New answers tagged homotopy-theory
8
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Does there exist a Bousfield localization of the category of spectra which makes the sphere unbounded below?
As discussed in the comments, the opposite of your premise seems to be true: Basically all interesting localisations seem to take non-bounded below values.
To substantiate this, let's classify the ...
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3
votes
Homotopy equivalence between certain loop spaces
The point is that $S^3$ is the universal $2$-connected cover of $S^2$, via the Hopf fibration $S^3 \to S^2 \to K(\mathbb Z, 2)$. As suggested by David Roberts, this map induces an isomorphism on $\...
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6
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What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
If $X$ is a smooth projective variety of dimension $r$ over $\mathbf{C}$ then the Leray spectral sequence of the (ordered) configuration space $F^n X$ of $n$ points on $X$ including into $X^n$ has ...
7
votes
What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
Any DG algebra $A$ over a field has a minimal model, which is a minimal $A_\infty$-algebra $(H,m_3,m_4,\dots)$. It consists of a graded algebra $B=H^*(A)$ equipped with multi-linear operations of ...
11
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What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
Some examples with one nonzero family of differentials:
The classical Adams spectral sequence for $j/p$, the connective image-of-J spectrum reduced mod $p$, collapses at $E_3$, by Theorems 4.5 (at $p=...
1
vote
Accepted
Why is the universal $n$-gerbe is universal? (HTT, 7.2.2.26)
(Turning my comments into an answer.)
The relevant claim is an assertion about a gerbe $f : \widetilde X \to X$. (Namely, the assertion that the mapping space $\mathrm{Map}_{\mathrm{Gerb}^A_n(\mathcal ...
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9
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What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
One example is a construction that is often used in the passage from smooth projective varieties to arbitrary varieties. There are various variants of this:
For a proper variety $X$, take a ...
10
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What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
The Serre spectral sequence for the path-loop fibration for an $n$-sphere is a positive answer to question 1, a negative answer to question 2. More generally, a fibration in which either the base or ...
8
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Accepted
Does a ring spectrum with even homotopy and even cells always have a polynomial algebra of homotopy groups?
Let $G$ be any discrete group, and let $MU[G] = MU \otimes \Sigma^\infty_+ G$ be the associated group algebra over $MU$. Additively, $MU[G] \simeq \bigoplus_{g \in G} MU$, and so it has both even ...
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4
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Left Kan extension and finite product preserving
Yes, but this is a completely general phenomenon unrelated to animated rings and sheaves. The general (surprising!) phenomenon is that the left Kan extension of any product preserving functor $C\to An$...
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10
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Does the Atiyah-Hirzebruch spectral sequence for $E^\ast(X)$ collapse whenever $E$ is complex-oriented and $X$ has even cells?
Yes, this is true.
As you commented, $X$ has finitely generated free homology, and so the $E_2$-term of the AHSS can be identified with
$$
H^*(X; E^*) \cong E^* \otimes H^*(X).
$$
To show collapse, we ...
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5
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What is the center of Morava $K$-theory?
As others pointed out in the comments, the topological Hochschild cohomology of $K(n)$ (i.e., the center of $K(n)$) was calculated by Angeltveit. Here is the paper: https://doi.org/10.2140/gt.2008.12....
4
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Derivations in the Steenrod algebra
I have a guess for question 1. Fix $n \geq 0$ and let $E(n)$ be the Hopf subalgebra dual to $\mathbb{F}_2 [\xi_{n+1}, \xi_{n+2}, \dots] / (\xi_i^{2^{n+1}})$. Every $x\in E(n)$ satisfies $x^2=0$, and ...
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13
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Derivations in the Steenrod algebra
The $D$ with $D(xy) = xD(y) + D(x)y$ are the primitives in the Steenrod algebra $A$, which are dual to the indecomposables $\xi_i$ in $A_* = F_2[\xi_i \mid i\ge1]$, so there is one such $D$ in each ...
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3
votes
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homotopic to a constant map
Let $X = \Bbb{RP}^n$ for $n \geq 2$. I claim that the map $X \to X^3 / M$ is nontrivial on mod-2 cohomology.
Here is some general material.
For $1 \leq i \leq 3$, let $p_i: X^3 \to X^2$ be the ...
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6
votes
Accepted
What is the group completion of finite sets with respect to cartesian product?
As already addressed in the comments:
Group completing the groupoid of finite pointed sets under the smash product gives a contractible space.
The groupoid of finite sets under the cartesian product ...
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3
votes
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Bar construction in commutative algebras is calculated by pushout
This is just to be explicit about the role of the bar construction in David's answer.
If $I$ is the category $a \leftarrow b \rightarrow c$ parametrizing pushout diagrams, then there is a functor $f: ...
- 51.5k
4
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“Geometric” vs Homotopical completion
In the affine case, this is more-or-less proved in [Bhargav Bhatt, Completions and derived de Rham cohomology]. More precisely, Kathryn Hess' completion is more akin to Carlsson's Adams completion, ...
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9
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Besides $F_q$, for which rings $R$ is $K_i(R)$ completely known?
In the ten years since this question was asked, there has been a lot of progress in algebraic $K$-theory. For example, Achim Krause, Ben Antieau, and Thomas Nikolaus came up with an algorithm to ...
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Homotopy theory and algebraic topology last 10 years. Is it a dying field?
If the criterion is “results using algebraic topology which shocked the mathematical community in the last 10 years”, then how about Abouzaid and Blumberg’s proof of the Arnol’d conjecture using ...
2
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Simplicial enrichment on unbounded algebras over an operad
There is no obstruction. If $M$ is a simplicial monoidal model category, and $O$ is an operad in $M$, then the category of $O$-algebras is simplicially enriched, tensored, and cotensored. If it's a ...
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2
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“Geometric” vs Homotopical completion
Yes, there's a way to relate the two. First, it's helpful to think of both in terms of universal properties. Since geometric completion is a fiber product, it's a pullback. Meanwhile, the homotopical ...
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2
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Homotopical Combinatorics
Back when this question was asked, several people suggested they were not sure what was meant by "homotopical combinatorics." Well, now there's a subfield known as homotopical combinatorics. ...
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8
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Homotopy theory and algebraic topology last 10 years. Is it a dying field?
No, it's not dying at all. If anything, now is the best time to do homotopy theory. Thanks to the recent work of Lurie and others, homotopy theory is easier than ever to get into (advances have ...
7
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Bar construction in commutative algebras is calculated by pushout
A way to see this which doesn't dive into the specifics of the simplicial diagram "$C\otimes D^{\otimes n}\otimes E$" is to apply 3.2.4.7 to the symmetric monoidal $\infty$-category $Mod_D(\...
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6
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Bar construction in commutative algebras is calculated by pushout
Welcome to MathOverflow! First, let me point out that what you're asking is already true at the 1-categorical level. The pushout in the category of commutative rings is computed by the tensor product. ...
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5
votes
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$\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $??(\Omega G)$
If $A$ is a braided ∞-group, the delooping $\def\B{{\sf B}}\B A$ is an ∞-group.
Consider the ∞-category of spaces equipped with an action of the ∞-group $\B A$.
Since $\B Ω G≃G$, this ∞-category is ...
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