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

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

Something that only satisfies the axioms of a category but whose homs are not sets I would call a "wild category".

So yes a category needs to have homsets otherwise you have to give it a number.  And -1-categories already have a name they are called preorders. 

I think the interesting notion for HoTT are categories whose Homs are types and which fullfill all the coherence conditions, i.e. they correspond to (\infty,1)-categories. Nicolai and Paolo gave a definition of univalent (\infty,1)-cats in 2-level HoTT as "complete Segal types". From this one can aslo derive non-univalent (\inft,1)-cats but in this case it is easier to define the univalent version because then you don't have to deal with degeneracies.

Thorsten

On 07/11/2018, 13:53, "homotopytypetheory@googlegroups.com on behalf of Thomas Streicher" <homotopytypetheory@googlegroups.com on behalf of streicher@mathematik.tu-darmstadt.de> wrote:

    It would appear to me as natural to call "categories" the most general
    notion and "univalent" those ones which validate the univalence axiom.
    In any case the word "category" should not imply univalence.
    
    But there is a point that one might call the most general notion
    "precategory" and "categories" those precategories whose hom types
    validate UIP, i.e. are sets.
    
    But then univalent categories are not categories but just precategories.
    Why not call the general notion "\infty-category" as usual?
    
    Thomas
    
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