category of sets, but it is still valid for presheaf categories. If is monadic, limits in are created by from limits in , and colimits can be computed in and then "lifted" to using that has a left adjoint.
Nathanael Arkor (Nov 22 2022 at 23:45):The result appears as Theorem 4.12 on the nLab page for [[locally presentable category]].
John Baez (Nov 24 2022 at 14:07):The rough idea of a 'monadic' functor, btw, is that it's a right adjoint that's the forgetful functor from the category of algebras of some monad on to itself. And then it's 'finitary' if this monad describes an algebraic gadget with finitary operations.
John Baez (Nov 24 2022 at 14:09):So, at an intuitive nonrigorous level, it's utterly plausible that the forgetful functor U : ωCat → Glob is finitary monadic. This just says that an ω-category is the algebra of some monad on globular sets, and that an ω-category is a globular set equipped with a bunch of finitary operations obeying some equations.