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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.
8
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1
answer
399
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Freely adding degeneracies does not change the homotopy type
Background: Let $S$ be a simplicial set. By freely adding degeneracies to $S$, I mean first applying the forgetfull functor from simplicial sets to semi-simplicial sets which forget the already existi …
- 42.4k
14
votes
1
answer
294
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Detecting weak equivalence on free loop space homology
Given $f:X \to Y$ a continuous map between two spaces (unpointed CW-complexes) such that $f$ induces an isomorphism in homology with integer coefficient, and $f$ induces an isomorphism on homology of …
- 42.4k
106
votes
4
answers
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What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?
In 1991, Kapranov and Voevodsky published a proof of a now famously false result, roughly saying that the homotopy category of spaces is equivalent to the homotopy category of strict infinity categori …
- 42.4k
10
votes
3
answers
948
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classifying $\infty$-toposes for topological/localic groups?
Let $G$ be a locally compact topological group (or more generally a localic group). Is there an infinity topos which classify principal $G$ bundles ?
More precisely, is there an $\infty$-topos $BG$ s …
- 42.4k
16
votes
2
answers
912
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Counter-example to the existence of left Bousfield localization of combinatorial model category
Is there any known example of a combinatorial model category $C$ together with a set of map $S$ such that the left Bousefield localization of $C$ at $S$ does not exists ?
It is well known to exists w …
- 42.4k
12
votes
3
answers
742
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On model categories where every object is bifibrant
Most model structures we use either have that every object is fibrant or that every object is cofibrant, and we have various general constructions that allow (under some assumption) to go from one sit …
- 42.4k
12
votes
0
answers
158
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Known obstruction for efficient computation of Stable homotopy groups?
Computation of stable homotopy groups (for example of sphere) is hard, but still, not as hard as unstable ones.
For unstable homotopy groups there are some results showing that there cannot be effici …
- 42.4k
11
votes
0
answers
329
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$\Gamma$-sets vs $\Gamma$-spaces
I know that every $\Gamma$-space is stably equivalent to a discrete $\Gamma$-space, i.e. a $\Gamma$-set.
For example, Pirashvili proves, as theorem 1.2 of Dold-Kan Type Theorem for $\Gamma$-Groups, …
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19
votes
2
answers
1k
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A "universally non Hypercomplete" $\infty$-topos via Goodwillie calculus?
My question is :
Is there a classifying $\infty$-topos for $\infty$-connected objects ? Does this $\infty$-topos has a nice description (as an $\infty$-category ) ?
What I mean by $\infty$-connected …
- 42.4k
7
votes
1
answer
295
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Is the canonical model structure on strict $\infty$-Cat left proper?
Is the canonical (or Folk) model structure on the category of (strict) $\infty$-categories as constructed by Lafont, Métayer and Worytkiewicz in A folk model structure on omega-cat left proper ?
All i …
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