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• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeJan 8th 2019

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeApr 14th 2020

Does anything prevent a codensity monad construction for the inclusion of FinCat into Cat?

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeApr 14th 2020

Mike’s asking about the inclusion of finitely presented $\infty$-groupoids, back here. Does changing ’finite’ to ’finitely presented’ make a difference to my question?

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeApr 14th 2020

I guess there’s also the inclusion of finite sets into $Cat$ as discrete categories. That might be nearer ultracategories.

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeApr 16th 2020

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeJun 23rd 2020

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.

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeJun 23rd 2020

Added a little more detail. Also that the Clementino-Tholen concept is different.