I am thinking about embedding of categories into topological spaces (or even manifolds).
I have found 2 connections:
one can construct the nerve/groupoid from a category and then use homotopy hypothesis to find the space for this groupoid. It is the representation of category that preserves the homotopical properties only.
I have found this article "Embedding of a free cartesian-closed category into the category of sets" by Djordje Čubrić, where he embeds $\mathrm{CCC}$ into the category of sets, and despite the fact that this is not a topological space as the category of open sets, but still I'd like something along these lines.
From the one side there is rich culture on efforts to embed graphs into manifolds (e.g. in the deep learning). From the other side there are some scarce (and old) results about embeddings of categories I mentioned.
Maybe someone suggest some topics, ideas, research efforts that advance those embeddings?