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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.
36
votes
Grothendieck's Homotopy Hypothesis - Applications and Generalizations
There are several ways to interpret the homotopy hypothesis.
Strictly speaking, Grothendieck's homotopy hypothesis is not a theorem yet: Grothendieck stated it in a very precise way in the very first …
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32
votes
Accepted
Is there an additive model of the stable homotopy category?
The answer is: no there isn't such a thing. Here is a rough argument (a full proof would deserve a little more care).
Using the main result of
S. Schwede, The stable homotopy category is rigid, Annals …
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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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25
votes
Are non-empty finite sets a Grothendieck test category?
The fact that G is a test category falls in large class of examples. Grothendieck proved that a small category $A$ is a local test category if and only if there exists a presheaf $I$ on $A$ which is a …
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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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23
votes
Accepted
Homotopy types of schemes
Any scheme which is separated of finite type, has at least a triangulation, hence is, in particular, a CW-complex. In fact, by a theorem of Lojasiewicz, this is true for any semi-algebraic set (one ca …
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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 …
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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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19
votes
Accepted
Is the simplicial completion of a localizer always a bousfield localization of the injective...
Let $A$ be a small category, and $W$ an $A$-localizer. Then we say that $W$ is regular if any presheaf $X$ over $A$ is canonically the homotopy colimit of the representable presheaves above $X$; see D …
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19
votes
Accepted
Do homotopy limits compute limits in the associated quasicategory in the non-combinatorial m...
Homotopy limits in any model category always coincide with limits in the associated $(\infty,1)$-category. To see this, you need to know the following (classical) facts:
1) given a cofibrant object $ …
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18
votes
Accepted
What's motivic about $\mathbb{A}^1$-homotopy theory? What's motivic about correspondences?
There is no need a priori to define these categories of motives starting from correspondences. The stable homotopy theory of schemes $SH$ may be characterized by a universal property saying that it is …
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16
votes
Accepted
Finiteness conditions on simplicial sheaves/presheaves
I don't know what Toën was talking about, but I suspect that it was about finiteness conditions for Artin stacks: the problem is that the usual finiteness conditions we look at for schemes (like the n …
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16
votes
Accepted
Why $K(X) \longrightarrow G (X)$ is a Poincaré duality for K-theory?
To my knowledge, one can only make this analogy fully consistent with Weibel's homotopy invariant $K$-theory $KH$ and $G$-theory (although the proofs of what I claim below rely heavily on our understa …
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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 …
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14
votes
Is there a homology theory that gives a *necessary and sufficient* condition for homotopy eq...
If $X$ and $Y$ are finite type nilpotent spaces, then $X$ and $Y$ are weakly equivalent if and only if their cochain complexes are quasi-isomorphic as $E_\infty$-algebras. Moreover, assuming $X$ and …
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