Timeline for Loop space of a category
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Jan 2, 2013 at 19:12 | comment | added | Mike Shulman | @Matthias: Okay, thanks, I'll have a look. (If you posted an answer describing Grothendieck's definitions and his theorems about them, I would be able to accept it...) | |
| Dec 26, 2012 at 10:24 | comment | added | Matthias Künzer | In Sec. 4 of Ch. VII, he constructs the path category _Ch_(X). In Sec. 12, he uses an alternative construction _Ch_oo(X) to produce an "analogue of the Cartan-Serre formalism". In Sec. 13, he constructs an equivalence of _Ch_(X) and _Ch_oo(X), saying the former is better. On page 106, then the (co)cone of a morphism appears. Of course, it's still in the form of a "mathematical diary", but the constructions are there. | |
| Dec 24, 2012 at 14:29 | comment | added | Mike Shulman | @Mathias, are you saying that there is an answer to this question buried in Grothendieck's work, or merely that it seems related? | |
| Dec 24, 2012 at 10:32 | comment | added | Matthias Künzer | Grothendieck has worked on catégories de chemins, see www.math.jussieu.fr/~maltsin/groth/Derivateursengl.html . | |
| Dec 5, 2012 at 14:11 | answer | added | Ilias A. | timeline score: 3 | |
| Dec 4, 2012 at 2:58 | comment | added | Mike Shulman | @Fedotov: Yes, I do need a concrete model, otherwise I wouldn't have asked. It's a good point that there must be some such category, namely the loop space object in the Thomason model structure, but I would like something more concrete, such as something made out of zigzags in C. | |
| Dec 4, 2012 at 2:56 | comment | added | Mike Shulman | @Bob: I don't see anything in the abstract you quoted which seems related to the question; can you explain why you think it has some connection? | |
| Nov 30, 2012 at 15:49 | comment | added | Ilias A. | Do you need a concrete model? The naive constructions will fail as was mentioned by Karlo Szumilo...essentially because the Thomason model structure on Cat is not a simplicial monoidal closed model category, but still you can use the derived internal hom in Cat (wich exists) Thomason model structure... The model that you obtain for $\Lambda C$ is very big. I don't know if you need more details. | |
| Nov 29, 2012 at 11:49 | comment | added | Bob Terrell | @Mike: the abstract says that: A method for associating to each topological category G, a principal top cat BXG is discussed. The first step is to associate with each G, a top 2-cat XG. The second is to apply the classifying space functor B. It is shown that there is a homomorphism BXG->G, which when restricted to the morphism spaces is a htpy equivalence of spaces over Ob G x Ob G. The particular example BX\Gamma^0 is considered and is shown to act on the disjoint union of (BA)^n, where A is a permutative category. | |
| Nov 28, 2012 at 1:29 | comment | added | Mike Shulman | @Bob: Well, I can't tell since Google books won't even let me read the table of contents. The title doesn't make me hopeful, though. | |
| Nov 27, 2012 at 18:11 | comment | added | Bob Terrell | I don't know whether this contains an answer or not: books.google.com/books/about/… but it does have some connection I think. | |
| Nov 26, 2012 at 20:37 | comment | added | Karol Szumiło |
@Dylan: this category has wrong homotopy type in general. It is actually isomorphic to the category of functors from the monoid $\mathbb{N}$ to $C$. If you take $C$ to be a category with two objects and two parallel arrows between them, then $C$ is homotopy equivalent to a circle, so its loop space should be countably infinite discrete (up to homotopy), but your construction gives a finite category.
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| Nov 26, 2012 at 18:30 | answer | added | Karol Szumiło | timeline score: 6 | |
| Nov 26, 2012 at 16:56 | comment | added | Dylan Wilson | Limits in the category of categories exist, so why not take the equalizer of the two maps Fun([1], C) evaluating at 0 and 1? The nerve respects limits so at least you'll get a good looking simplicial set. If I recall May correctly, geometric realization preserves pullbacks... So this seems like a good candidate. | |
| Nov 26, 2012 at 16:41 | history | asked | Mike Shulman | CC BY-SA 3.0 |