# Abelian versions of straightening and unstraightening functors

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 $$X$$?

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.