Timeline for Reference for the Brown representability theorem in the case of locally presentable (∞,1)-categories
Current License: CC BY-SA 4.0
31 events
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| Oct 24, 2022 at 17:22 | comment | added | Kevin Carlson | @DavidRoberts Thanks for spotting that! I've replaced the target with UCLA's official repository, which I assume will be there indefinitely. | |
| Oct 24, 2022 at 17:21 | history | edited | Kevin Carlson | CC BY-SA 4.0 |
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| Oct 24, 2022 at 10:43 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
Wayback machine link
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| Oct 24, 2022 at 10:41 | comment | added | David Roberts♦ | @Kevin your thesis link has rotted! I wanted to add a link to it under your recent answer about derivator axioms, and searching led me here... I found the Wayback machine had a copy, so I popped the link to it in. | |
| Feb 19, 2021 at 1:16 | comment | added | Kevin Carlson | @DavidRoberts I was wondering about that! I just use the copy from Ravenel’s site, but thanks for being less lazy than me. | |
| Feb 19, 2021 at 1:02 | comment | added | Kevin Carlson | @DmitriPavlov I quite agree that the gap between the basic intuition and the explicit counterexample is unexpectedly large. I remain hopeful there’s some way to make this all much easier. As for your question, no, I don’t claim that you can promote the cogroup generators from the $\alpha$-presentable to the finitely presentable $\infty$-category. But I don’t know for sure that you can’t. | |
| Feb 19, 2021 at 0:08 | comment | added | David Roberts♦ | No worries. I added the doi link for your latest edit (it doesn't turn up on the first page of a google search for me, so perhaps Elsevier is not trying to SEO Topology so much, now it's not a current journal and everything in it can be read for free) | |
| Feb 19, 2021 at 0:05 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
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| Feb 19, 2021 at 0:03 | comment | added | Dmitri Pavlov | Thanks, this is all quite instructive and I did not expect the counterexample for α-filtered colimits to be so complicated. Just to confirm, does the new version imply the desired representability statement for any locally α-presentable (∞,1)-category with a generating set of α-compact cogroups? The part that is not entirely obvious to me is whether any such category is a reflective localization of a locally finitely presentable category with a generating set of compact cogroups, i.e., whether we can choose localizations so that a generating set of compact cogroups exists. | |
| Feb 18, 2021 at 23:13 | history | edited | Kevin Carlson | CC BY-SA 4.0 |
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| Feb 18, 2021 at 23:06 | history | edited | Kevin Carlson | CC BY-SA 4.0 |
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| Feb 18, 2021 at 22:49 | comment | added | Kevin Carlson | @DmitriPavlov Sure, though unfortunately I don't think I can do stable. I'll add this to the answer. | |
| Feb 18, 2021 at 22:34 | comment | added | Dmitri Pavlov | @KevinArlin: Sorry, I misunderstood your last comment. This sounds like a potential counterexample to the proposed formulation in the main post. Do you have a simple example that would show this? (Preferably for (∞,1)-categories satisfying the assumptions in the main post, e.g., locally presentable stable (∞,1)-categories?) | |
| Feb 18, 2021 at 21:56 | comment | added | Kevin Carlson | @DmitriPavlov Everything I said two of my comments ago was in reference to the $(\infty,1)$-colimits. Those indexed by uncountable ordinals are simply not sent to weak limits by representable or potentially representable functors into sets. | |
| Feb 18, 2021 at 21:32 | comment | added | Dmitri Pavlov | @KevinArlin: Yes, that's what I mentioned in my previous comment, filtered colimits in Ho(C) are different from filtered (∞,1)-colimits in C. However, it seems that from a practical perspective, it is (∞,1)-colimits that matter for the formulation of the Brown representability theorem; in particular, Lurie's formulation says that certain types of (∞,1)-colimits in C map to weak limits in Set, and my proposed formulation extends Lurie's statement by adding another type of an (∞,1)-colimit, namely, ordinal-indexed (∞,1)-colimits. | |
| Feb 18, 2021 at 20:35 | comment | added | Kevin Carlson | Dan and I address the natural next question: given that the colimits from the model won’t work, are there any choices at all of weak colimits in the homotopy category that might serve for the inductive step in the representability proof? We give a diagram for which no weak colimit thinks finite complexes are compact, which shows that this attempt to salvage the construction also fails. | |
| Feb 18, 2021 at 20:35 | comment | added | Kevin Carlson | @DmitriPavlov Yes, the problem appears already with ordinal-indexed colimits. Your construction works fine if your assumptions are satisfied, it’s just that they’re never satisfied—colimits indexed by $\alpha$ will never be sent to weak limits in Set. You can see this easily by considering a representable functor: it would say that a naively homotopy commutative cone under an $\alpha$-indexed diagram should always cohere into some map from the colimit, which is obstructed by (in the stable situation, at least) the derived functors of lim. | |
| Feb 18, 2021 at 20:10 | comment | added | Dmitri Pavlov | To add to the previous comment, I just realized that my formulation really meant filtered (or ordinal-indexed) colimits in the (∞,1)-category C, not in its homotopy category Ho(C). I clarified this aspect in the main post. | |
| Feb 18, 2021 at 20:06 | comment | added | Dmitri Pavlov | It should probably be mentioned that the relevant case of α-filtered colimits here is colimits indexed by ordinals, since this is precisely what we need for transfinite compositions. Does your observation also apply to (∞,1)-colimits indexed by ordinals? The specific construction I have in mind would work like in Lurie's proof, but would use transfinite induction, and the passage to limit ordinals would be achieved by invoking the property about α-indexed (∞,1)-colimits mapping to weak limits. (Which part of your paper with Christensen addresses this issue, by the way?) | |
| Feb 18, 2021 at 19:37 | comment | added | Kevin Carlson | Sorry for missing your intent with the question. The bolded question suggested to me quite strongly that you weren't interested in restricting to the special case of cogroup generators. | |
| Feb 18, 2021 at 19:35 | comment | added | Kevin Carlson | @DmitriPavlov No, I don't claim that--the third paragraph is describing how you might try to prove a statement along your lines, and that such an approach won't work because weak $\alpha$-filtered colimits in the homotopy category behave badly. In fact, $\alpha$-filtered $(\infty,1)$-colimits don't need to be respected at all by functors to sets (or groupoids) which respect coproducts and $(\infty,1)$-pushouts. In particular, they have too many coherence conditions to become even weak colimits in the homotopy category. | |
| Feb 18, 2021 at 19:24 | comment | added | Dmitri Pavlov | Moving on to mathematics, in your third paragraph you seem to imply that any functor of the type discussed in the main post that sends coproducts to products and (∞,1)-pushouts to weak pullbacks also sends α-filtered (∞,1)-colimits to weak α-cofiltered limits? How can one see this and is there a reference for such a statement? | |
| Feb 18, 2021 at 19:20 | comment | added | Dmitri Pavlov | "since you didn't ask for any pointedness etc in your categories": I did ask for it when I said "satisfying certain conditions" in the first paragraph and then referenced Theorem 1.4.1.2 in Lurie's Higher Algebra. I guess this should have been made more explicit, though, so an explicit statement now appears in the main post. | |
| Feb 18, 2021 at 19:16 | comment | added | Kevin Carlson | @TimCampion Rosicky's version relies on Neeman's theorem for well-generated triangulateds, which is the only proof of any kind of $\alpha$-Brown representability I'm aware of that doesn't use an embedding in something finitely presentable a la my argument above. So, yes, I'd say there is Brown's theorem, though I wouldn't say it's false in general but rather that not every homotopy 1-category satisfies its assumptions, and Neeman's theorem, and they aren't directly comparable. | |
| Feb 18, 2021 at 19:13 | comment | added | Kevin Carlson | @DmitriPavlov No, I didn't (mean to) claim these are counterexamples to Lurie, Jardine, Neeman, or anybody. That was my point in the sentence "Arguably, the whole focus..." They're counterexamples to a literal reading of what you wanted, though, since you didn't ask for any pointedness etc in your categories. You can get an $\alpha$-version of Brown in the pointed world as a corollary to Lurie's version, or what have you, by the same argument I gave that applies in the unpointed world. | |
| Feb 18, 2021 at 17:13 | comment | added | Dmitri Pavlov | @TimCampion: Given the continuous confusion as to what exact formulation is being meant, I added a precise statement to the original post. (I did give a precise reference to Lurie, but apparently it wasn't enough.) | |
| Feb 18, 2021 at 16:37 | comment | added | Dmitri Pavlov | I was aware of the unpointed counterexamples when I wrote the question, e.g., the work of Heller and also your work with Christensen. What I don't understand, though, is your claim that these are counterexamples to Theorem 1.4.1.2 in Lurie's Higher Algebra (or Jardine's version). Indeed, the very requirement of the existence of a cogroup structure on generators seems to immediately exclude all of your counterexamples. | |
| Feb 18, 2021 at 15:05 | comment | added | Tim Campion | Just thinking out loud in case I'm missing something: this apparently means that there are at least 2 meanings of "abstract Brown representability" in the literature -- the direct generalization of the usual statement on the one hand, and Rosicky's version on the other hand. The former is false in general (but true for certain categories -- perhaps stably for instance?), and the latter is true. | |
| Feb 18, 2021 at 13:07 | comment | added | Kevin Carlson | Thanks for the edits, @DavidRoberts. | |
| Feb 18, 2021 at 7:24 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
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| Feb 18, 2021 at 6:00 | history | answered | Kevin Carlson | CC BY-SA 4.0 |