Timeline for An abstract nonsense proof of the Hurewicz theorem
Current License: CC BY-SA 3.0
12 events
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| Oct 12, 2017 at 18:05 | comment | added | Anton Fetisov | Finally, I describe a general approach to such algebraic questions. It may be an overkill for Hurewicz, but it is more powerful. E.g. I show that it proves Freudenthal, there are probably other interesting algebras which would be amenable to the method. | |
| Oct 12, 2017 at 18:02 | comment | added | Anton Fetisov | It's not a matter of innate complexity but a matter of historical precedence and inperfect axiomatization that $\infty$-categorical constructions are regarded as something vastly more difficult than classical ones. I also don't need that the universal $E_\infty$-group is THE sphere spectrum or BPQ theorem, it is mentioned only for comparison with other answer. The fact that groups can be delooped is a fundamental property of $\infty$-topoi, there is no getting around that one. Jeff also requires either that or Dold-Kan to construct $K(A, n)$. (...) | |
| Oct 12, 2017 at 17:56 | comment | added | Anton Fetisov | @DylanWilson Well, the OP has asked for an abstract nonsense proof. I have delivered on both points, didn't I? More seriously, the essential part of the proof is in the last paragraph. Most of the rest is required for the comparison to classic singular homology. The proof is trivial if you can define homology $\infty$-categorically in a simple way, e.g. if you know some simple description of the abelian operad (I don't). I consider Day convolution and operads to be a trivial matter. You're not terrified by Lawvere theories or coends, are you? (..cont) | |
| Oct 12, 2017 at 17:33 | comment | added | Dylan Wilson | I find this absolutely ridiculous. | |
| Oct 12, 2017 at 17:33 | comment | added | Dylan Wilson | (contd) varius $\infty$-operads have nice explicit descriptions in the 1-categorical setting, the fact that grouplike $E_k$-spaces are equivalent to $k$-fold loop spaces (which includes checking somehow that $\Omega^k$ preserves sifted colimits as a functor on connected enough spaces- a fact which seems dangerously close to the Hurewicz theorem already), the claim that the sphere spectrum (and I mean like.. THE sphere spectrum) is the universal $E_{\infty}$-group (including defining what that means), which is roughly the Barrat-Priddy-Quillen theorem, and the ($\infty$-)Dold-Kan correspondence | |
| Oct 12, 2017 at 17:29 | comment | added | Dylan Wilson | As someone who likes and uses $\infty$-categories, this answer terrifies me. If I am correctly interpreting your insistence on using only native $\infty$-categorical concepts (which means you aren't allowed to cheat and get the symmetric monoidal structure on $Sp$ using a model structure or anything) you have used all of the following things to prove the Hurewicz theorem: the fact that spectra has the structure of symmetric monoidal $\infty$-category, the definition and basic properties of various $\infty$-operads, $\infty$-categorical Day convolution, the fact that algebras over | |
| Oct 12, 2017 at 17:09 | comment | added | Anton Fetisov | @Oscar If you mean that I need Hurewicz or Freudenthal to prove that $\pi_0(\mathbb S) = \mathbb Z$, I don't. By the sphere spectrum I mean the universal $E_\infty$-group. It's 0-truncation is a universal abelian group, QED. If you wish, you can define $\mathbb S$ as the group completion of the groupoid of finite sets. | |
| Oct 12, 2017 at 16:12 | comment | added | Anton Fetisov | @OscarRandal-Williams No, because it's a single space of coefficients rather than homology itself. The real content of the proof is how you go from a single space to a colimit of spectra. Jeff Strom's answer relies on Freudenthal's theorem and stabilization, I proceed directly from the definition of free algebras. But of course all approaches will be somewhat related. I rely only on general category theory, thus I believe the proof works in any topos and for any algebraic theory, not only $\mathbb Z$-homology of spaces. | |
| Oct 12, 2017 at 9:56 | comment | added | Oscar Randal-Williams | "Here $H\mathbb{Z}$ is an $E_{\infty}$-ring spectrum which is the 0-truncation of the sphere spectrum." Isn't this observation essentially the content of the Hurewicz theorem, as explain by Jeff Strom? | |
| Oct 11, 2017 at 20:24 | history | edited | Anton Fetisov | CC BY-SA 3.0 |
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| Oct 11, 2017 at 19:54 | history | edited | Anton Fetisov | CC BY-SA 3.0 |
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| Oct 11, 2017 at 19:43 | history | answered | Anton Fetisov | CC BY-SA 3.0 |