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Let $X$ be a quasi-category (an inner Kan complex), let $\mathfrak{C}(X)$ be its rigidification (its associated simplicial category). J. Lurie in "Higher Topos Theory" proved the following theorem 2.1.2.2 (I give a particular case when $\phi=Id$) :

"Straightening and unstraigntening functors determine a Quillen equivalence $$ St_{X} : (sSets)_{/X} \rightleftarrows sSets^{\mathfrak{C}(X)} :Un_{X}$$ where $(sSets)_{/X}$ is endowed with the contravariant model structure and $sSets^{\mathfrak{C}(X)}$ with the projective model structure."

I was wondering wether one could get a similar statement for $sAb^{\mathfrak{C}(X)}$ endowed with the projective model structure (here $sAb$ is the category of simplicial abelian groups), more generally for simplicial $R$-modules. Where we replace the category $(sSets)_{/X}$ by the category of simplicial algebraic gadgets over $X$.

Morever can we stabilize this theorem, in order to identify the category $spectra^{\mathfrak{C}(X)}$ together with a category of spectra over $X$?

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Let $\mathcal{C}$ be a presentable $\infty$-category, and let $X$ be a Kan complex. Let $\mathcal{C}_{/X} = \mathrm{Fun}(X, \mathcal{C})$. Then $\mathrm{Sp}(\mathcal{C}_{/X}) = \mathrm{Sp}(\mathcal{C})_{/X}$. Indeed, this follows from the facts that $\mathrm{Sp}(\mathcal{C}_{/X}) = \mathcal{C}_{/X} \otimes \mathrm{Sp}$ and $\mathcal{C}_{/X} = X \otimes \mathcal{C}$. If $\mathcal{C}$ is the $\infty$-category $\mathcal{S}$ of spaces, then this implies that $\mathrm{Sp}_{/X} = \mathrm{Sp}(\mathcal{S}_{/X})$. Googling tells me that a keyword is "parametrized spectra", and that the above argument appears in Proposition 3.4 of https://arxiv.org/pdf/1112.2203.pdf.

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  • $\begingroup$ Thank you for your answer, can you give some references please? $\endgroup$ – David C Apr 7 '20 at 8:05
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    $\begingroup$ @DavidC Edited. $\endgroup$ – skd Apr 7 '20 at 15:15

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