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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.
11
votes
1
answer
599
views
Do h-coequalizers and coproducts give all h-colimits?
It is well known that if a category has all coequalizers and all (small) coproducts then in fact it has all (small) colimits. More important is the proof which shows that every colimit can be built by …
- 24.2k
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 …
- 1,446
8
votes
Accepted
A step in Lurie's treatment of $L$-theory
Here is a way to see it which might constitute "just unfolding the definitions" (at least if one were sitting in on the class at Harvard when it was being taught).
Set $Z(T) = Y(T^c)$, (compliment tak …
- 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 …
- 64k
9
votes
1
answer
316
views
Framed version of the "copants bordism"?
The "pants" bordism in dimension n is a bordism which goes from $S^n \sqcup S^n$ to $S^n$ witnessing the connected sum operation - equivalently by attaching 1-handle to the trivial bordism, equivalent …
- 409
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
7
votes
Accepted
Categorical models for truncations of the sphere spectrum
I don't understand what you mean about the "directed sphere" so will focus on the other questions.
The free Picard $n$-category on one object has a description as a bordism $n$-category. Specifically …
- 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
10
votes
Accepted
Classifying Space of a Group Extension
Yes. The principal bundles are the same and your guess that $BA$ is an abelian group is exactly right. A good reference for this story, and of Segal's result that David Roberts quotes, is Segal's pape …
- 6,023
7
votes
Accepted
Arcwise-connectedness generalized to higher connectivity?
No, there is no generalization to "n-arcwise connected" that you ask for.
Take $X= \mathbb{R}^3$. This space is as nice a space as you could ever hope for. It is also contractible, so in particular …
- 27.5k
6
votes
Accepted
Space of sections of a fibration under weak homotopy equivalence
This is not true in general, unless you assume the base is sufficiently nice (eg a CW-complex). Here is a counter-example.
Let $B = \mathbb{Q}$, the rationals with its topology as a subspace of the …
- 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 …
- 22.3k
12
votes
Is any CW complex with only finitely many nonzero homology groups homotopic to a finite dime...
As requested I am writing this as an answer.
No there are spaces with vanished homology which are not homotopy equivalent to finite CW-complexes.
For example if $G$ is an acyclic group, then the cl …
- 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 …
- 5,565
8
votes
Accepted
Understanding model independently the equivalence of two ways of obtaining homotopy types fr...
Here is an argument, which is basically Denis Nardin's comment.
To have a model independent proof you need model independent definitions of the hocolim and of the localization. You can define them …
- 27.5k