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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.
36
votes
3
answers
3k
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For which spaces is homology (or cohomology) determined by the Eilenberg-Steenrod axioms
This is a spinoff of Can anyone give me a good example of two interestingly different ordinary cohomology theories? . By an ordinary homology theory, I mean a functor on topological spaces which sati …
- 27.2k
37
votes
3
answers
3k
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Are there pairs of highly connected finite CW-complexes with the same homotopy groups?
Fix an integer n. Can you find two finite CW-complexes X and Y which
* are both n connected,
* are not homotopy equivalent, yet
* $\pi_q X \approx \pi_q Y$ for all $q$.
In Are there two non-homo …
- 27.2k
5
votes
1
answer
503
views
Explicit description of a free braided monoidal groupoid with inverses
Let G be a braided monoidal groupoid: it does no harm to suppose that the monoidal product on G is strictly associative, so I'll do that.
"With inverses" means that for every object $X$ of G, there …
- 27.2k
8
votes
0
answers
338
views
Reference request: Whitehead product and the Borel construction
This is a question about signs.
Fix
a based space $(X,x_0)$,
a topological group $G$
acting on $X$ from the left, so that the basepoint $x_0$ is fixed,
a based map $\alpha\colon S^p\to G$ ($p\geq1 …
- 27.2k
20
votes
2
answers
1k
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Are non-empty finite sets a Grothendieck test category?
A "test category" is a certain kind of small category $A$ which turns out to have the following property: the category $\widehat{A}$ of presheaves of sets on $A$ admits a model category structure, whi …
- 27.2k
52
votes
2
answers
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What's an example of an $\infty$-topos not equivalent to sheaves on a Grothendieck site?
My question is as in the title:
Does anyone have an example (supposing one exists) of an
$\infty$-topos which is known not to be equivalent to sheaves on a
Grothendieck site?
An $\infty$-topos is as …
- 27.2k