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    • CommentRowNumber1.
    • CommentAuthorhepworth
    • CommentTimeOct 22nd 2013
    • (edited Oct 22nd 2013)

    This is a request for a reference for a universal property of Day convolution.

    Everything below takes place in the setting of categories enriched over an appropriate symmetric monoidal category 𝒱. All symmetric monoidal functors are taken to be lax, not strong and not strict.

    The universal property: Let π’œ be a small symmetric monoidal 𝒱-category. The 𝒱-category [π’œ,𝒱] of all 𝒱-functors π’œβ†’π’± can be made symmetric monoidal under the Day Convolution:

    (Ξ¦*Ξ¨)(a)=∫b1,b2Ξ¦(b1)βŠ—Ξ¨(b2)βŠ—A(b1βŠ—b2,a)

    In this situation the evaluation functor

    [π’œ,𝒱]Γ—π’œβŸΆπ’±

    is symmetric monoidal, and so induces a functor

    e:SymMon(ℬ,[π’œ,𝒱])⟢SymMon(β„¬Γ—π’œ,𝒱)

    between categories of symmetric monoidal 𝒱-functors and symmetric monoidal 𝒱-natural transformations. Here ℬ is any symmetric monoidal 𝒱-category. The universal property states that the functor e is an equivalence.

    My Question: Does anybody here know of a reference for the universal property (that e is an equivalence) stated above?

    A simple case: Let us take ℬ=1, the β€œunit” symmetric monoidal 𝒱-category. In this case the universal property reduces to an equivalence between the category of commutative monoids in [π’œ,𝒱] and the category of symmetric monoidal functors π’œβ†’π’±. This MathOverflow question identifies a couple of references to this fact.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeOct 23rd 2013

    Surprisingly, I don’t recall having seen this before.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeDec 27th 2018

    This came up again on mathoverflow and I suggested a somewhat more abstract way to prove it (though I still don’t know a reference).

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