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Results tagged with homotopy-theory
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user 41291
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
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1
answer
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$\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $??(\Omega G)$
This is a crosspost (with minor alterations).
For a topological group $G$, assigning to a $G$-space $X$ the (canonical) map $EG\times_GX\to BG$ establishes an equivalence between the homotopy category …
- 18.4k
9
votes
1
answer
228
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Does the (1-)topos structure on simplicial sets have any homotopy-theoretic significance?
To give an example of a peculiar feature of simplicial sets that I cannot remember encountering anywhere in the context of homotopy theory: every simplicial set $X$ possesses partial map classifier $X …
- 18.4k
5
votes
0
answers
232
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Is this a stack?
A continuous map $f:X\to Y$ and a vector bundle $E\to X$ seem to give rise to a presheaf of groupoids on $Y$ along the following lines. For an open $U\subseteq Y$, each section of $f$ over $U$ (i. e. …
- 18.4k
4
votes
0
answers
111
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Is there a simple algebraic setup to accomodate fibres and cofibres at the same time?
If I understand it correctly, there are two mutually dual "leading principles" in homotopy theory:
never perform quotients, add structure instead;
never require subobjects, take fibres instead.
Al …
- 18.4k
5
votes
1
answer
308
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Conceptual and practical reasons and consequences of inverting weak equivalences
Although dealing with this in one or other form for many years, to my shame this question only struck me now.
One of the most radical differences between categories of "algebraic" and "topological" ki …
- 18.4k
26
votes
1
answer
926
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Are (semi)simple Lie groups some sort of "homotopy quotient groups" of their maximal tori?
Warning: non-specialist writing, some rubbish possible.
The formula $h^*(BG)\cong h^*(BT)^W$ valid for complex oriented cohomology of the classifying space of a compact Lie group $G$ with maximal tor …
- 18.4k
11
votes
2
answers
551
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The cofibration/fibration $\leftrightarrow$ epi/mono confusion
A recent question Why do we need model categories? reminded me of this long-standing confusion of mine -- I mentioned it in an answer there, and then decided to ask a separate question about it. I eve …
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11
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3
answers
916
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Can Homotopy Type Theory or algebraic geometry deal with homotopy fibers in terms of families?
((In conclusion)
It was hard to choose which answer to accept. I decided for the one which addressed most of the various aspects of the question.
)
(Later addon)
I now decided to put a bounty on t …
- 18.4k
1
vote
1
answer
186
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Comparing self-equivalences of a triangulated category and automorphisms of its Grothendieck...
There is a homomorphism from the group of (isomorphism classes of) self-equivalences of a triangulated category to the automorphism group of its Grothendieck group. Is this homomorphism surjective? If …
- 18.4k
22
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3
answers
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Stable homotopy type theory?
This is a combined question, strictly speaking I am asking three questions concerning, respectively, homotopy type theory, stable homotopy theory and Yetter-Drinfeld modules. But I believe in the pres …
- 18.4k
9
votes
0
answers
370
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Which of the physics dualities are closest in essence to the Spanier-Whitehead duality (with...
First of all, what I want to ask is slightly more elaborate than what stands in the title (hence the subquestion).
I am telling this since as it is, the title contains a meaningful question, but it s …
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8
votes
2
answers
687
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"Economic" Eilenberg-MacLane topological abelian groups
This might be regarded as a sequel to my previous "Economic" CW-structure for Eilenberg-MacLane spaces? However the content seems to be quite different.
I believe it is easy to prove that for any top …
- 18.4k
6
votes
0
answers
157
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Are there versions of highly connected covers of Lie groups with highly periodic homotopy gr...
There is much activity around the study of highly connected covers of Lie groups (well, of their "infinite rank" versions like $\displaystyle{\lim_{N\to\infty}} \ O(N)$, say).
Looking at the resultin …
- 18.4k
8
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3
answers
646
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Homotopy type of some lattices with top and bottom removed
The only reaction to this question on math.SE was 21 views in 4 days, so I decided to repost it here. I am not changing anything.
There was an interesting question on MO which OP removed by some reas …
- 18.4k
4
votes
1
answer
224
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A Dold-Thom style construction of a cohomology class from a sphere bundle
Re-reading my comment to the question https://mathoverflow.net/q/195912/41291 I suddenly realized that I do not understand something crucial about it. For the purposes of that crucial thing let me ref …
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