ScalingStacks

0NQM

Proposition 9.48. The spectral functor F~R→FR\tilde{F}_{R}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F_{R} is a weak equivalence of spectral categories, and there is an equivalence of A∞A_{\infty} ring spectra EndF~R∞⁡(R∨∞)≃L​R\End_{\tilde{F}^{\infty}_{R}}(R^{\lor\infty})\simeq LR.

0NQN

Proof. As the functor is actually surjective on objects, it is enough to show that it is fully faithful. This follows from the fact that mapping spectra in the homotopy pullback spectral category are computed as the homotopy pullbacks of the mapping spectra. Applying the long exact sequence to the homotopy pullback

MapF~R∞​(R∨m,R∨n)\textstyle{\mathrm{Map}_{\tilde{F}^{\infty}_{R}}(R^{\lor m},R^{\lor n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}MapFR∞​(R∨m,R∨n)\textstyle{\mathrm{Map}_{{F}^{\infty}_{R}}(R^{\lor m},R^{\lor n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}MapH​π0​F~R∞​(R∨m,R∨n)\textstyle{\mathrm{Map}_{H\pi_{0}\tilde{F}^{\infty}_{R}}(R^{\lor m},R^{\lor n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}MapH​π0​FR∞​(R∨m,R∨n)\textstyle{\mathrm{Map}_{H\pi_{0}{F}^{\infty}_{R}}(R^{\lor m},R^{\lor n})}

implies the desired equivalence. A similar computation with m=n=∞m=n=\infty implies the second statement. ∎

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

Andrew J. Blumberg, David Gepner, Goncalo Tabuada

Original source: arXiv:1001.2282v4

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