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Results tagged with homotopy-theory
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user 4177
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.
28
votes
Accepted
Conjectures in Grothendieck's "Pursuing stacks"
To dash off a quick answer, Pursuing Stacks is composed of (if memory serves correctly) three themes. The first was homotopy types as higher (non-strict) groupoids. This part was first considered in G …
- 35.4k
24
votes
Accepted
Who computed the third stable homotopy group?
The error is that Rokhlin claimed that $\pi_6(S^3)=\mathbb{Z}/6$, but Hilton, in his review, points out that the paper instead shows that $\pi_6(S^3)/\pi_5(S^2) = \mathbb{Z}/6$. The error lies in a pr …
- 35.4k
16
votes
What is the homotopy theory of categories?
This is just to answer your first question. The second one I don't know about.
The homotopy theory of categories is not quite as you envisage it. Really Grothendieck is thinking of the Thomason model …
- 35.4k
15
votes
When did the Joyal model structure on simplicial sets originate?
My suspicion is now that it was some time between 2004 and 2006. I have a lot more citations in this blog post, but I note three points, in reverse chronological order:
Multiple experts are referring …
- 35.4k
11
votes
Accepted
Why is the string group not a Lie group?
The result is that a compact, connected simple Lie group $G$ has $\pi_3(G) = \mathbb{Z}$. Simple covering space or subgroups arguments should get you to $\mathrm{SO}(n)$ which is all that matters. For …
- 35.4k
10
votes
Why is the string group not a Lie group?
To follow up, there is now an infinite-dimensional Lie group model of String:
Thomas Nikolaus, Christoph Sachse, Christoph Wockel, A Smooth Model for the String Group, Int. Math. Res. Not. 16 (2013) …
- 35.4k
10
votes
In which situations can one see that topological spaces are ill-behaved from the homotopical...
The geometric realisation functor (read: homotopy colimit for nice situations) from simplicial spaces to $Top$ preserves pullbacks only when you take the $k$-ification of the product in $Top$, or work …
9
votes
0
answers
367
views
Topologies (and sheaves) on Cat and CAT
I've been wondering lately what sort of Grothendieck (pre)topologies there are on $Cat$ (the category of small categories) and $CAT$ (the v. large category of large categories - to forestall criticism …
- 35.4k
8
votes
The higher Van Kampen Theorems and computation of the unstable homotopy groups of spheres
Ronnie and collaborators' HHvK theorems are essentially all for crossed complexes and similar. These are, as I'm sure you're aware, partially linearised homotopy types. In particular, for a simply con …
- 35.4k
8
votes
2
answers
647
views
Is there an interesting definition of a category of test categories?
Given a pair of test categories $C_1$ and $C_2$ (in the sense of Grothendieck - weak or strict or otherwise), has anyone defined an interesting notion of morphism between them? Or are ordinary functor …
- 35.4k
8
votes
0
answers
227
views
Holomorphic contractibility of GL(H)?
Kuiper's theorem is well-known to give the triviality of the homotopy groups of ${\rm GL}(\mathcal{H})$ for $\mathcal{H}$ a (separable) infinite-dimensional complex Hilbert space. Work of Palais later …
- 35.4k
8
votes
What properties make $[0,1]$ a good candidate for defining fundamental groups?
I only have a partial answer for 1. and a hopefully non-confusing answer to 2.
To start with, let us work with the fundamental groupoid, which is more, ahem, fundamental and better suited to generali …
- 35.4k
8
votes
1
answer
484
views
First mention of the fundamental bigroupoid of a space?
The fundamental bigroupoid $\Pi_2(X)$ of a space $X$ was independently described by Hardie, Kamps and Kieboom (paywall) and Stevenson (arXiv) around the year 2000. HKK cite Baez-Dolan's seminal HDA0, …
- 35.4k
7
votes
Accepted
What's the name of this flavor of n-category?
Ronnie Brown has a related idea, contained in this article:
Moore hyperrectangles on a space form a strict cubical omega-category
arXiv
discussed briefly here at the nLab.
If you are instead …
- 35.4k
7
votes
3
answers
448
views
Conclusion of Hurewicz for $H_3$ without vanishing fundamental group?
Fix a space $X$, which I want to assume is a manifold. Under the assumption of simple-connectivity, Hurewicz's theorem tells us that
$$
\pi_3(X)\to H_3(X,\mathbb{Z})\qquad (*)
$$
is surjective, hence …
- 35.4k