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Results tagged with homotopy-theory
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user 184
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.
73
votes
Do we still need model categories?
Here are some rough analogies:
Model Category :: $(\infty, 1)$-category
Basis :: Vector space
Local coordinates :: Manifold
I especially like the last one. When you do, say, differential geometry …
- 27.5k
68
votes
Accepted
Is there an accepted definition of $(\infty,\infty)$ category?
One thing that might interest you is my result with Clark Barwick which gives an axiomatiation + uniqueness result for the homotopy theory of higher categories:
arXiv:1112.0040
(i.e. $(\infty,n)$-cat …
- 27.5k
49
votes
What are surprising examples of Model Categories?
The category of sets admits precisely nine model category structures, no more no less.
I learned this fact from Tom Goodwillie's comments on a different MO question. It always shocks people when I m …
- 27.5k
45
votes
Difference between represented and singular cohomology?
This is a good question because it really hits on a subtle issue. It turns out that Johannes and Ben are both correct and incorrect at the same time unless we settle some very subtle issues. Let me ex …
- 27.5k
40
votes
Accepted
Classifiying sphere eversions
Answer Summary
The fundamental group of the space of immersions of $S^2$ into $\mathbb{R}^3$ is
$$ \pi_1 Im(S^2, \mathbb{R}^3) \cong \mathbb{Z}/2 \times \mathbb{Z}$$
This means that there are infini …
- 27.5k
31
votes
3
answers
2k
views
Is the counit of geometric realization a Serre fibration?
Recall that a Serre fibration between topological spaces is a map which has the homotopy lifting property (HLP) for all CW complexes (equivalently for all disks $D^k$). The Serre fibrations are the fi …
- 27.5k
30
votes
Accepted
"Homotopy-first" courses in algebraic topology
I was a heavily involved TA for such a graduate course in 2006 at UC Berkeley.
We started with a little bit of point-set topology introducing the category of compactly generated spaces. Then we move …
24
votes
6
answers
2k
views
Simplicial model of Hopf map?
The Hopf fibration is a famous map $S^3\to S^2$ with fiber $S^1$, which is the generator in $\pi_3(S^2)$. We can model this map in terms simplicial sets by taking the singular simplicial sets of these …
- 27.5k
22
votes
4
answers
2k
views
Functorial Whitehead Tower?
The Whitehead tower of a (pointed) space is a tower of spaces which successively kills the bottom homotopy groups. The first two spaces can be constructed functorially (at least for suitably nice spac …
- 27.5k
21
votes
Accepted
Is super-vector spaces a "universal central extension" of vector spaces?
This will be a little imprecise, but hopefully if you need a precise result like this you can fill in the details.
First of all Vect has not only the symmetric monoidal structure but also the direct …
- 27.5k
20
votes
Accepted
For what topological groups $G$ can we take $EG \rightarrow BG$ to be of the form $S^{\infty...
I like to think of $EG$ and $BG$ in terms of configuration spaces.
The space $BG$ can be identified with the following configuration space. It consists of configurations of finitely many points in the …
- 27.5k
20
votes
2
answers
1k
views
How many model categories have the same weak equivalences?
There are many situations which arise where one might consider different Model categories with the same underlying category. For example in (left) Bousfield localization you start with a model categor …
- 27.5k
19
votes
4
answers
3k
views
What are the fibrant objects in the injective model structure?
If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category in two natural ways which are compatible with the usual model structure on simplicial set …
- 27.5k
18
votes
Accepted
What is the free symmetric monoidal $\infty$-category on one object?
Yes, it is the same as $\mathbb{F}$.
As John Baez points out, it is the same as the free symmetric monoidal $\infty$-groupoid on one object. (This can also be seen by playing around with the adjoints …
- 27.5k
18
votes
1
answer
2k
views
A Model Category of Segal Spaces?
So in Julie Bergner's work on $(\infty, 1)$-categories arXiv:0610239, she considers several model categories which model $(\infty, 1)$-categories, which are known to be equivalent. I'm guessing that t …
- 27.5k