Re: [HoTT] Re: Precategories, Categories and Univalent categories

[Ulrik Buchholtz <ulrikbuchholtz@gmail.com>]

[-- Attachment #1.1: Type: text/plain, Size: 1604 bytes --]

On Wednesday, November 7, 2018 at 2:58:28 PM UTC+1, Thorsten Altenkirch 
wrote:
>
> As I tried to say, I find that precategory is the novel concept, and that 
> both strict category and univalent category should be familiar to category 
> theorists. (They have a mental model for when one notion is called for or 
> the other, but we can make the distinction formal.)
>
>
> This is too clever!
>
>  
>
> If you just transcribe the traditional definition of a category in type 
> theory you end up with what in the HoTT book is called precategory. This is 
> confusing for the non-expert even though you can justify why it should be 
> so.
>

No, you get the notion of a strict category, which in some sense is all 
that you directly have in set theory. (To get the (2,1)-category of 
univalent categories, you take the homotopy (2,1)-category of the folk 
model structure on the category(?!) of strict categories.)

As I said before, the notion of strict category is useful, and encompasses 
the examples that Erik was missing from univalent categories.

But from a HoTT/UF point of view, the notion of strict category has the 
drawback that you cannot get a strict category of sets (or groups or…) 
without assuming the axiom (a homotopical taboo) that sets cover.

-- 
You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.
To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.

[-- Attachment #1.2: Type: text/html, Size: 2176 bytes --]