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Is 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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    $\begingroup$ If you haven't seen it, look at McClure and Smith, "Cosimplicial objects and little n-cubes, I" arxiv.org/abs/math/0106024 . I don't think they answer your question, but they do produce monoidal structures on cosimplicial spaces, which I believe have the property that the functor $Spc\to Spc^\Delta$ sending a space to the constant cosimplicial diagram on that space is strongly monoidal. $\endgroup$
    – Charles Rezk
    Commented Oct 20, 2010 at 17:54
  • $\begingroup$ Thanks, I will take a look at it although I really need the Tot functor being part of the adjunction. By the way, the correct link for the McClure and Smith paper mentioned above is arxiv.org/abs/math/0211368 $\endgroup$
    – Toribio Smith
    Commented Oct 21, 2010 at 13:57

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