Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with homotopy-theory
Search options answers only
not deleted
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.
3
votes
Accepted
proving that an inclusion map from a subcomplex is a homotopy equivalence
By taking K = a simplex and L = its boundary you can show that |A| -> |X| is an isomorphism on all homotopy groups (do surjectivity and injectivity separately). Then apply Whitehead's theorem.
- 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
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
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
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
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
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
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
4
votes
triangulated vs. dg/A-infinity
One nice thing about the (∞,1)-categorical point of view is that being a stable (∞,1)-category is a property of an (∞,1)-category, and not extra structure. So whatever you have, you can probably find …
- 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
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
7
votes
For which spaces is homology (or cohomology) determined by the Eilenberg-Steenrod axioms
Clark Barwick suggested to me that this should be true for any Δ-generated space, and that this is a strictly larger class than spaces homotopy equivalent to a CW-complex. I haven't attempted to veri …
- 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
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
7
votes
Accepted
Equivalences in Model Categories
Yes. The isomorphism in $\mathrm{Ho}(\mathcal{M})$ is represented by a morphism in $\mathcal{M}$ from a cofibrant replacement for $A$ to a fibrant replacement for $B$. The "converse to the Whitehead …
- 25.2k