You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
Kevin Arlin (Jul 14 2023 at 02:54):Riehl and Verity's book on infinity-category theory is framed by the concept of an -cosmos, a kind of simplicially enriched category that behaves in certain ways like an -category of -categories.
But only in certain ways, because the axioms on an -cosmos are really quite weak: the hom-simplicial sets must be quasicategories, there are simplicially enriched products, and cotensors with simplicial sets; then, there's a class of "isofibrations", which must be representably isofibrations of quasicategories, closed under composition, products, pullback, limits of countable towers, and cotensoring with monos of simplicial sets (i.e. the Leibniz cotensor of an isofibration by a mono is still an isofibration.)
The interesting thing for me is that a complete 2-category seems to have all these properties. becomes enriched over quasicategories by taking the nerve of its hom-categories; products and terminal objects arise from those in the 2-category (in the enriched sense); finite completeness is enough, by a paper of Lack, to get a model structure where the fibrations are the representable isofibrations, and any right class in a weak factorization system has all the properties required of the isofibrations of an -cosmos by general nonsense, except for the last condition on cotensoring with monos of simplicial sets.
Riehl and Verity do prove that the 2-category of categories becomes an -cosmos in this way, including a direct technical argument for the cotensoring condition. But then it seems to me we can get this tricky condition into any complete 2-category with its representable isofibrations representably, since the cotensors for are sent to cotensors in and similarly for the pullback defining the Leibniz cotensor.
In short, it looks to me as if every complete 2-category is an -cosmos when equipped with its representable isofibrations. This would be pretty amazing, because Riehl and Verity go on to construct a virtual equipment out of whose underlying 2-category is the homotopy 2-category of which is just itself in the degenerate case discussed above. The general idea is that a 2-category isn't really enough to do category theory, you need a virtual equipment at least, but this would show that very light assumptions allow you to do all of formal category theory in any 2-category. I was hoping for a sanity check since this doesn't seem to be mentioned in the book and smells like too strong a conclusion. Anybody have a reaction?
Mike Shulman (Jul 14 2023 at 04:20):Small correction: Lack's model structure requires finite cocompleteness too. But you don't need the whole model structure for this argument.
Mike Shulman (Jul 14 2023 at 04:22):It is true that every finitely complete 2-category gives rise to a virtual equipment. Geoff and I observed this in section B.14 of our 2010 paper, but the fundamental fact (predating the notion and terminology of "virtual equipment") goes back much further, e.g. Street's "Fibrations and Yoneda's lemma in a 2-category" (1974) which was an important inspiration for Riehl-Verity.
Mike Shulman (Jul 14 2023 at 04:26):So this does give you a way to do a fair amount of formal category theory in any 2-category. One issue is that you may not be able to compose two-sided discrete fibrations, i.e. your virtual equipment may not be an equipment. This is less problematic than people used to think, but it does sometimes matter. Of course, you may also want to have "presheaf objects", but those are an extra assumption in a virtual equipment too.
Mike Shulman (Jul 14 2023 at 04:31):The more important issue with this approach is that the virtual equipment you get is not always "correct". In particular, for a nice monoidal category , the 2-category of -enriched categories is finitely complete, but the internal two-sided discrete fibrations therein are not the same as -enriched profunctors.
Street also realized ("Fibrations in bicategories", 1980) that you can fix this by dualizing: the -enriched profunctors can be found inside as the two-sided codiscrete cofibrations, which can be defined in any finitely cocomplete 2-category. However, those don't automatically form a virtual equipment: they form a covirtual coequipment, which is much less useful than a virtual equipment for doing formal category theory. So then you really do need to attack the problem of how to compose them to make an actual equipment. Various authors (including Street) have tried various formal ways to do this, with varying degrees of success; the nLab page [[codiscrete cofibration]] includes some references, and also describes a slightly improved approach involving a choice of factorization system.
Nathanael Arkor (Jul 14 2023 at 06:34):There's a recent paper of Bourke and Lack entitled On 2-categorical ∞-cosmoi, where they show that every 2-category with flexible limits induces an -cosmos whose distinguished class of isofibrations are the representable isofibrations supporting a normal cleavage.
Nathanael Arkor (Jul 14 2023 at 06:35):(I don't know how this relates to Mike's answer, though.)
Nathanael Arkor (Jul 14 2023 at 06:38):However, those don't automatically form a virtual equipment: they form a covirtual coequipment, which is much less useful than a virtual equipment for doing formal category theory. So then you really do need to attack the problem of how to compose them to make an actual equipment.
I had wondered about this, but not found the time to work it out, so I'm happy to discover you already thought about it.
Mike Shulman (Jul 14 2023 at 07:17):Nathanael Arkor said:
(I don't know how this relates to Mike's answer, though.)
Hmm... after glancing at the paper, I think what's going on is that a given 2-category could be made an -cosmos in more than one way, by choosing some class of representable isofibrations that among other things are closed under (strict) pullbacks. In a 2-category with (strict) finite limits, I expect you can choose all the representable isofibrations, and the resulting virtual equipment would then be the one I mentioned. In a 2-category with only flexible finite limits, not every representable may have strict pullbacks, so this may not work -- but Bourke and Lack have a lemma that every normal isofibration does have strict pullbacks in a 2-category with flexible finite limits, so you can choose these to be the isofibrations in an -cosmos structure. If a 2-category has finite limits (hence flexible finite limits) and not every isofibration is normal, this might give two different -cosmos structures and hence two different virtual equipments. However, in many naturally-occurring 2-categories every isofibration is normal (although this is a form of excluded middle).
Kevin Arlin (Jul 14 2023 at 17:36):Oh, that's really great, Mike, synthesizing a bunch of things I almost knew for me. Thanks a lot.
Kevin Arlin (Jul 14 2023 at 17:40):And oh my Nathanael, I can't believe I missed that yesterday–I was literally on Steve Lack's research page. I guess his official page doesn't list preprints.
Kevin Arlin (Jul 14 2023 at 17:44):Speaking of multiple cosmos structures, the 2-category I actually have in mind is that of multicategories. The modules in the equipment from your paper with Cruttwell, Mike, involve heteromorphisms with multiple inputs, which stops a module from having a Grothendieck construction fibered over On the other hand the modules coming from the plain 2-category structure and representable isofibrations involve only unary heteromorphisms but by definition have nice Grothendieck constructions and stuff. It's not clear to me yet whether this equipment is "wrong" in something like the same sense as the naive equipment for V-categories; for instance I'm particularly trying to figure out how initial multifunctors differ in the two equipments.
Mike Shulman (Jul 14 2023 at 17:47):I've never found an equipment of multicategories that I was completely satisfied with. Good luck!