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Results tagged with homotopy-theory
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user 20233
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.
4
votes
localization and $E_{\infty}$-spaces
I think all three are true (assuming $X$ simply connected in 2):
For 1, the key is the fact that the product of two homology equivalences is a homology equivalences (Künneth). So the map $L(X\times Y …
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15
votes
Accepted
Sheaves of complexes and complexes of sheaves
I'm going to restrict the discussion to Grothendieck abelian categories, because I'm not sure what can be said more generally.
The main reference for what follows is Appendix C in Lurie's book Spectra …
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5
votes
Necessity of hypercovers for sheaf condition for simplicial sheaves
This might become clearer when you know that you can obtain the category of sheaves from the category of presheaves by localizing: you invert maps that are isomorphisms locally for the topology. The l …
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8
votes
do spectra have diagonal maps?
The answer is no in general. So if $E$ is a ring spectrum, $E^\ast(X)$ need not be a ring, unless $X$ is a suspension spectrum. It is only a module over $E^\ast(pt)$. The ring structure on $E$ only gi …
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16
votes
Can we define homotopy groups using Tannakian categories
There certainly is a notion of higher Tannakian category which would have meaningful higher homotopy groups. I'm not sure how much of the theory has been worked out already, but higher Tannakian duali …
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6
votes
Aspheric functors and Grothendieck fibrations
This seems to be a special case of Prop. 4.1.2.15 in Higher Topos Theory:
Let $p: X \to S$ be a coCartesian fibration of simplicial sets. Then $p$ is smooth.
Take $X$ and $S$ to be the nerves of …
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22
votes
Accepted
When is a topological space the homotopy colimit of an open covering?
It is true in complete generality that $X$ is the homotopy colimit of $C_U$ (and hence that the fat realization computes the homotopy colimit in this case). This is a special case of Lurie's version o …
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4
votes
Accepted
Exact sequences in homotopy categories
I don't think the Seifert-van Kampen theorem follows from these kinds of considerations. Rather, it is the statement that the fundamental groupoid functor $\tau_{\leq 1}$ preserves homotopy colimits. …
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7
votes
Accepted
When is the Thom spectrum of a virtual vector bundle effective?
Yes.
A bit more generally, if $\xi$ is a perfect complex of rank $\geq 0$, then $Th(\xi)$ is effective (even very effective): the question is Nisnevich-local on $X$ and $\xi$ is locally a complex of …
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25
votes
Why is the motivic category defined over the site of smooth schemes only?
It's worth noting first that smooth schemes are essentially the smallest possible category from which one can define the motivic homotopy category: to make sense of $\mathbb A^1$-homotopy and of the N …
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6
votes
Accepted
Defining Hochschild homology of non-commutative DG-algebras with animated rings with a circl...
No, this does not work in the non-commutative case. In general we have $HH(A)=A\otimes_{A\otimes A^{\mathrm{op}}} A$, and this is only a $k$-module, not an algebra. If $A$ is commutative, the tensor p …
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14
votes
What's with equivariant homotopy theory over a compact Lie group?
Regarding 2, there is no difficulty in defining $G$-spectra in the setting of $\infty$-categories. The only complications I can think of are that (1) the orbit category $\mathrm{Orb}^G$ is now an $\in …
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29
votes
Accepted
Voevodsky's six functor formalism VS Lucas Mann's
There may be some confusion in this question about what exactly Voevodsky/Ayoub and Mann do, as they do very different things.
Mann's thesis constructs a formalism of six operations in the setting of …
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5
votes
On triangulated categories of pro-objects
Take this answer with a grain of salt since I can only provide vague references. Nevertheless, I claim that the properness of $M$ should be sufficient. If $C$ is a stable $\infty$-category, then $Pro( …
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7
votes
Accepted
Long exact sequences for parametrized cohomology
$\require{AMScd}$Yes. It is better to consider the more general situation of a pushout square of types, since pushout squares (unlike cofiber sequences) are stable under base change. Then the differen …
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