# Questions tagged [parameterized-homotopy]

The parameterized-homotopy tag has no usage guidance.

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 $...

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 ...

7
votes

1
answer

290
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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 ...

10
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2
answers

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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 ...

3
votes

1
answer

227
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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{...

2
votes

1
answer

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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 ...

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 ...

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 ...

12
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2
answers

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### 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 ...

7
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0
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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 $E$...

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 ...

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 ...