Toën's comparison conjecture for commutative ring spectra

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Let SpΣSp^{\Sigma} be the category of symmetric spectra and let Comm(SpΣ)Comm(Sp^{\Sigma}) be the model category of commutative ring spectra. Let LSpΣLSp^{\Sigma} be the tensor Segal category obtained by localizing SpΣSp^{\Sigma}, and define the Segal category of commutative monoids in a tensor Segal category AA by

Comm(A):=RHom‾⊗(FS,A),Comm(A):=\mathbb{R}\underline{Hom}^{\otimes}(FS,A),

where FSFS is the symmetric monoidal category of finite sets under disjoint union. Toën's comparison conjecture. The natural morphism

L(Comm(SpΣ))⟶Comm(LSpΣ)L(Comm(Sp^{\Sigma}))\longrightarrow Comm(LSp^{\Sigma})

is an equivalence of Segal categories. This compares commutative ring spectra with commutative monoids in the tensor Segal category of spectra and is presented as an important conjectural comparison.

References

Primary source

Betrand Toen, “Homotopical and Higher Categorical Structures in Algebraic Geometry”, arXiv:math/0312262 (2003).

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