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Is there any natural condition on a pullback of categories-with-fibrant-objects ensuring that it induces a pullback of their homotopy -categories? Obviously one such condition that works for any pullback of relative categories is that it is a homotopy pullback in the model structure on relative categories — e.g. if the relative categories are fibrant and one of the functors is a fibration — but that is not very explicit. Is there any more explicit condition in the special case when we have categories with fibrant objects?
There is a criterion in my thesis, namely that it is a pullback along a “fibration” (Definition 1.9). There is a fibration category of fibration categories and such pullbacks are homotopy pullbacks in there. The functor I introduce on p. 44 is an exact functor from this fibration category to the (fibration category underlying the) Joyal model structure so it induces a homotopy pullback of resulting quasicategories.
Excellent, thank you! This is exactly the sort of thing I was looking for, and it’s not impossible that one of the functors I’m looking at is a fibration in your sense. It’s certainly an isofibration, and I think it has the property that any acyclic fibration lifts to an acyclic fibration , which I think means that if it has your lifting property for factorizations then it also has your lifting property for pseudofactorizations. Moreover, I think I know that it lifts certain (weak equivalence, fibration) factorizations (namely, those for which the weak equivalence is an acyclic cofibration in some underlying model category — I don’t know whether the whole model structure lifts to either of the fibration categories in question). (I’m saying “I think” here because I haven’t actually written down these arguments yet, which always makes them suspect.)
So, I wonder whether it is essential for your results that all factorizations be liftable? That seems to be used in your Prop. 1.10 (C5), but that could perhaps be avoided in my case by either knowing that a particular pullback exists for other reasons, or knowing that the special factorizations that lift happen to also be preserved by the functor . Your Prop. 3.4, which is the result I really care about, doesn’t seem to depend on properties of fibrations other than that pullbacks along them exist. So maybe it is ok?
I am also curious (and can’t tell from my cursory glance over the thesis): where is the lifting property for pseudofactorizations used?
I think it has the property that any acyclic fibration lifts to an acyclic fibration , which I think means that if it has your lifting property for factorizations then it also has your lifting property for pseudofactorizations.
That’s right.
So, I wonder whether it is essential for your results that all factorizations be liftable? That seems to be used in your Prop. 1.10 (C5), but that could perhaps be avoided in my case by either knowing that a particular pullback exists for other reasons, or knowing that the special factorizations that lift happen to also be preserved by the functor .
That also seems OK.
Your Prop. 3.4, which is the result I really care about, doesn’t seem to depend on properties of fibrations other than that pullbacks along them exist. So maybe it is ok?
The question seems to be: is a pullback along such a not-quite-fibration still a homotopy pullback? Proposition 3.4 might be useful here, but it doesn’t quite work as stated since it’s not clear that a not-quite-fibration is carried to a fibration in Joyal’s model structure.
It may be that the proofs of Propositions 3.12 and 3.14 still work in this case, but deciding that for sure may take some effort since these arguments build on a general theory of fibrations. I don’t see it right away.
As a side note, Section 1.3 contains some general ways of recognizing fibrations, specifically Lemmas 1.20 and 1.21. Perhaps they are useful in your situation.
I am also curious (and can’t tell from my cursory glance over the thesis): where is the lifting property for pseudofactorizations used?
It is really only used in the proof of Proposition 1.11 which is needed to show that acyclic fibrations are stable under pullbacks. This is a little subtle and I don’t really have a good explanation for why we want these pseudofactorizations other than they happen to work. The argument boils down to the property (App2) of Proposition 1.8. These “Approximation Properties” were originally introduced by Waldhausen as a criterion for equivalence on K-theory and his version had just a triangle with a single weak equivalence. The extra equivalence was introduced by Cisinski who needed it for the if-and-only-if statement characterizing equivalences of homotopy categories. Ultimately, I think of pseudofactorizations as a “factorizations into a cofibration and a zig-zag of two weak equivalences” and they somehow account for the fact that not every morphism in the homotopy category is induced by a single morphism of the underlying category.
Ah, right, thanks. Tracing through your proofs, the first place I see a potential problem is that Proposition 3.12 relies (I think, although it doesn’t state this explicitly) on the fact that every acyclic fibration between (co)fibration categories is literally surjective on objects — whereas without the lifting property for pseudofactorizations, Proposition 1.11 doesn’t apply, so that Lemma 1.21(3) only gives us a not-quite-fibration that is also a weak equivalence, rather than an acyclic fibration. Does that seem right?
By the way, this is the first time I’m looking carefully at your construction, and it seems quite neat, though I don’t fully understand it yet. One thing that currently puzzles me is: suppose I have a good old left homotopy in a cofibration category (or a right homotopy in a fibration category) defined in terms of a cylinder object; how does it get incarnated as a 2-simplex in your ?
A left homotopy defines a diagram of shape (= the poset of non-degenerate simplices of ) in which most of the arrows are degenerate. There’s a nice picture on p. 45; there, take , take to be the cylinder, and take . Then take a Reedy-cofibrant resolution of this diagram to get a 2-simplex of the quasicategory of frames.
Zhen Lin is exactly right about left homotopies. For some reason this fact didn’t make it as a lemma of its own, but the argument is spelled out in the latter half of the proof of Lemma 4.10 (using notation rather specific to the situation there). In fact, it is stated for generalized left homotopies that also work when the target is not fibrant (Definition 1.2(2)).
Proposition 3.12 relies (I think, although it doesn’t state this explicitly) on the fact that every acyclic fibration between (co)fibration categories is literally surjective on objects
Yes, and I think this is somehow unavoidable. If a functor of cofibration categories induces an inner isofibration after applying , then it is “surjective on weak equivalence classes”. Meaning that if some object is in its image, then all weakly equivalent ones also are (on the nose).
Lemma 1.21(3) only gives us a not-quite-fibration that is also a weak equivalence, rather than an acyclic fibration.
The functor in the conclusion of this lemma is certainly a weak equivalence since is an honest acyclic fibration. Whether it is a not-quite-fibration will really depend on what “not-quite-fibration” means exactly.
Thanks Zhen; I was staring at that diagram, but failed to guess which of the objects should be the cylinder. I guess I must have been asleep, because obviously it has to be the one that’s equivalent to the domain.
Can the proof of Proposition 3.12 be beta-reduced to make it more explicit? E.g. suppose is a fibration and we have a 2,1-horn in , which is a resolution of a double cospan that is Reedy cofibrant — which I think means that and are cofibrations (plus similar conditions on the rest of the resolution) — together with an extension of its image in to a 2-simplex. What does the lifting of this 2-simplex to look like? Maybe if I understand what the technology is doing “under the hood”, I’ll be better able to guess whether it ought also to work with what I have.
Ah, I see what was confusing me. You described how to view a left homotopy as a 2-simplex whose 2-face is degenerate; but a homotopy between two maps in a quasicategory can equivalently be seen as a 2-degenerate 2-simplex or as a 0-degenerate 2-simplex. How do I view a left homotopy as a 2-simplex whose 0-face is degenerate?
What you are asking in #9 is a very good question. In some sense the proof of Proposition 3.12 is a little too slick for its own good and quite a lot is hidden under the hood. Originally, I tried much more direct approaches but I always ended up with absolutely horrendous diagrams to work with. I tried unravelling the proof a little bit, but I didn’t have much time to concentrate on it yesterday. I will think about it more.
Also, I don’t see a canonical way of producing a 2-degenerate homotopy. Of course you can get one by filling in a horn resulting from the 0-degenerate one. (And perhaps if I manage to expand the proof of 3.12 some semi-canonical filler of this horn will come up.)
Here is another question. So far, I don’t see how to prove that the functor between fibration categories that I’m looking at is actually a fibration in your sense. However, there’s some chance I could prove that it has the following properties:
(Here the “acyclic cofibrations” are a subclass of the weak equivalences, such that every map has a “good” factorization into an acyclic cofibration followed by a fibration — no lifting properties are assumed. Also, all the exact functors in sight preserve acyclic cofibrations.)
Would this be enough? I’m looking at your Proposition 1.11 and it seems that it might be, for that result at least: at the end of the proof, we can lift to by (4) above, and then we can lift to by (3); it doesn’t seem to matter whether we can also lift . So if that is the only place where lifting of pseudofactorizations is used, then these conditions might be enough.
I do think that this actually works and this would seem to give an answer to your original question even without revising the results of Chapter 3.
The upshot of your argument is that acyclic not-quite-fibrations are stable under pullback. It should follow that they satisfy the Gluing Lemma. So whenever you have a pullback along a not-quite-fibration you can factor it via an actual fibration and then the Gluing Lemma says that the orignal pullback is weakly equivalent to the new one so both are homotopy pullbacks. In particular, carries both to homotopy pullbacks of quasicategories. (Even though we don’t know whether of a not-quite-fibration is a fibration of quasicategories.)
Thanks! Sorry, apparently I’m being dense right now; how does the Gluing Lemma follow?
Sorry, I said that it should follow but I didn’t mean to imply that it was immediate or even that we had nailed down the necessary assumptions.
I was thinking that we should have some version of a fibration category of fibration categories with not-quite-fibrations as fibrations. (The main unresolved thing is saying what the underlying category is exactly.) However, it appears to me that the only potentially tricky axiom was stability of acyclic fibrations which your argument above addresses. The rest of the proof of Theorem 1.17 should go through almost verbatim. The the Gluing Lemma holds just as in any fibration category. Then of course there is a question of what happens exactly when we carry out factorization in my (slightly different) category. If we can arrange that this factorization doesn’t lead out of your category (and if I’m right that the proof of factorization goes through almost verbatim, it shouldn’t) the conclusion will follow.
On the other hand, there may be problems. I’m assuming that you are working with some variant of “type theoretic fibration categories”. Actually, Chris Kapulkin and I tried constructing a fibration category of tribes in the sense of Joyal and we didn’t manage to make the factorization work. However, you said that your “acyclic cofibrations” are just a distinguished class of morphisms that have no lifting properties. That’s the reason why I said that the proof may go through almost verbatim. Our problems were arising around trying to construct a path object on a tribe whose “anodyne extensions” behave correctly.
Ah, I see.
Actually, the categories I’m looking at right now are not tribes, but categories of algebras for “cofibrant” monads on a tribe, which arise when considering semantics of HITs. The limits, fibrations, weak equivalences, and acyclic cofibrations are created downstairs in the tribe, and all the functors are induced by monad morphisms. The not-quite-fibrations are the functors induced by “cofibrations” of monads.
I just glanced over your Prop 1.13 and Theorem 1.14, and so far I think I agree that the proof may go through almost verbatim.
OK. I will be very happy if this works out and turns out to be useful.
I have one more remark (I hope this is explained sufficiently clearly in my thesis but I doesn’t hurt to reiterate). The construction of the path object on a fibration category has a little twist. It is the category of spans of weak equivalences that are “fibrant” but not necessarily Reedy fibrant and the fibration structure is a sort of unusual “partial Reedy structure”. This is necessary for the “constant path functor” to be a well defined exact functor. The price to pay is that we lose control over “acyclic cofibrations” which was a non-issue in my original theorem. However, it is a source of great frustration when trying to generalize to tribes.
Yes, I did notice that, but thanks for emphasizing it.
Ok, I wrote down some details and I still believe it. The category in question has, as objects, fibration categories equipped with a subclass of the weak equivalences called “acyclic cofibrations” such that every map has a “good” factorization into an acyclic cofibration followed by a fibration. The morphisms are exact functors that also preserve acyclic cofibrations. The fibrations are the maps satisfying conditions (1)–(4) from #12 above; let me call them “ac-fibrations” for the nonce. I believe your proof of Theorem 1.14 goes through straightforwardly to show that this is a fibration category, once we know that acyclic ac-fibrations are pullback-stable. (We define the acyclic cofibrations in levelwise.)
Moreover, the path object projection is both a fibration and an ac-fibration (though in general, neither of those conditions seems to imply the other). Thus, any map in this category factors as a weak equivalence followed by a map that is both a fibration and an ac-fibration. So if we have a pullback of an ac-fibration , we can factor in this way and then apply the ordinary cogluing lemma in the ac-fibration fibration category to compare the pullbacks of and , where the pullback of induces a homotopy pullback of quasicategories by your result since is also a fibration. Does that seem right?
What you are describing looks exactly like the best case scenario that I was picturing in #15. I didn’t go into any details but I went through the checklist and everything seems to be fine.
So yes, I think this works.
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