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Results tagged with homotopy-theory
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Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
4
votes
Accepted
Higher-dimensional version of the "Magic Cube Lemma" for homotopy pushouts/pullbacks
I don't think so. Take $n=2$, and consider a map $f\colon b\to a$ between objects of $\mathrm{Fun}(I^2, \mathcal{S})$. If $a$ and $b$ are pullback squares, then any map $f$ between them is relativel …
- 27.2k
9
votes
Are the real and complex Adams operations compatible under the inclusions $U(n) \rightarrow ...
I think the answer is yes, it commutes.
The identities $rc=2$, $\psi^k c=c\psi^k$, and additivity of these operations already implies that $\epsilon:= \psi^k r-r\psi^k$ satisfies $2\epsilon =0$ in $[ …
- 27.2k
12
votes
What is the generator of $\pi_9(S^2)$?
Since my old answer is referenced here What is the generator of $\pi_9(S^3)$?, I spent a little time trying to figure out what it says about this. "Toda's sequence" is a $p$-local fiber sequence ($p$ …
- 27.2k
10
votes
Classification of bundles, Postnikov towers, obstruction theory, local coefficients
I'll try to answer question 1. Unfortunately, I know of no especially convenient reference for the case of understanding general Postnikov towers; however, everything I say below is "well known".
I …
- 27.2k
4
votes
HKR generalized character theory question regarding a certain construction
I think what you understand is correct. Write $A_r= L_r(E^*)$, which fit into a direct system $A_r\to A_{r+1}\to \cdots$. Let $G=\mathrm{Aut}(\mathbb{Z}_p^n)$, which acts compatibly on every $A_r$ ( …
- 27.2k
3
votes
Accepted
Reference request for statement on nlab: Reedy (co)fibrancy of (co)simplicial objects
The claim would be true if it were the case that for any simplicial object $X_\bullet$ in some (cocomplete) category $C$, the maps $L_nX\to X_n$ from the latching objects are always monomorphisms. Th …
- 27.2k
14
votes
Accepted
Are (semi)simple Lie groups some sort of "homotopy quotient groups" of their maximal tori?
Here's the "answer" that I started writing, then put away for a while. The short answer is: although "T//W" is not the same as G, they "look sort of the same" from the point of view of certain genera …
- 27.2k
11
votes
Accepted
Proper model category of simplicial rings revisited
This paper proves some things about left properness for categories of simplicial algebras. The context of the paper is in terms of "algebras for a simplicial algebraic theory", which certainly includ …
- 27.2k
12
votes
Accepted
Monomorphisms, epimorphisms, (co-)images and factorizations in $\infty$-categories
An $n$-monomorphism is a map $A\to B$ for which $Map(X,A)\to Map(X,B)$ has all homotopy fibers $n$-truncated, for all $X$; a space (=$\infty$-groupoid) is $n$-truncated if its homotopy groups all vani …
- 27.2k
5
votes
Accepted
The homotopy fibre of an map $f \colon S^{2n-1} \to S^n$
In the following, any identity may actually be up to a sign. I don't want to keep track of that.
We are thinking about the fibration sequence
$$ \Omega S^n \to F \to S^{2n-1} \xrightarrow{f} S^n.$$
…
- 27.2k
27
votes
What is the intuitive meaning of the coskeleton of a simplicial set?
It should be added that if $X$ is a Kan complex, then $X\to \mathrm{cosk}_n X$ computes a model for the $(n-1)$-truncation of the homotopy type of $X$. That's a fact you can easily read off using ele …
- 27.2k
30
votes
Accepted
Errata on Rezk's paper
It looks like I completely missed this.
Here's what I guess happens: although the original 2.19 was wrong, there is a weaker version that is true (I'll just state it for simplicial sets): If $X$ i …
4
votes
Accepted
pullback square in Goerss-Jardine
I would understand this proof as describing the limit of a "deleted 3-cube" in two different ways.
We have a square involving maps $X_n\to X_{n-1}$, $X_n\to M_{n,k}X$, $X_{n-1}\to M_{n-1,k}X$, and …
- 27.2k
12
votes
Accepted
Is the hom-simplicial set in the hammock localization a nerve?
The nLab description is not correct.
For each "shape" of zig-zag, there is a "hammock category" for it (not a groupoid, and the nLab page I am looking at never mentions groupoids here), whose objec …
- 27.2k
6
votes
Accepted
Glueing a property via homotopy colimits
I don't think you want to apply 2.6 here. This is supposed to follow from "descent", or more precisely, from the fact that hocolims are stable under base change.
Let me write $|F|$ for $\mathrm{hoco …
- 27.2k