Questions tagged [parameterized-homotopy]
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12
questions
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$\infty$-local systems
Let $X$ be a "nice" topological space, $R$ a ring. I believe that there is an equivalence of $\infty$-categories betweeen:
the full subcategory of $D(X,R)$ (derived category of sheaves of $...
- 38.3k
7
votes
0
answers
290
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Funtoriality of twisted K-theory
I posted this question on math.stackexchange, but received no answer there.
In order to avoid the XY problem I will first state what I want, then what I think is the solution and how that failed until ...
- 301
7
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1
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Are these two notions of unstable localization suitably equivalent?
It seems to me that although homological localization (i.e. formally inverting $E$-homology equivalences for some $E$) is a reasonable thing to do to a spectrum, it's a pretty brutal thing to do to a ...
- 55.4k
10
votes
2
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433
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Rational parameterized spectra
Consider the homotopical category of rational dg-modules. I suppose this ought to present the rationalization of the homotopy theory of parameterized spectra. Has such "rational parameterized stable ...
- 19.2k
3
votes
1
answer
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Morphisms of parametrized ring spectra
This is a follow-up to this question, in which Denis Nardin nicely explained that
$$
\operatorname{Map}_{\operatorname{Fun}(X, \operatorname{Sp})}(E_X, E'_X)
\simeq
\operatorname{Map}(X, \operatorname{...
- 1,911
2
votes
1
answer
162
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Are morphisms of parametrized spectra themselves parametrized morphisms of spectra?
Let $X$ be a fixed parametrizing space. Let $E$ and $E'$ be two spectra and let $E_X$ and $E'_X$ be their trivial parametrized versions. Intuitively I imagine that the morphisms of parametrized ...
- 1,911
8
votes
1
answer
420
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Parametrized Dold-Kan correspondence?
The stable Dold-Kan correspondence says that for every commutative ring $R$, there is an equivalence of $\infty$-categories between the category $Ch(R)$ of (unbounded) chain complexes of $R$-modules ...
- 2,240
5
votes
1
answer
336
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$\Omega X$-action on spectral $X$-bundles
I can generalize the notion of a space over $X$ (where $X$ is a based and connected space) to the notion of a spectrum over $X$ by considering functors of quasicategories $X\to Mod_\mathbb{S}$. In the ...
- 9,855
12
votes
2
answers
540
views
Why does $Mf$ always support an $Mf$-orientation?
Let $f:X\to BGL_1(\mathbb{S})$ be a morphism of $E_n$-spaces and determine a principle $GL_1(\mathbb{S})$-bundle over $X$. Then it can be shown in the classical case that there is always a Thom ...
- 9,855
7
votes
0
answers
157
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Spectral Sequences of Parametrized Spectra
I apologize if this question is of the form "what are some interesting problems in bla" but I was wondering if anybody have studied the following set-up:
Suppose that I have a parametrized spectra $E$...
- 2,237
8
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Parametrized cancelations in stable Morse theory
Let $B$ be a closed manifold. Let $\pi : M\to B$ be a submersion such that each fiber is a manifold without boundary. Let $f : M \to \mathbb{R}$ be a function such that the restrictions $f_x$ to each ...
- 2,550
9
votes
1
answer
377
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Shriek push-forward for parameterized spectra
In May and Sigurdsson's Parameterized Homotopy Theory, Proposition 2.2.11, four isomorphisms of functors are given. For a pullback square of base spaces $C=holim(A\overset{f}\to B\overset{j}\leftarrow ...
- 9,855