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Symmetric monoidal structure on cosimplicial spacesIs there a monoidal structure on the category $Spc^{\Delta}$ of cosimplicial spaces such that in the adjunction $$ \Delta^{\bullet}\otimes-\colon Spc\leftrightarrows Spc^{\Delta}\colon Tot(-) $$ the left adjoint is (strong) symmetric monoidal? If this is true, can it be generalized by replacing $Spc$ by any symmetric monoidal category?
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