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Results tagged with homotopy-theory
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user 43054
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.
3
votes
Importance of $E_n$-algebras over ring structures on $\pi_*(E)$
This might be a naive answer, but I think it is more than a comment: $E_n$-algebra structures on $E$ give rise to (increasingly commutative) monoidal structures on the category of modules. In general …
- 16.5k
3
votes
Accepted
Simplicial version of the A-infinity operad
In my opinion the construction in "the geometry of iterated loop spaces" by P. May should carry through the simplicial world. If you want a completely simplicial treatment you can find it in theorem 5 …
- 16.5k
7
votes
Accepted
Thom isomorphism from the ABGHR perspective
Let's try to see what a lift $\tilde f : X\to R\textrm{-triv}$ is. Recall that $R\textrm{-triv}$ is the $\infty$-groupoid of $R$-lines with a specified isomorphism with $R$, so a lift $\tilde f$ corre …
- 16.5k
3
votes
Homotopy pullbacks/relative homotopy groups vs homotopy pushouts/relative homology groups
A first observation is that homology groups are themselves homotopy groups. Precisely there is a functor $\mathbb{Z}[-]$ from spaces to pointed spaces (it is easier to describe when you think of space …
- 16.5k
5
votes
Accepted
Invariance of Thom spectra
Yes this is true. This follows implicitly from the fact that the $E_1$-structure on $Mf$ can be identified with the canonical $E_1$-structure on $\mathrm{colim}_Xf$ (see for example here) and that is …
- 16.5k
2
votes
Accepted
Transfer map of simplicial sets
In the more general version with compact (finitely dominated) fibers, this is called the Becker-Gottlieb transfer. You can find a long list of references on the nlab. Here are a few of them:
Becker, …
- 16.5k
8
votes
Accepted
Loop space of a Simplicial Abelian group
You can find such a map, but it goes the other way: $\Gamma(Y)\to \Omega X$. It is always wise to keep one's right adjoints on the right hand side.t
But first, let me note that every simplicial abeli …
- 16.5k
13
votes
CW-complex of Eilenberg-MacLane spaces
There is a completely explicit simplicial set realizing to $K(A,n)$ for each $n$, coming from the Dold-Kan correspondence.
It is defined as
$$\bar{A}[S^n]=\mathrm{ker}(A[S^n]\to A[*])$$
(the kernel …
- 16.5k
3
votes
Accepted
Formula relating the cup product in dimensions n and n+1
It is a bit of a roundabout way of proving it, but what you're looking for is immediately implied by the fact that the Eilenberg-Mac Lane spectrum is a commutative ring spectrum, as proven for example …
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1
vote
Accepted
Localization, Slice Tower, and Motivic Spectra
The stable motivic category is a presentable symmetric monoidal ∞-category so smashing with any motivic spectrum preserves all homotopy colimits. On the other hand smashing with a spectrum need not to …
- 16.5k
8
votes
Accepted
A "non-abelian excision" statement for mapping out of a space
It depends on what exactly you mean by "subspace" and "fiber". Let me put some theorems down for you:
Theorem: Let $U,V\subseteq X$ open subspaces. Then the following is a homotopy pushout square:
$ …
- 16.5k
9
votes
Accepted
Reference request for K-Theory linearization
I claim that for every $A_\infty$-space $A$, there is a canonical $A_\infty$-ring structure on $\Omega^\infty\Sigma^\infty_+A$.
First, $\Sigma^\infty_+$ from spaces to spectra is symmetric monoidal. …
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12
votes
Accepted
How to construct the Moore spectrum?
What you're missing is that $[\mathbb{S},\mathbb{S}]=\mathbb{Z}$. Let now $R$ be the infinite matrix of integers representing $\rho$. Note that since $\rho$ takes value in $\bigoplus_{I_1}\mathbb{Z}\s …
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3
votes
Accepted
Pullbacks and fibers in the $\infty$-category of spaces
Well, I guess I can write as an answer what I wrote as a comment.
Any pullback square where $C$ is not discrete will yield a counterexample. For simplicity let $B=G=\ast$ and $C=S^1$. Then $E=H=\Omeg …
- 16.5k
9
votes
Example of a non-$\infty$-category whose homotopy category is a groupoid
Since the homotopy category depends only on the 2-skeleton of the simplicial set, the easiest thing to do is to take the 2-skeleton of an ∞-groupoid. For example, let $E$ be the nerve of the contracti …
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