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2 votes

Is there a recognition principle for $\mathbb{E}_{\infty}$-spaces with zero?

I see no real point in considering $E_{\infty}$ spaces with $0$ except in the context of the multiplicative structure of an $E_{\infty}$ ring space or, essentially equivalently, the multiplicative ...
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4 votes
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Diagonal maps, Goodwillie calculus, and $T(n)$ local homotopy theory

This is an elaboration on my comment above. Let us consider the natural transformation induced by the diagonal $$\Sigma^\infty\Delta\colon \Sigma^\infty X \to \Sigma^\infty X\wedge X.$$ The natural ...
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3 votes

Spaces homotopy dominated by $S^2 \times S^2\times S^2$

You're actually right. The cohomology algebra of $X=S^2\times S^2\times S^2$ with coefficients in $\mathbb{Z}$ is $H^*(X)=\mathbb{Z}[x,y,z]/(x^2,y^2,z^2)$ with $|x|=|y|=|z|=2$. The cohomology algebra ...
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6 votes
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Spaces homotopy dominated by $S^2 \times S^2\times S^2$

Put $$ R(n)=H^*((S^2)^{\times n}) = \mathbb{Z}[x_1,\dotsc,x_n]/(x_1^2,\dotsc,x_n^2) $$ A key point is that if $u\in R(n)$ with $|u|=2$ then $u^2=0$ iff $u=0$ or $u=m\,x_i$ for some $m\in\mathbb{Z}\...
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4 votes

Connection between Grothendieck's homotopy hypothesis and Lie's second and third theorems?

There are analogues of Lie's theorems in homotopy theory, primarily for rational and $p$-adic homotopy types, as well as Lie ∞-groupoids, which can be seen as smooth homotopy types. In the rational ...
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0 votes

Explicitly calculating the homotopy fiber of a 3-cocycle in the category of simplicial sets

For anyone interested, I've found a solution. It is the central construction in https://arxiv.org/abs/2203.11392.
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7 votes
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$BP$-Adams Novikov Spectral Sequence or Homotopy groups of $S/3$

For $3$-primary homotopy of $S$ there is early work by ...
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4 votes

What is $TP(\mathbb{Z}_p)$?

The calculation of $\pi_* TP(\mathbb{F}_p) = \pi_* THH(\mathbb{F}_p)^{tS^1} = \pi_* \widehat{\mathbb{H}}(S^1, THH(\mathbb{F}_p))$ (the notation has changed over the years) was first published by ...
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  • 7,126
8 votes
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Homotopy invariance of $\ell$-adic cohomology

Proof of homotopy invariance: This follows from a base change/Kunneth type statement and the calculation of the cohomology of $\mathbb A^1$. Specifically, Lemma 7.6.7 of Lei Fu's etale cohomology ...
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  • 115k
9 votes
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Is there a Dold-Kan theorem for circle actions?

No, they are not equivalent, even for $C = Sp$. Indeed, the category of spectra with $S^1$-action is also the category of $\mathbb S[S^1]$-modules, and is compactly generated by a single object. On ...
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  • 8,160
5 votes
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Fibrant replacement of an injective model category of enriched diagrams

Section 8 of my paper All (∞,1)-toposes have strict univalent universes shows that under fairly general conditions, injective fibrant replacements can be given by cobar constructions (e.g. the dual of ...
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11 votes

Can the Bousfield class of projective space be computed directly?

Here is an easy argument which sometimes works. I have updated and extended it to incorporate comments from Dylan Wilson and Maxime Ramzi. For any finite spectrum $X$, we have (co)unit maps $S\...
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4 votes

What are Koszul dualities?

I have finally found a source which puts together the pieces in a satisfactory way, at least in the stable setting, here: Amabel, Araminta. "Poincaré/Koszul Duality for General Operads." ...
0 votes
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Maps that preserve winding numbers

I found an answer to this question: the question of how a continuous function changes the winding number of a closed curve can be studied quite generally. The important concept here is the degree of ...
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  • 1,109
10 votes

Is the decomposition of the homotopy type of a complex into a product and into a smash product unique?

Let me write $S^n/p$ for the cofibre of $p$ times the identity map on $S^n$, or in other words the mod $p$ Moore space with homology in degree $n$. If $p$ and $q$ are coprime then $S^n/p\wedge S^m/q$ ...
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