Timeline for Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Apr 16 at 20:46 | vote | accept | Dmitri Pavlov | ||
| Apr 16 at 13:47 | answer | added | John Rognes | timeline score: 5 | |
| Apr 16 at 11:24 | answer | added | Georg Lehner | timeline score: 3 | |
| Jan 30 at 9:13 | comment | added | Georg Lehner | I have added an answer to another mathoverflow post that is very much related. mathoverflow.net/a/462976/76299 To spare you the click: In the case of a symmetric monoidal groupoid, the S^-1 S construction is the unstraigthening of the diagonal action of S on S x S. This characterization can be completely generalized in the setting of infinity-categories | |
| Oct 20, 2017 at 18:51 | comment | added | Tim Campion | Here's a motivating example: let $S$ be the topological groupoid of finite-dimensional real inner-product spaces. The category $S^{-1}S$ (and not just its geometric realization) is used by Sagave and Schlichtkrull, to construct certain graded orthogonal ring spectra. I think in other work they do the analogous thing for symmetric spectra. $\mathbb{Z}$-graded ring spectra are important -- related to power operations and log structures -- but surprisingly delicate, since $\mathbb{Z}$ is a strictly symmetric monoidal category ($m+n \to n+m$ is the identity). | |
| Jul 13, 2017 at 18:29 | comment | added | Dmitri Pavlov | @FrankScience: Yes, Quillen's S⁻¹S construction is an elaboration on this theme (and Segal credits Quillen for this section). For special Γ-spaces this indeed produces a very special Γ-space, which is a fibrant replacement. | |
| Jul 13, 2017 at 14:58 | comment | added | user20948 | Hello, I am also learning this material. I don't know whether you have read Segal's paper Categories and Cohomology Theories. It seems to me that the construction on page 304-305 of section 4 is closely related to $S^{-1}S$, which could be viewed as a kind of fibrant replacement. I cannot work out a precise argument for the moment. | |
| Nov 30, 2015 at 21:03 | comment | added | Ilias A. | I am quite pessimistic about a positive answer to your question with such constraints :) | |
| Nov 30, 2015 at 20:57 | history | edited | Dmitri Pavlov | CC BY-SA 3.0 |
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| Nov 30, 2015 at 20:50 | comment | added | Dmitri Pavlov | @IliasAmrani: I now see where the misunderstanding comes from. My intended meaning of “even” was that even if one does know that S⁻¹S is the homotopy group completion of S, there is still the question of which model structures to choose on both sides so that S⁻¹S preserves (acyclic) cofibrations (and hence is a left Quillen functor). | |
| Nov 30, 2015 at 20:44 | comment | added | Ilias A. | No problem, that happens :). You wrote "I cannot $\mathbf{even}$ find a reference in the literature that shows that S⁻¹S represents (in the Thomason model structure) the homotopy group completion of S, so any references on this matter will be appreciated." That is why i concluded that you were asking also if this functor is homotopy left adjoint under hypothesis that you know that it represent complition... | |
| Nov 30, 2015 at 20:36 | comment | added | Dmitri Pavlov | @IliasAmrani: I guess I misunderstood the intended meaning of “translate … E_∞-spaces”: showing that some other well-known formula for the homotopy group completion (e.g., ΩB) implements the homotopy group completion functor is not a part of the question. I am interested exclusively in S⁻¹S, not some other construction. | |
| Nov 30, 2015 at 15:16 | history | edited | Dmitri Pavlov | CC BY-SA 3.0 |
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| Nov 30, 2015 at 15:15 | comment | added | Dmitri Pavlov | @IliasAmrani: I might be wrong, but I think the implicit folklore understanding is that S⁻¹S computes the homotopy left adjoint. I'll add “derived” to the post. I will also be happy with an interpretation using group-like E_∞-spaces of any kind. | |
| Nov 30, 2015 at 10:37 | comment | added | Ilias A. | do you want to see $S\mapsto S^{-1}S$ as left adjoint or as homotopy left adjoint ? Would you be satisfied if you translate every thing in terms of $E_{\infty}$-spaces (since you have mentioned Thomason model structure). | |
| Nov 29, 2015 at 12:30 | history | asked | Dmitri Pavlov | CC BY-SA 3.0 |