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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 π±. All symmetric monoidal functors are taken to be lax, not strong and not strict.
The universal property: Let π be a small symmetric monoidal π±-category. The π±-category [π,π±] of all π±-functors πβπ± can be made symmetric monoidal under the Day Convolution:
(Ξ¦*Ξ¨)(a)=β«b1,b2Ξ¦(b1)βΞ¨(b2)βA(b1βb2,a)In this situation the evaluation functor
[π,π±]ΓπβΆπ±is symmetric monoidal, and so induces a functor
e:SymMon(β¬,[π,π±])βΆSymMon(β¬Γπ,π±)between categories of symmetric monoidal π±-functors and symmetric monoidal π±-natural transformations. Here β¬ is any symmetric monoidal π±-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 β¬=1, the βunitβ symmetric monoidal π±-category. In this case the universal property reduces to an equivalence between the category of commutative monoids in [π,π±] and the category of symmetric monoidal functors πβπ±. This MathOverflow question identifies a couple of references to this fact.
Surprisingly, I donβt recall having seen this before.
This came up again on mathoverflow and I suggested a somewhat more abstract way to prove it (though I still donβt know a reference).
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