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I am reading Lurie's Elliptic Cohomology II and it claims (Section 4.1.3) that for an $\mathbb{E}_\infty$-ring $A$ "there is an essentially unique symmetric monoidal functor $\mathcal{S} \to \operatorname{Mod}_A$ which preserves small colimits", where $\mathcal{S}$ is the category of spaces.

At first I thought this would be the map $X \mapsto \Sigma^\infty X \wedge A$. As spectra, it preserves small colimits, but I am not sure this is the case as $A$-modules. Does anyone have a more explicit description of this functor?

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Your description of the functor is correct. The suspension spectrum functor is homotopy colimit preserving, as you say. Smashing with A is left adjoint to the forgetful functor from A-modules to spectra, and this means that it also preserves homotopy colimits.

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  • $\begingroup$ As a remark: the forgetful fuctor $Spectra \to Mod_A$ is also conservative, and has a right adjoint (the function spectrum $X \mapsto F(A,X)$). This means that not only does the forgetful functor preserve colimits, but also that colimits in $Mod_A$ can be detected by checking if they are colimits in $Spectra$. $\endgroup$
    – Tyler Lawson
    Commented Mar 31, 2022 at 13:33
  • $\begingroup$ Do you perhaps mean the forgetful functor $Mod_A \to Spectra$? $\endgroup$
    – Sofía Marlasca Aparicio
    Commented Apr 4, 2022 at 8:40
  • $\begingroup$ @SofíaMarlascaAparicio Yes, sorry. $\endgroup$
    – Tyler Lawson
    Commented Apr 7, 2022 at 16:11

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