[HoTT] "type-theoretic model structures"

[HoTT] "type-theoretic model structures"

[Thomas Streicher]

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

[Michael Shulman]

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

[Thomas Streicher]

Haven't taken pains to examine Andre's treatise at least for the old
display map categories there was no axiom assuring that for every
object X there there is a chain of display maps from X to 1.
So tribes/display map cats are more general model cats, isn't it?

As a model of type theory I don't see any need to have infinitary
axioms as are common in model cats.

Thomas

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

[Michael Shulman]

Every object in a tribe is fibrant.  (A tribe is not just a display
map category; it also has the categorical structure corresponding to
identity types.)  For purposes of modeling type theory, the
non-fibrant objects are of course irrelevant, since every concrete
context does have a chain of display maps to 1.  And yes, of course,
one doesn't need infinitary structure to model type theory; as I said,
that's one of the differences between a tribe and a type-theoretic
model category, that the latter has infinitary structure but the
former doesn't.

On Mon, Feb 18, 2019 at 12:30 PM Thomas Streicher
<streicher@mathematik.tu-darmstadt.de> wrote:
>
> Haven't taken pains to examine Andre's treatise at least for the old
> display map categories there was no axiom assuring that for every
> object X there there is a chain of display maps from X to 1.
> So tribes/display map cats are more general model cats, isn't it?
>
> As a model of type theory I don't see any need to have infinitary
> axioms as are common in model cats.
>
> Thomas
>
> --
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Re: [HoTT] "type-theoretic model structures"

[Thomas Streicher]

I don't see why identity types require every object to be fibrant.
But Andr'e has this requirement also for clans which really are
display map cats with a few more additional requirements.

Thomas

> Every object in a tribe is fibrant.  (A tribe is not just a display
> map category; it also has the categorical structure corresponding to
> identity types.)  For purposes of modeling type theory, the
> non-fibrant objects are of course irrelevant, since every concrete
> context does have a chain of display maps to 1.  And yes, of course,
> one doesn't need infinitary structure to model type theory; as I said,
> that's one of the differences between a tribe and a type-theoretic
> model category, that the latter has infinitary structure but the
> former doesn't.

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

[Michael Shulman]

Identity types do not by themselves require all objects to be fibrant.
Sorry, my parenthetical was intended to mention *another* way in which
tribes are more specific than display map categories, since you seemed
to be conflating the two.  I think a clan is just a display map
category in which all objects are fibrant.  A tribe adds to this a
weak factorization system suitable for modeling identity types.

On Mon, Feb 18, 2019 at 12:57 PM Thomas Streicher
<streicher@mathematik.tu-darmstadt.de> wrote:
>
> I don't see why identity types require every object to be fibrant.
> But Andr'e has this requirement also for clans which really are
> display map cats with a few more additional requirements.
>
> Thomas
>
> > Every object in a tribe is fibrant.  (A tribe is not just a display
> > map category; it also has the categorical structure corresponding to
> > identity types.)  For purposes of modeling type theory, the
> > non-fibrant objects are of course irrelevant, since every concrete
> > context does have a chain of display maps to 1.  And yes, of course,
> > one doesn't need infinitary structure to model type theory; as I said,
> > that's one of the differences between a tribe and a type-theoretic
> > model category, that the latter has infinitary structure but the
> > former doesn't.
>
> --
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