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Stream: learning: questions

Topic: Question about a perspective on fibrations

Patrick Nicodemus (Jan 07 2024 at 01:44):

Consider the 2-category $\mathbf{Cat}$ of small categories. There are multiple 2-category structures on $\mathbf{Cat}/\mathbb{B}$ for a choice of small category $\mathbb{B}$; there is the obvious "strict" one or we could also consider the 2-category where morphisms $p\Rightarrow q$ are given by oplax cells. A Grothendieck fibration $p$ over $\mathbb{B}$ is an object in $\mathbf{Cat}/\mathbb{B}$ with the property that every oplax cell into $p$ can be strengthened to a strict cell. I think this should make the category $\mathbf{Cat}/\mathbb{B}_{\mathrm{strict}}(q,p)$ into a coreflective subcategory of $\mathbf{Cat}/\mathbb{B}_{\mathrm{oplax}}(q,p)$ when $p$ is a fibration, and I was wondering if there were other interesting situations where you have two reasonable 2-category structures on the same set of objects, with the first one being a sub-2-category of the second one, and a class of fibrant objects characterized by the fact that the inclusion functor on homs admits a right adjoint.
I think this is maybe related to the fact that fibrations are algebras of Kock-Zoberlein monads but I really don't know anything about those, I just learned about them the other day.