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Results tagged with homotopy-theory
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user 126667
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.
65
votes
Accepted
Analogue to covering space for higher homotopy groups?
There's certainly a homotopy-theoretic analogue. A universal cover of a connected space $X$ is (up to homotopy) a simply connected space $X'$ and a map $X' \to X$ which is an isomorphism on $\pi_n$ fo …
- 25.2k
43
votes
Are there two non-homotopy equivalent spaces with equal homotopy groups?
All of these examples involve a bit of cleverness, so I thought I'd point out a more straightforward way to construct counterexamples. If $X$ is any space, we can build a space $X' = K(\pi_0 X, 0) \t …
- 25.2k
26
votes
Accepted
Counter-example to the existence of left Bousfield localization of combinatorial model category
A surprisingly effective way to construct counterexamples in model category theory is to just write down all the objects and morphisms involved and try to give the resulting (finite!) diagram the stru …
- 25.2k
24
votes
Accepted
Homotopy pullbacks and homotopy pushouts
You can think of the pushout of two maps f : A → B, g : A → C in Set as computing the disjoint union of B and C with an identification f(a) = g(a) for each element a of A. We could imagine forming th …
- 25.2k
23
votes
Accepted
Functorial Whitehead Tower?
The nth stage of the Whitehead tower of X is the homotopy fiber of the map from X to the nth (or so) stage of its Postnikov tower, so you can use your functorial construction of the Postnikov tower pl …
- 25.2k
18
votes
Accepted
Are non-empty finite sets a Grothendieck test category?
That your G is a test category is stated in the last sentence of 4.1.20 in the paper of Cisinski you mention. This case is also treated in more detail in section 8.3, where it is shown that the left …
- 25.2k
16
votes
Accepted
How to think about model categories?
Model categories are 1-categorical presentations of (∞,1)-categories, which you can just think of as categories enriched in topological spaces, such as the category of spaces itself. (Actually, there …
- 25.2k
14
votes
Accepted
Do h-coequalizers and coproducts give all h-colimits?
There is an analogue, but one should replace coequalizers by geometric realizations (homotopy colimits over Δop). If $F : I \to M$ is a diagram in a model category, one has
$$\operatorname{hocolim}_I …
- 25.2k
13
votes
Why do finite homotopy groups imply finite homology groups?
It sounds like maybe you can prove your first statement for simply connected spaces. In that case, you can use the homotopy orbit spectral sequence (the Serre spectral sequence associated to the fibr …
- 25.2k
13
votes
"Models" in homotopy theory
Maybe a "lower" analogue would be helpful.
An ordered pair is an object that contains two pieces of data, the first component and the second component.
Suppose we want to make this precise in the lang …
- 25.2k
12
votes
Definition of homotopy limits
You can fix it by making η a homotopy coherent diagram (a map to each object of D, a homotopy for each arrow of D, a homotopy-between-homotopies for each commuting triangle in D, ...) and also replaci …
- 25.2k
9
votes
Accepted
Homotopy Pushouts via Model Structure in Top
Question 1: The model category $\mathcal{C}$ should be left proper, i.e. the pushout of a weak equivalence along a cofibration is again a weak equivalence. (Dually, there is a notion of right proper. …
- 25.2k
9
votes
Accepted
Localizing Model Structures
(Your question is basically about presentable (∞,1)-categories, so I will take the liberty of writing my answer in that language. Hopefully the translations to model category language will be straigh …
- 25.2k
9
votes
Accepted
$(\infty,1)$-categories and model categories
Mostly I refer you to my answer here and also this question.
To answer the question about (co)fibrations: No, there is no notion corresponding to (co)fibration in the (∞,1)-category associated to a m …
- 25.2k
8
votes
Reference for iterated homotopy fixed points?
The statement XhG = (XhH)hG/H is true for any G-object X of any complete (∞,1)-category C. An object of C with a G-action is the same as a functor BG → C where BG represents the category (or (∞,1)-ca …
- 25.2k