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 -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. For more options, visit https://groups.google.com/d/optout.

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 > > -- > You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. > To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. For more options, visit https://groups.google.com/d/optout.

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 -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. For more options, visit https://groups.google.com/d/optout.

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 > > -- > You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. > To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. For more options, visit https://groups.google.com/d/optout.

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. -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. For more options, visit https://groups.google.com/d/optout.

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. > > -- > You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. > To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. For more options, visit https://groups.google.com/d/optout.