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Concerning direct/inverse categories we let direct category take the lead, with a small note about the opposite concept. Is there a reason for preferring to phrase things here in terms of the inverse side? The example I’ve included is the opposite of an orbit category.
Clark Barwise wrote
The bases in which we’re most interested — and have the most to say — are atomic orbital ∞-categories that admit a conservative functor to a poset (so that they are ’EI ∞-categories’). Examples include: the orbit category of a group, the category of finite sets and surjective maps, any ∞-groupoid, the cyclonic orbit category, and the total ∞-category of any G-space.
The page should also speak of exit-path ∞-categories of stratified spaces. But then are statifications always well-founded?
I prefer to talk about covariant diagrams on an inverse category than contravariant diagrams on a direct category, since I don’t have to be constantly reversing arrows in my head: the arrows in the inverse category go the same direction as the morphisms that they induce in a diagram. This is the same reason I prefer to define a derivator as a functor $Cat^{op}\to CAT$ rather than $Cat^{coop}\to CAT$, with the represented ones being $Hom(-,C)$ rather than $Hom((-)^{op},C)$. I guess you could say I prefer to avoid “duality redexes”. (-:O But other people seem to have different preferences.
added missing redirects and hyperlinks (for instance: of course we have an entry for well-founded relation)
Presumably Mike’s Univalence for inverse EI diagrams relates somehow to the modal dependent type theory towards the end of his HoTTEST talk with its inverse categories of mode contexts.
For example, can we see the truncation modalities as arising from the building up by stages of the first article?
Perhaps I’d need to know more about this hoped for coherence theorem:
its [HoTT’s] most “direct” notion of model is actually a kind of structured 1-category (a comprehension category, category with families, etc.); a coherence theorem then (hopefully) relates these to the intended (∞,1)-categorical models. (What is an n-theory?)
I’ve been thinking on and off about some such relationship, but haven’t nailed down anything in particular yet. The uses of inverse categories are different: in the first case, type theory is modeled in the category of diagrams on one fixed inverse (EI) category, while in the second case, diagrams on varying inverse categories are used to describe the contexts in one model of (modal) type theory. My feeling is that a sort of modal type theory that “mixes” the mode theory with the object theory should be able to include “diagrams” as a sort of basic judgmental objects, and that this might be helpful in talking about semisimplicial types etc., but I haven’t managed to make it quite come out yet.
in the first case, type theory is modeled in the category of diagrams on one fixed inverse (EI) category
Does anything interesting happen by taking a (2-)category of inverse EI-categories to generate a varying collection of type theories?
Probably yes, I think that’s basically what I was talking about in #7.
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