I'm a bit confused by your message, Peter: HoTT doesn't have a naive set 
interpretation and is inconsistent with UIP, so I'm not sure how that 
should guide us. (Maybe if we're working in good old (agnostic?) MLTT?)

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.)

On Wednesday, November 7, 2018 at 12:10:10 PM UTC+1, Peter LeFanu Lumsdaine 
wrote:
>
> Ulrik’s email nicely lays out the three key notions (pre-category, strict 
> category, univalent category), and the argument for the 
> Ahrens–Kapulkin–Shulman / HoTT book terminology, with “category” meaning 
> “univalent category” by default.
>
> For my part I lean the other way: I think it’s too radical to use 
> “category” for a definition which doesn’t come out equivalent to the 
> traditional definition under the naïve set interpretation (or under the 
> addition of UIP to the type theory).  Choosing terminology that actively 
> clashes with traditional terminology makes it much harder to compare 
> statements in HoTT with their classical analogues, and see what difference 
> HoTT really makes to the development of topics.
>
> Based on that criterion, I strongly prefer taking category to mean 
> “precategory”.  A big payoff from this is that if you formalise something 
> using “category ” to mean “precategory” in type theory without assuming UA, 
> then you can read the result either as valid in HoTT, or (under the 
> set-interpretation) as ordinary arguments in classical category theory, 
> with all the terms meaning just what they traditionally would.
>
> Univalence of categories is an important and powerful property, but not an 
> innocuous one; it changes the character of the resulting “category theory” 
> in interesting ways.  Making the restriction to univalent categories tacit 
> is misleading to readers who aren’t fully “insiders”, and obscures 
> understanding its effects.
>
> –p.
>
>

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