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

[Thorsten Altenkirch <Thorsten.Altenkirch@nottingham.ac.uk>]

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Strict cats are no good because the first example of a category, the category of sets isn’t one. And we always say homSETS but not sets of objects (ok this is mainly for size but the two are not unrelated).

Btw, there are strict univalent cats but they are quite rare…

Thorsten

From: <homotopytypetheory@googlegroups.com> on behalf of Ulrik Buchholtz <ulrikbuchholtz@gmail.com>
Date: Wednesday, 7 November 2018 at 14:14
To: Homotopy Type Theory <homotopytypetheory@googlegroups.com>
Subject: Re: [HoTT] Re: Precategories, Categories and Univalent categories

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.
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