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 not deleted
user 22810
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.
10
votes
Accepted
For which $n$ does there exist a closed manifold of (chromatic) type $n$?
After discussing this with Tim we came up with the following answer:
The first steifel whiteny class $\omega_1$ of $M$ can be written as the following composition:
$$M \to BO(n) \to BO \to BAut(\mathb …
- 7,799
6
votes
0
answers
308
views
An adjunction between monads on $\mathcal{C}$ and presentable categories under $\mathcal{C}$
Fix a regular cardinal $\kappa$ and let $\mathcal{C}$ be a $\kappa$-presentable $\infty$-category (comments about the 1-categorical case are welcome as well!).
I'm looking for a reference for the fol …
- 7,799
6
votes
0
answers
245
views
Uniqueness of the $(2,2)$-category theory of $(\infty,1)$-categories?
The question, as in the title, may be very simply stated as follows:
Main Question: Can the homotopy $(2,2)$-category of $(\infty,1)$-categories be characterized as the unique $2$-category upto eq …
- 7,799
12
votes
0
answers
403
views
The $\infty$-category of $n$-manifolds and open embeddings determined homotopically from tha...
Let $\mathrm{Diff}_n$, $\mathrm{PL}_n$, $\mathrm{Top}_n$ denote the $\infty$-categories of $n$-manifolds which are respectively smooth/PL/topological, and open embeddings (for instance by taking the h …
- 7,799
6
votes
1
answer
341
views
Compact objects in the $\infty$-category presented by a simplicial model category
Let $\mathsf{M}$ be a simplicial model category presenting an $\infty$-category $\mathcal{M}$. I'm interested in a general statement relating compact objects in $\mathcal{M}$ (in the $\infty$-categori …
CommunityBot
- 1
5
votes
0
answers
335
views
A compendium of weak factorization systems on $sSet$
A (weak) factorization system on a category $\mathcal{C}$ consists of a pair of classes of morphisms in the category $(L,R)$ satisfying
Every morphism $f:x \to y \in \mathcal{C}$ can be factored (n …
- 37.8k
5
votes
0
answers
219
views
Is there a systematic way to "bound" the $d_n$'s of ASS's by "pairing" them with elements in...
All details in the question are for the case $p=2$ though I expect the answer shouldn't be that different for odd primes.
Adams showed (i think it was him) the following statement:
The element …
- 7,799
12
votes
1
answer
853
views
The (fiber of the) cofiber of the fiber of a map of spaces
Consider a fiber sequence of spaces
$$F \overset{i}{\to} E \to B$$
The cofiber $C(i)$ of the inclusion of the fiber comes with a canonical map $C(i) \to B$. Its possible to show (using some point se …
- 13.5k
5
votes
1
answer
548
views
Defining hom spaces in the derived category as limits of hom spaces in the homotopy category
Let $C$ be an abelian category and $K(C)$ the homotopy category of complexes in $C$. I've seen the following claimed in several sources (without proof):
A. The following isomorphisms hold:
$$\li …
- 7,377
2
votes
2
answers
210
views
Spelling out explicitly the data of a two step filtration in terms of pieces and gluing data
Let $V$ be an object of some stable infinity category (nothing is lost by taking spectra but I see no reason to state the question in this way as it is irrelevant) and suppose we have a two step filtr …
- 9,491
8
votes
2
answers
533
views
A map of spaces implementing the Pontryagin Thom collapse map? (collapse maps in families)
Let $M$ be an $n$ dimensional smooth manifold and let $j: M \to \mathbb{R}^{m}$ be an embedding. Associated to this embedding we can form the "collapse map" which is a pointed map from a sphere to the …
CommunityBot
- 1
5
votes
0
answers
75
views
Bounding the dimension of the euclidean space in which any $n$-manifold embeds "$k$-uniquely...
(The question will be interesting for topological/Pl as well but in order to not be too vague I will restrict the meaning of manifold to smooth manifold without boundary).
I'm interested in the funct …
- 7,799
6
votes
0
answers
344
views
Categorical formalism for higher non-abelian group cohomology / obstruction theory for gerbes?
I'm sure this is very well known but I haven't found any references for this searching the internet so hence the question:
What's the neat abstract framework for obstruction theory for non-abelian …
- 7,799
14
votes
2
answers
774
views
Interpretation of the cohomology of compact lie groups and their classifying spaces in DAG?
I'll be using homological grading throughout this question.
Let $G$ be a compact connected lie group. The following isomorphisms are classical and can be proven using several methods:
$$H^{\bullet}( …
- 63.9k
3
votes
1
answer
149
views
A "non-abelian excision" statement for mapping out of a space
Let $U \subset A \subset X$ be spaces (in the sense of homotopy theory).
For every pointed space $Y$ restriction maps induce the following canonical map between mapping spaces:
$$fiber(Map(X,Y) \to …
- 16.5k