Re: [HoTT] "type-theoretic model structures"

[Michael Shulman <shulman@sandiego.edu>]

Regarding the first sense, there is an important distinction between
the model category on the one hand and the underlying tribe of fibrant
objects on the other.  In particular, a tribe is defined by *finitary*
structure and operations, and can be small (e.g. the syntactic
category of type theory is a tribe), whereas a model category is
necessarily large, with infinitary colimits and so on.  Also, a model
category (usually) contains non-fibrant objects, whereas a tribe
doesn't.  So we do need two different terms.

In my paper "Univalence for inverse diagrams and homotopy canonicity"
I referred to them as "type-theoretic model categories" and
"type-theoretic fibration categories" respectively.  Nowadays the
momentum seems to be with "tribe" for the latter, which among other
advantages is eleven syllables shorter.  If we want to maintain some
terminological parallelism and avoid confusion with Sattler/Cisinski
model structures, we could start referring to the former as "tribal
model categories".

Note by the way that whatever we call these model categories, there is
not (at the moment) really a unique definition of them, but a
collection of properties that tend to be added or subtracted as needed
(some are listed at
https://ncatlab.org/nlab/show/type-theoretic+model+category).


On Mon, Feb 18, 2019 at 2:25 AM Thomas Streicher
<streicher@mathematik.tu-darmstadt.de> wrote:
>
> I was a bit imprecise in my mail about "type-theoretic model structures".
> I think there are (at least) 2 different uses of the word. The first
> is as certain model structures whose fibrations give rise to a model
> of type theory. In the old days these were called "categories with
> display maps" which have got rebaptized by Joyal as "tribes" which is
> a nice name since it's about families which interact in a certain way.
>
> Another use seems to be for particular model structures on categories
> (of presheaves) whose fibrations provide a model of type theory. Sometimes,
> e.g. for simplicial and cubical sets these are minimal Cisinski model
> structures where "minimal" means "fewest anodyne cofibrations", typically
> generated by open box inclusions.
>
> But not every (minimal) Cisinski model structure provides a model of
> type theory and, thus, it is not at all a good idea to call them
> "type-theoretic model structues".
>
> Thomas
>
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