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Results tagged with homotopy-theory
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user 14447
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.
3
votes
Accepted
Explicit contraction for the universal simplicial bundle WG
Pages 75-81 of Appendix A of On the theory and applications of differential torsion products, Memoirs AMS 142 (1974), by V.K.A.M. Gugenheim and myself, gives a detailed treatment of the $W$-constructi …
- 35.5k
3
votes
Historical transition from classical homotopy to modern homotopy theory
This is not an answer, but a little too long for a comment.
Scott, what a list: 3 dead, 2 retired, and me; also I think Mike and I are always on the same side of the fence. As to the original questi …
- 30.4k
1
vote
Operad terminology - Operads with and without O(0).
A 2020 paper Operads, monoids, monads, and bar constructions by Ruozi Zhang, Foling Zou, and J.P. May revisits the foundations of operad theory and discusses several possible choices for what $\mathca …
- 5,309
81
votes
Accepted
Why should have Peter May worked with CGWH instead of CGH in "The Geometry of Iterated Loop ...
I'm not quite certain what Peter May had in mind 40 years ago,
but probably he had in mind the fact that pushouts are a lot better
behaved in CGWH than in CGH. Specifically, CGWH is closed
under push …
- 30.3k
2
votes
Is there a recognition principle for $\mathbb{E}_{\infty}$-spaces with zero?
I see no real point in considering $E_{\infty}$ spaces with $0$ except in the context of the multiplicative structure of an
$E_{\infty}$ ring space or, essentially equivalently, the multiplicative str …
- 30.4k
21
votes
Accepted
Serre fibration vs Hurewicz fibration
A short paper with references to several early counterexamples proves that (in the good category of compactly generated weak Hausdorff spaces) a Serre fibration in which the total space and base space …
- 35.5k
7
votes
Accepted
Different ways to “deloop” a (topological) $A_\infty$-algebra
There are two old papers that address this topic in some detail: R. W. Thomason. Uniqueness of delooping machines. \url{https://projecteuclid.org/euclid.dmj/1077313403}
Z. Fiedorowicz. Classifying …
- 30.4k
4
votes
Why study infinite loop spaces?
There is so much much more. For one historical starting point among many, you see that many spaces of interest are infinite loop spaces and that tells you how to calculate things about them. For jus …
- 30.4k
19
votes
Accepted
Homotopy extension of $E_{\infty}$-spaces
This is probably belaboring the obvious, but just take seriously the equivalence between grouplike $E_{\infty}$ spaces and connective spectra. See for example
Equivalence between $E_\infty$-spaces …
- 30.4k
7
votes
Accepted
Homotopy of functors
I'm surprised this has been up for days with nobody telling Atticus the obvious. Let $I$ be the unit interval category with two objects, $0$ and $1$, and one non-identity arrow $0 \to 1$. A natural …
- 30.4k
4
votes
Accepted
Group completion of $E_k$-algebras
Group completion and the answer to your questions (for $k\geq 2$) are probaby best understood homologically. An ancient definition is that a map $X\rightarrow Y$ of homotopy commutative $H$-spaces …
- 30.4k
4
votes
For which categories of spectra is there an explicit description of the fibrant objects via ...
David has answered 1-3, and I agree with him in the abstract. However, I would like to say more and specifically address 4, since there is a huge difference between the fibrant objects in the two mai …
- 30.4k
20
votes
Accepted
Is the $\infty$-category of spectra “convenient”?
My colleague Dylan answered first (I keep telling him not to spend too much time on this toy :) but I both agree and disagree with his "Yes of course". The same words are used with different meanings …
- 30.4k
23
votes
Why do homotopy theorists care whether or not $BP$ is $E_\infty$?
I suppose I should try to answer since the question of whether or not $BP$ is an $E_{\infty}$ ring spectrum
was Problem 1 of "Problems in infinite loop space theory'', http://www.math.uchicago.edu/~ma …
- 30.4k
4
votes
Is there a map of spectra implementing the Thom isomorphism?
I just noticed this question, so my apologies for a very belated answer. Proposition 20.5.5 of http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf states that a $k$-orientation of a spherical fibra …
- 30.4k