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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 $R$-modules) spanned by objects with locally constant cohomology
  • dg modules over $C_\ast(\Omega X,R)$ (or "$\infty$-local systems", or "parametrized $HR$-module spectra")

Is something like this true, under some conditions? What's a reference?

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    $\begingroup$ (I assume $X$ is connected) Can you start with $\mathsf{Psh}(X;\mathsf{Sp}) \simeq \mathsf{Mod}_{\mathbb{S}[\Omega X]}$ since the left hand side has the 'skyscraper sheaf at the basepoint' as a compact generator with endomorphism spectrum given by $\mathbb{S}[\Omega X]:= \Sigma^{\infty}_+\Omega X$? (Alternatively, descent gives a similar equivalence before stabilizing). Then tensor both sides with $\mathsf{Mod}_R$. $\endgroup$ Sep 12, 2021 at 0:26
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    $\begingroup$ or maybe I'm skipping a step and you're more interested in going from "sheaves with locally constant cohomology" to "presheaves on the underlying groupoid"? That sounds like the territory of section A.1 and A.4 in Higher Algebra $\endgroup$ Sep 12, 2021 at 0:31
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    $\begingroup$ Thanks for the pointer Dylan! Yes exactly, I want to go from cohomologically locally constant sheaves to presheaves on the underlying groupoid. $\endgroup$ Sep 12, 2021 at 0:46
  • $\begingroup$ As it turns out, this question has been asked before. mathoverflow.net/questions/153344 $\endgroup$ Sep 12, 2021 at 20:33

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This is to record that Dylan's suggestion in the comments was on the money: Sections A.1 and A.4 of Higher algebra are precisely what is needed. Specifically there is an equivalence like this whenever $X$ is a topological space locally of singular shape, and $\mathrm{Shv}(X)$ is hypercomplete, e.g. if $X$ is a metric ANR locally of finite dimension.

Section A.1 defines in general what it means to have a locally constant sheaf valued in an $\infty$-category $C$ on a topological space $X$ (or more generally an $\infty$-topos). Now $D(X,R)$ is the $\infty$-category of hypercomplete $D(R)$-valued sheaves on $X$. The full subcategory of locally constant objects in Lurie's sense coincides precisely with the full subcategory of objects with locally constant cohomology. So if $\mathrm{Shv}(X)$ is hypercomplete then the $\infty$-category defined in my first bullet point is simply the category of locally constant $D(R)$-valued sheaves on $X$, as defined in Lurie.

What Lurie then explains/defines in A.4 is that for $X$ locally of singular shape, there is an equivalence of $\infty$-categories between locally constant $C$-valued sheaves on $X$, and functors $\mathrm{Sing}(X) \to C$. When $C=D(R)$ these are precisely parametrized $HR$-module spectra as in the second bullet point.

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    $\begingroup$ Note that by corollary A.1.17 in ibid. you don't even need to assume $\mathrm{Shv}(X)$ hypercomplete, since locally constant objects are automatically hypercomplete in an ∞-topos of locally constant shape. $\endgroup$ Sep 12, 2021 at 16:46
  • $\begingroup$ However I do not understand why the category of hypersheaves with locally constant cohomology is equivalent to the category of locally constant sheaves of complexes. I can imagine that every $H_n$ is constant on some open neighborhood $U_n$ of a point but $\bigcap_n U_n$ is not a neighborhood of the point.. $\endgroup$ Sep 12, 2021 at 20:51

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