Timeline for Homotopical algebra is not concrete
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| when toggle format | what | by | license | comment | |
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| Sep 23, 2018 at 2:21 | comment | added | Alec Rhea | More generally I wonder if the non-concreteness of these categories is a result of working in a set theoretical universe that is 'too weak' like $ZFC$, in the sense that they can be made 'class concrete' in a stronger background universe, or if something more subtle is going on here. | |
| Sep 23, 2018 at 2:17 | comment | added | Alec Rhea | Apologies for the delay, the system never pinged me - by the Cayley representation of $\mathcal{C}$ I had in mind the category whose objects are the Hom-sets of $\mathcal{C}$ and whose arrows are the functions induced by composition in Hom-sets, but the Yoneda embedding should also suffice -- the question I posed at the end was a non-sequitur wondering how exactly the Isbell condition fails. The legitimacy of these constructions seems to depend on which set theoretical universe we work in, in Ackermann set theory or $MK+CC$ they're legitimate -- see arxiv.org/pdf/0810.1279.pdf p.25-26. | |
| Jun 17, 2018 at 10:18 | comment | added | fosco | I can't deduce what you mean by 'Cayley representation' from your notation, so please explain. In case you mean the Yoneda embedding, keep in mind that the category of functors $[A,B]$ between two large categories is not legitimate because each hom-class is proper. | |
| Jun 16, 2018 at 22:51 | comment | added | Alec Rhea | Please forgive my ignorance, but I was under the impression that each category was isomorphic to its ‘Cayley representation’ which is itself a concrete category. How does this fail for the examples above? Is $S({\bf Ho}(\mathcal{M}),A)$ a proper class for some $A$? | |
| Apr 11, 2017 at 13:01 | answer | added | fosco | timeline score: 10 | |
| Mar 1, 2017 at 18:30 | comment | added | Ivan Di Liberti | That is properly where the question have been motivated. | |
| Feb 19, 2017 at 16:37 | comment | added | Qfwfq | Possibly relevant for context and motivation: mathoverflow.net/questions/21667/… and mathoverflow.net/questions/190578/… | |
| Feb 5, 2017 at 23:14 | comment | added | Tim Campion | I guess I'm naively expecting that if $C$ is a model category presenting a stable $\infty$-category $C_\infty$ with a $t$-structure, then the full subcategory $C^0$ on objects of degree 0 should be a model category presenting the heart $C^\heartsuit = C^0_\infty$ of the $t$-structure on $C_\infty$. I guess I'm assuming that this will in particular mean that that the homotopy category of $C^0$ is exactly $C^\heartsuit$. And the point is that $C^\heartsuit$, though constructed as an $\infty$-category, is in fact an ordinary category, and it is concrete in the usual sense. | |
| Feb 5, 2017 at 19:19 | comment | added | fosco | I'd like to have a reference for the same result in stable model categories (how comes that the heart ${\cal C}^\heartsuit$ is equivalent to $ho({\cal C}')$?); also, what's a $t$-structure on a stable model category, precisely? I would say it's a a normal torsion theory in the sense of Bousfield's homotopy factorization systems, but I remember it is still a non well-understood question how to define them properly. | |
| Feb 5, 2017 at 19:14 | comment | added | fosco | (cont.) let $\cal C$ be a stable presentable quasicategory; and $t=({\cal C}^\ge, {\cal C}^<)$ a $t$-structure. Now (HA 1.3.5.23) the heart ${\cal C}^\heartsuit$ is presentable abelian and then "concrete" (but what's a "concrete" $(\infty,1)$-category???), and moreover equivalent to its "fundamental" category $ho({\cal C})$. That's truly inspiring (afaict, this answers a slightly different question) | |
| Feb 5, 2017 at 19:09 | comment | added | fosco | Funny, I never tought about this. This is related to the fact that the heart of the $t$-structure is abelian, and hence equivalent to its $(\infty,1)$-category (in whatever model), right? I'm more comfortable seeing a $t$-structure as a factorization system, on stable quasicategories, as you may remember; so let's state what you said in different terms (cont.) | |
| Feb 5, 2017 at 18:55 | comment | added | Tim Campion | I think a nice source of examples of "concrete homotopical algebra" comes from the hearts of $t$-structures on triangulated categories. I'm not sure, (someone please correct me if this is wrong!) but I suspect that if $C$ is a stable combinatorial model category with a reasonable $t$-structure, then the heart of the $t$-structure ought to be the homotopy category of another combinatorial stable model category, while at the same time being a locally presentable category, and thus concrete. There are also trivial examples, e.g. $(L,C,R)$ for a wfs $(L,R)$ on $C$ (so $ho(C)$ is trivial). | |
| Feb 4, 2017 at 6:00 | comment | added | David Roberts♦ | Also: I changed your SD direct link into a doi link, since that is more permanent. | |
| Feb 4, 2017 at 5:56 | history | edited | David Roberts♦ | CC BY-SA 3.0 |
Converted ScienceDirect link to doi, and gave minimal human-readable reference to the paper
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| Feb 4, 2017 at 5:51 | comment | added | David Roberts♦ | Silly observation: In the first and third bullets of 2., you'd need some sort of non-triviality condition. And ditto for some of the others, else you can localise at the isomorphisms. | |
| Feb 3, 2017 at 16:31 | history | edited | fosco | CC BY-SA 3.0 |
added 16 characters in body
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| Feb 3, 2017 at 16:20 | history | edited | fosco | CC BY-SA 3.0 |
deleted 51 characters in body
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| Feb 3, 2017 at 16:09 | history | asked | fosco | CC BY-SA 3.0 |