On Fri, 9 Nov 2018, at 10:18 AM, Ulrik Buchholtz wrote:
> 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.

Let's say we define a *dependently-sorted algebraic theory* to just be a 
CwF. A model for it is a CwF-morphism into Set (or more generally into a 
CwF). The theory of categories would then be the opposite of Cat_{fp} (I 
mean finitely presentable, *not* locally finitely presentable), or 
alternatively you can define it inductively as the syntax of a type theory 
(but then you have to prove an initiality theorem!).

I'm not sure how this kind of presentation translates to ∞-categories 
though.

> 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?)

Nice question!

I guess by model here you mean a left exact ∞-functor T → Space, where T is 
regarded as an ordinary category with finite limits. I think such models 
are equivalent to Segal spaces, but without completeness. Or at least, I 
don't see where completeness would come from.

I find it quite an illuminating way to look at Segal spaces, by the way. 
You get the space of n-simplices by mapping the iterated pullback of the 
object of arrows in T (i.e. the object that stands for sequences of 
composable arrows). The Segal condition is the preservation of that 
pullback. It's essentially the same idea as in the definition of Γ-Spaces.

Best,
Paolo

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