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Results tagged with homotopy-theory
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user 65
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.
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Something like Yoneda's lemma
This is inspired by The Whitehead for maps question.
Consider two maps f, g: X\to Y which happen to induce the same maps (of discrete spaces) [Z, X] \to [Z, Y] for every Z. Does this mean f and g are …
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Whitehead for maps
I think the proper Whitehead for maps says that if the cone of the map has trivial homotopy groups, then the map is a homotopy equivalence.
Edit: see also the discussion of Whitehead theorem in the c …
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Something like Yoneda's lemma
I found it myself: the image of id \in [X, X] under both maps will be the same class in [X, Y], which is the definition of homotopy between f and g, so the ansewr is yes.
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Killing the torsion in homotopy
Origin
This question was asked by John Baez in This Week's Finds in Mathematical Physics (Week 286). Therefore, please don't upvote this question (unless you really want to), but do upvote the answer …
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Homotopy theory of schemes examples
Is it possible give an example of (or explain) how the Voevodsky et al.'s homotopy theory of schemes computes higher Chow groups?
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Formalism of homotopy theory of schemes
I have some vague knowledge about the philosophy that schemes should be thought of as similar to topologic spaces, and we should divide everything by homotopy, and that the space should be actually sh …
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What is the "intuition" behind "brave new algebra"?
This is a general phrase that refers to the direction of
higher category theory, per Lurie (you know references)
scheme homotopy theory, per Voevodsky
derived spaces, per Ben-Zvi and Nadler (0706.032 …
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$E_\infty$ spectrum corresponding to $\Bbb Z_p$
First of the questions about derived algebraic geometry from a noobie.
The way I understand it, every discrete ring $R$ corresponds to some ring spectrum whose $\pi_0$ is $R$. Now consider $p$-adic nu …
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How to get product on cohomology using the K(G, n)?
This came up in the question about Eilenberg-MacLane spaces. Given the definition of K(G, n), it's easy to prove that there is a map K(G,n) x K(G,n) --> K(G,n) that endows cohomology with an additive …
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Cohomology and Eilenberg-MacLane spaces
Indeed, the statement is that homotopy classes of continuous maps of pointed spaces $[X, K(G, n)]$ are in 1-1 correspondence with the elements of singular homology $H^n(X, G)$ for a CW-complex $X$.
T …
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Spectrum of the Grothendieck ring of varieties
I think I'll be collecting references I found in this answer, rather then in the original (already large) post:
http://www-fourier.ujf-grenoble.fr/~peters/hodge.f/peters-proc.pdf (Wayback Machine)
I …
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Spectrum of the Grothendieck ring of varieties
Here's a problem that may ultimately require just simple algebraic-geometry skills to be solved, or perhaps it's very deep and will never be solved at all. From the comments, some literature and my me …
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6
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How to think about model categories?
I've read about model categories from an Appendix to one of Lurie's papers.
What are the examples of model categories? What should be my intuition about them?
E.g. I understand the typical examples …
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