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Does anything prevent a codensity monad construction for the inclusion of FinCat into Cat?
Mike’s asking about the inclusion of finitely presented $(\mathcal{X}_i, \mathcal{O}_{\mathcal{X_\infty$-groupoids, back here. Does changing ’finite’ to ’finitely presented’ make a difference to my question?
I guess there’s also the inclusion of finite sets into $Cat$ as discrete categories. That might be nearer ultracategories.
Added a reference
I was thinking ultracategories should be a kind of generalized multicategory, and it seems they (or least some form of them) are.
I don’t know why they say
we introduce a new structure called ultracategory which simultaneously generalizes the notions of topological space and of category
when Makkai had already coined the term. Seeing that Lurie’s notion is slightly different from the latter’s, there’s then the question of where this third notion fits.
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