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Results tagged with homotopy-theory
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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.
9
votes
Accepted
Homotopy excision for structured ring spectra -- reference?
Theorems 1.4–1.11 in Ching and Harper's paper “Higher homotopy excision and Blakers-Massey theorems for structured ring spectra” (arXiv:1402.4775)
give higher homotopy excision and Blakers-Massey (and …
- 37.8k
2
votes
When are (weak) homotopy equivalence testable on open covers?
The claim about weak equivalences follows as soon as one proves that the cocartesian squares generated by U←U∩V→V and U'←U'∩V'→V' are also homotopy cocartesian.
To this end one can use Lurie's Seifer …
- 37.8k
5
votes
Accepted
Are simplicial abelian sheaves fibrant?
Fibrant in what model structure?
Simplicial abelian sheaves (and presheaves) are fibrant
in the projective model structure because
all simplicial abelian groups are fibrant.
Simplicial abelian sheav …
- 37.8k
6
votes
Accepted
Kan fibrant replacement for a sphere
Computing homotopy groups for spheres are fundamentally hard, and I believe the problem lies in the difficulty of finding their Kan fibrant replacement.
Computing the fibrant replacement for simp …
- 37.8k
5
votes
Does lifting correspondence hold for principal bundles too?
Let P be a (nontrivial) principal bundle over the base space R^4
All principal bundles over R^4 are trivial because R^4 is contractible.
Or at least can I assert that all such horizontal lifts end …
- 37.8k
5
votes
Accepted
Localizing $\mathrm{CombModCat}$ at the Quillen equivalences
The answer is affirmative and is provided by the paper
Combinatorial model categories are equivalent to presentable quasicategories.
Among other things, it proves that the relative categories of combi …
- 37.8k
5
votes
Accepted
Why is this condition necessary for the existence of a transferred simplicial model structure?
Since Goerss and Jardine do not give a (full) proof of this theorem, it is unclear how exactly this condition was intended to be used.
However, this type of construction (where weak equivalences and f …
- 37.8k
3
votes
Accepted
Limit of weak equivalences in a Bousfield localization
No. For a counterexample to your claim, consider the model category M
of simplicial presheaves on a small site S equipped with the projective
model structure.
Its fibrant objects are presheaves of Ka …
- 37.8k
4
votes
Accepted
Quillen equivalent module categories
The counit map is cocontinuous in M, so using the fact
that any cofibrant object is a retract of a transfinite composition
of cobase changes of generating cofibrations of A-modules,
combined with the …
- 37.8k
6
votes
Accepted
Do infinite products commute with trivial cofibrations, for simplicial sets?
This fact admits a much easier proof.
To show that for any simplicial fibrant sheaf F and open sets U⊆V the map F(V)→F(U) is a fibration
it suffices to show that F(V)→F(U) has a right lifting property …
- 37.8k
10
votes
Accepted
How to compute Homotopy Pullback
Are there any explicit ways of calculating it, similar to the methods, working for the bog down homotopy pullbacks and pushout?
Yes, in fact the same formula continues to work in this case.
Consider …
- 37.8k
3
votes
Is an A-infinity thing the same the same as strict thing viewed through a homotopy equivalence?
When is an A_∞ structure of this type - i.e. is there always an equivalent strict version?
Risking unsolicited self-advertising, I would like to point out
Proposition 10.1.1 (and the more general …
- 37.8k
3
votes
Accepted
Monochromatic infinity operads as algebras over the "operad operad"
Yes, combine Corollary 9.4.1 and Theorem 7.11 of arXiv:1410.5675, for example.
This topic is also examined more explicitly
in the work of Chu and Haugseng, arXiv:1707.08049.
Corollary 5.1.13 shows tha …
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1
vote
Why does this construction give a weak factorization system in the category of span diagrams?
they seem to implicitly use the fact that (in their notation) if a map f is such that fa, fb, and fc are acyclic cofibrations, then ia(f) and ic(f) are again acyclic cofibrations.
No, that's not …
- 37.8k
1
vote
Accepted
Restricting spectra to finite $n$-truncated/$n$-connected pointed spaces
$\renewcommand{\S}{\mathcal{S}}\newcommand{\l}{\langle}\newcommand{\r}{\rangle}\newcommand{\op}{\mathsf{op}}\newcommand{\fin}{\mathrm{fin}}$can we view nonconnectivity as arising from enlarging Segal …
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