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 10605
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
An abstract nonsense proof of the Hurewicz theorem
Here is a sketch of the proof, some details filled below. All categories
are $(\infty,1)$-categories and all functors are $(\infty,1)$-functors
unless specified otherwise. The notion of a topological …
- 4,780
7
votes
What is the homotopy fiber of a fold map?
I will consider overcategories of the form $Top/S$, $S\in Top$. Their objects are morphisms $R\to S$, their morphisms are triangles, commutative up to a (specified) homotopy, etc. For details see this …
- 4,780
3
votes
Special $\Gamma$-categories and symmetric monoidal categories
Tom Leinster's book is very old. For higher category theory 2003 is like a previous epoch. In those times there were many competing definitions of higher category theory and higher algebra, for most o …
- 4,780
13
votes
2
answers
1k
views
Proving that a space cannot be delooped.
Suppose we have some pointed connected topological space $X$. How can we determine if there exists a space $BX$, called delooping of $X$, such that its space of based loops $\Omega BX$ is homotopy equ …
- 4,780
10
votes
Accepted
Automorphisms of Eilenberg-Mac Lane spaces and semidirect products (and the odd line)
As per Qiaochu Yuan's comment we need to only understand the space of based maps between $K(A,n)$ with a chosen base point.
The loop-deloop pair of functors establish an equivalence between the categ …
- 4,780
11
votes
Accepted
Topology of categories, very basic facts surrounding Quillen's Higher Algebraic K-Theory I
N.B.: I have reread your question and it occured to me that you a probably asking something entirely different. However since I'm unclear what exactly is your question and since I don't want to delete …
- 4,780
3
votes
Homotopy classes of maps to Lie groups
Notice that since $G$ is a topological group, we have
$$\pi_0[X,Y] = \pi_1 [X, BG]$$
Here $[X,Y]$ denotes the mapping space with standard compact-open topology, $BG$ is the classifying space for grou …
- 4,780
6
votes
Accepted
Can Homotopy Type Theory or algebraic geometry deal with homotopy fibers in terms of families?
Yes, there is a certain sense in which your statements are true. As Mike Shulman and Qiaochu Yuan said, the strict fiber of a map cannot be defined in HoTT and doesn't make sense, but you can work fro …
- 4,780
3
votes
3
answers
3k
views
Exact sequences in homotopy categories
I am not really familiar with homotopical category theory, so please forgive me if I make rude mistakes. I know quite a bit of common category theory, as well as familiar with algebraic topology.
How …
- 4,780
8
votes
Can one make a category concrete by "enlarging the universe"?
As already noted above, any category can be considered concrete after a base change to a suitably large universe. However, doing so would be completely missing the point of concreteness. The underlyin …
- 4,780
11
votes
Accepted
Connective spectra and infinite loop spaces
Ok, this discussion has grown beyond the level of comments so I'll collect the facts here. A bit of terminology: a $(-1)$-connected space is a space with a choice of basepoint and the category of $(-1 …
- 4,780
9
votes
1
answer
744
views
Equivariant homotopy, simplicially
It is a classic result of Kan that the homotopy categories (with appropriate model structures) of simplicial sets and of topological spaces (in fact, one could only care about CW-complexes) are equiva …
- 4,780