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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.
7
votes
Is there an interesting definition of a category of test categories?
I guess that it might be slightly more interesting to look for a notion of morphism of local test categories. A first natural candidate is given by the notion of locally constant functor: a functor $u …
- 13.6k
7
votes
CW-complex of Eilenberg-MacLane spaces
This article of C. Berger gives a general principle to compute the number of $n+k$-cells of $K(\pi,n)$ for an abelian group $\pi$ in terms of "generalised Fibonacci numbers" (see section 4.10 in loc. …
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6
votes
Example of a weak basic localizer which is not a basic localizer?
As for basic localizers: they are all strongly saturated (Prop. 4.2.4 in Astérisque 308). I do not know any weak basic localizer which is not a basic localizer, but it is quite likely that there are s …
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20
votes
Accepted
Non standard (?) model category structure on (co)chain complexes.
This is well known, but formulated in a slightly different way.
Recall that a Frobenius category is an exact category which has enough injectives as well as enough projectives, and such that an objec …
- 13.6k
3
votes
cosimplicial algebras to dg-algebras
This construction is closely related to de Rham cohomology, and used to compute the continuous cohomology of Lie groups.
This is part of the tools to compute Beilinson's regulator, as explained in
J. …
- 13.6k
13
votes
Accepted
"Models" in homotopy theory
Yes, this is absolutely related to model categories (although the concept of models for an homotopy theory is much more general): the notion of model category was introduced by Quillen exactly to expr …
- 13.6k
11
votes
Examples of Brown (co)fibration categories that are not Quillen model categories?
Consider an abelian category $A$ (or, more generally, an exact category in the sense of Quillen), then the category of complexes of $A$ is a category of cofibrant objects with the quasi-isomorphisms a …
20
votes
Accepted
Derivators (in English)
For a few references in English, there are the papers of Heller, the main one being:
A. Heller, Homotopy theories, Mem. Amer. Math. Soc. 71 (383) (1988)
There is also a paper I wrote with A. Neeman, …
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6
votes
Accepted
Unicity up to homotopy of simplicial enrichments
If, given any fixed cofibrant object $A$, there is a funtor $map(A,-)$ from $M$ to simplicial sets which preserves weak equivalences between fibrant objects and commutes with homotopy limits up to can …
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29
votes
Accepted
Homotopy theory of schemes examples
To keep things simple, let us assume we work over a perfect field.
The easiest part of motivic cohomology which we can get is the Picard group (i.e. the Chow group in degree 1). This works essentially …
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2
votes
Pullback-stable model of fibrewise suspension of fibrations (in simplicial sets, or similar ...
This is a long comment which becomes longer and longer while I answer to the comment’s comments.
Take $U$ to be Voevodsky’s univalent universe classifying Kan fibrations with small fibers. There is t …
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15
votes
Accepted
Which motivic spectra are dualizable?
If $X$ is noetherian of dimension $>0$, there is always a compact object of $SH(X)$ which is not dualizable: e.g. $j_\sharp$ of the sphere spectrum where $j:U\to X$ is any dense open immersion with no …
- 13.6k
11
votes
For which spaces is homology (or cohomology) determined by the Eilenberg-Steenrod axioms
I don't know the precise answer (because it depends on the precise definition of an homology theory you may prefer), but I would ask the question in a slightly different way (giving out the CW-structu …
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6
votes
Why the Bousfield localization of spectra at topological K group is important?
The work of Akhil Mathew et al. really is a continuation of Thomason's: the goal is to consider algebraic $K$-theory and its sibblings (such as $TC$) and to see them as cohomology theories of (affine) …
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24
votes
Formalism of homotopy theory of schemes
Let $E(S)$ be the category of Nisnevich sheaves on the site of smooth schemes over some base $S$. Then Morel and Voevosy's homotopy category $\mathrm{H}(S)$ is obtained as a localization of the catego …
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