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Nov 14, 2019 at 2:30 comment added Tim Campion I might think about some of the properties of the homotopy 2-category of $\infty$-categories used by Riehl and Verity. They show that several models are equivalent, though I don't think they give an axiomatization.
Nov 13, 2019 at 21:33 comment added Saal Hardali @KevinCarlson I also just realized a further problem that it's bot clear how to get the correct 2-morphisms i.e. the natural transformations out of this.
Nov 13, 2019 at 21:27 comment added Kevin Carlson @SaalHardali That will only get you to the right place if you restrict to cofibrant topological categories. They don't all have the correct mapping spaces.
Nov 13, 2019 at 21:17 comment added Saal Hardali @CharlesRezk Actually now when I think about this why don't I just take the 2-category of topological categories and localize w.r.t. the essentially surjective functors inducing weak equivalences on all mapping spaces. This might already give me the homotopy 2-category of $\infty$ categories and it seems like the simplest and most direct way of combining ordinary category theory and the homotopy hypothesis. I'm not sure why I didn't notice this before...
Nov 13, 2019 at 20:58 comment added Saal Hardali @CharlesRezk If I had to guess I would bet not. What I do think may be possible (probably being naive here) is to get homotopy category of $\infty$-categories by melding together in a clever way the category of topological spaces with their weak equivalences and the 2-catgory of 1-categories. We already know it can be done in a myriad of different ways many of which give equivalent resulting 2-category. The question is whether we can do this using only 2-catgorical arguments, or rather whether the construction has a certain 2-categorical universal property if you like.
Nov 13, 2019 at 20:46 comment added Charles Rezk Is there a list of properties that characterizes the homotopy 1-category of $\infty$-groupoids among all 1-categories?
Nov 13, 2019 at 18:28 comment added Andrej Bauer It looks like the Skeptic is such an annoying person nobody wants to talk to them.
Nov 13, 2019 at 16:53 history edited Saal Hardali CC BY-SA 4.0
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Nov 13, 2019 at 16:42 history edited Saal Hardali CC BY-SA 4.0
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Nov 13, 2019 at 16:41 comment added Saal Hardali @CharlesRezk I'm sorry you had to read through all that just because I was afraid of being shut down. I hope future readers will not have to go through this travesty.
Nov 13, 2019 at 16:36 history edited Saal Hardali CC BY-SA 4.0
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Nov 13, 2019 at 16:30 comment added Saal Hardali @CharlesRezk I completely agree with what you said. I was afraid if I would phrase the question as simply as I did in the title it would be dismissed as imprecise and voted to be closed. Let me add a remark at the start pointing that out.
Nov 13, 2019 at 16:27 comment added Charles Rezk I can't really figure out what you are asking here. It sounds like you are asking "Is there a list of properties that completely characterizes the homotopy 2-category of $\mathrm{Cat}_\infty$ among all homotopy 2-categories?", but if that is the case, why not just ask that? You give a long list of steps I'm supposed to follow, but I don't see why I should be constrained by that.
Nov 13, 2019 at 16:05 history edited Saal Hardali CC BY-SA 4.0
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Nov 13, 2019 at 15:56 history edited Saal Hardali CC BY-SA 4.0
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Nov 13, 2019 at 15:35 history edited Saal Hardali CC BY-SA 4.0
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Nov 13, 2019 at 15:29 history edited Saal Hardali CC BY-SA 4.0
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Nov 13, 2019 at 15:26 comment added Saal Hardali @DylanWilson I was under the impression that in this paper one starts off with a working theory of $(\infty,1)$-categories then builds on that. Maybe I'm missing your point.
Nov 13, 2019 at 15:22 comment added Dylan Wilson The unicity theorem of Barwick-Schommer-Pries (arxiv.org/pdf/1112.0040.pdf) gives a characterization of the $\infty$-category of $(\infty, n)$-categories. This $\infty$-category is cartesian closed and hence canonically enriched over itself; so we get a canonical enhancement to an $(\infty, n+1)$-category of $(\infty, n)$-categories. In other words, their unicity theorem already characterizes this $(\infty, n+1)$-category up to equivalence.
Nov 13, 2019 at 15:22 history edited Saal Hardali CC BY-SA 4.0
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Nov 13, 2019 at 15:04 history asked Saal Hardali CC BY-SA 4.0