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Results tagged with homotopy-theory
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user 22131
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.
47
votes
What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?
It's been more than a year and a half since I asked this question and I had a lot of thought about it so I decided I will post my own answer.
First I entirely agree with Yonatan that the main problem …
- 42.4k
36
votes
Accepted
Current status of Grothendieck's homotopy hypothesis and Whitehead's algebraic homotopy prog...
The problem is that the question is highly dependent on the definition of $n$-groupoids. The notion of strict $n$-groupoid is very clear and precise but we know very well (and Grothendieck knew that) …
- 42.4k
23
votes
What is the intuition for higher homotopy groups not vanishing?
So far the discusion mostly focused on a geometric explanation, I'd like to mention the algebraic one as well:
One way to formulate it involves the delooping machinery: up to delooping, $\mathbb{S}^n …
- 42.4k
22
votes
Accepted
Useful ideas in category theory which violate the principle of equivalence
I would think about this question in this way: If you have a construction that violates the equivalence principle then either (A) it is a strictified or simplified version of something that is compati …
18
votes
Why do we need model categories?
Today we understand that what we are really interested in when we talk about "homotopy theory" are in the end "$\infty$-categories".
In fact I have even heard some peoples claim that maybe in the fut …
- 42.4k
14
votes
Why is Kan's $Ex^\infty$ functor useful?
Another thing for which Kan Ex$^{\infty}$ functor is useful is actually in the construction of the Kan-Quillen model structure. It morally gives a purely algebraic version of the simplicial approximat …
- 42.4k
13
votes
sSet-enriched categories, quasi-categories and the model-independent theory
This has not been done, and there are good reasons for it: While $sSet$-enriched categories are indeed very good to easily get examples of $\infty$-categories, they are very bad at understanding what …
- 42.4k
13
votes
Correspondence between classes of model categories and classes of $\infty$-categories
Regarding (1) :
A) Every model category has an associated $\infty$-category, obtained for example by taking the Dwyer-Kan localization at the class of all equivalence, (But there are other more expli …
- 42.4k
12
votes
Accepted
Simple example of nontrivial simplicial localization
For any $1$-category $C$ the localization $C[C^{-1}]$ at all arrows is an $\infty$-groupoid homotopy equivalent to the nerve of $C$, so it can be any $\infty$-groupoid.
For example take $C$ to be the …
- 42.4k
10
votes
The cofibration/fibration $\leftrightarrow$ epi/mono confusion
This is probably not a full answer to your question, but I think it is a remark worth to make:
It is actually a couple of remarks:
1)If you have a weak factorization system where either the left cla …
- 42.4k
9
votes
A possible alternative model for $\infty$-groupoids
It is known that the category of finite non-empty set is a test category, in particular there exists a model structure on the category of presheaf on $Fin_+$ whose cofibrations are the monomorphisms a …
- 42.4k
9
votes
Conservative cocompletion of categories of geometric shapes for homotopy theory
I don't really see a way to give a single answer here, these are 7 different questions (well actually a lot more than 7 if we count all the different flavours of cubes, and of the other ones, probably …
- 42.4k
8
votes
Accepted
Is the suspension of a finite fibration again finite?
Assuming you work with unpointed spaces (but the example can easily be adapted to the pointed case) the map $1 \to 2$ gives a counterexample : its fiber are $1$ and $\varnothing$ so they are both fini …
- 42.4k
7
votes
What are the advantages of simplicial model categories over non-simplicial ones?
I believe the main reasons enriched model category are simpler boils down to:
Tensoring and co-tensoring by $\Delta[1]$ gives very well behaved path objects and cylinder objects adjoint to each other …
- 42.4k
6
votes
Accepted
Localisation of categories, but instead of "isomorphisms" I want "morphisms with right inver...
I think there is a relatively good reason why such a thing shouldn't exists.
In general when you freely add right inverse or inverse, the general arrows of the resulting category will be zig-zag in th …
- 42.4k