All Questions
Tagged with big-list homotopy-theory
12 questions
3
votes
0
answers
159
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Proofs of the loop-suspension adjunction in infinity-categories
$\DeclareMathOperator{\Map}{Map}$$\DeclareMathOperator{\Fun}{Fun}$$\DeclareMathOperator{\const}{const}$$\DeclareMathOperator{\colim}{colim}$$\DeclareMathOperator{\lim}{lim}$In Elements of $\infty$-...
- 348
-8
votes
2
answers
859
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Homotopy theory and algebraic topology last 10 years. Is it a dying field? [closed]
I'm under the impression that algebraic topology is a dying field in mathematics. That was my impression but I think I'm wrong. As every person I do need some evidence that my impression is not ...
28
votes
1
answer
2k
views
Useful ideas in category theory which violate the principle of equivalence
Or an alternate title: using evil for the greater good.
In category theory, the principle of equivalence says that statements about things should be invariant under the appropriate notion of thing-...
10
votes
1
answer
414
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Big list: barycentric subdivision of simplicial sets
I'm preparing a seminar on the barycentric subdivision of simplicial sets and I'm looking for some examples of this construction appearing in the literature. Since it's a useful technique (at least in ...
0
votes
1
answer
146
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Examples of cartesian-closed model categories
One of the main settings of my research are Cartesian-closed model categories. I would like to know as many interesting and/or important examples of such categories as possible. "Interesting"...
- 6,962
42
votes
12
answers
7k
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Why is the definition of the higher homotopy groups the "right one"?
If someone asked me the question for the fundamental group, I would answer as follows:
The connection to classification of covering spaces.
The fundamental group of many spaces is an object of ...
46
votes
2
answers
4k
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What are the potential applications of perfectoid spaces to homotopy theory?
This year's Arizona Winter School was on perfectoid spaces, and there were quite a few homotopy theorists in the audience. I'd like to get a "big list" of reasons homotopy theorists might care about ...
24
votes
2
answers
3k
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Roadmap to Hill-Hopkins-Ravenel
How does one go from an understanding of basic algebraic topology (on the level of Allen Hatcher's Algebraic Topology and J.P. May's A Concise Course in Algebraic Topology) to understanding the paper ...
- 3,309
44
votes
9
answers
3k
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Homotopy as a general organizing principle
One of the realizations that led to the development of Homotopy Type Theory (HoTT) is that the ideas of homotopy theory have very broad applicability in mathematics. Indeed, Quillen model categories ...
11
votes
3
answers
1k
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A category with weak equivalences that is not a model category
I'm only considering complete and cocomplete categories. A pair $(\mathfrak{X} , \mathfrak{W}) $ is, by definition, a category with weak equivalences if $ \mathfrak{X} $ is a category and $ \mathfrak{...
- 875
23
votes
8
answers
3k
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How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there?
Offhand I can think of two ways in classical homotopy theory:
Show that $\pi_k(S^n)=0$ for $k\lt n$ by deforming a map $S^k\to S^n$ to be non-surjective, then contracting it away from a point not in ...
39
votes
3
answers
4k
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In which situations can one see that topological spaces are ill-behaved from the homotopical viewpoint?
In the eighties, Grothendieck devoted a great amount of time to work on the foundations of homotopical algebra.
He wrote in "Esquisse d'un programme": "[D]epuis près d'un an, la plus grande partie ...