On Friday, November 9, 2018 at 5:43:15 AM UTC+1, Andrew Polonsky wrote:
>
> The intuitive notion of a category is given by the dependently-sorted 
> algebraic theory of categories.
>

Maybe for you! But not for me: I don't even know what a dependently-sorted 
algebraic theory is, where and how such a thing has models, nor what the 
particular theory you're thinking of is.

But that reminds me of a MATHEMATICAL question. Consider the well-known 
essentially algebraic theory T whose models in Set are strict categories. 
What are the models of T in infinity-groupoids? (Or in an arbitrary 
(infinity,1)-topos?)
 

> However, more modern type theories, like HoTT or CubicalTT, have a 
> different, "native" conception of equality, so the "natural" notion of 
> category in these theories will accordingly be different as well.
>

In cubical type theories, the two notions of equality (path types and the 
inductive Id-family) are equivalent, so they give equivalent notions 
wherever they are used.

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