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1. • CommentRowNumber1.
   • CommentAuthorhepworth
   • CommentTimeOct 22nd 2013
   • (edited Oct 22nd 2013)
   • PermaLink
   Author: hepworth
   Format: MarkdownItexThis 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 $\mathcal{V}$. All symmetric monoidal functors are taken to be lax, not strong and not strict. **The universal property:** Let $\mathcal{A}$ be a small symmetric monoidal $\mathcal{V}$-category. The $\mathcal{V}$-category $[\mathcal{A},\mathcal{V}]$ of all $\mathcal{V}$-functors $\mathcal{A}\to\mathcal{V}$ can be made symmetric monoidal under the Day Convolution: $$(\Phi\ast\Psi)(a) = \int^{b_1,b_2} \Phi(b_1) \otimes \Psi(b_2) \otimes A(b_1\otimes b_2,a)$$ In this situation the evaluation functor $$ [\mathcal{A},\mathcal{V}]\times \mathcal{A}\longrightarrow \mathcal{V} $$ is symmetric monoidal, and so induces a functor $$ e\colon SymMon(\mathcal{B},[\mathcal{A},\mathcal{V}]) \longrightarrow SymMon(\mathcal{B}\times\mathcal{A},\mathcal{V}) $$ between categories of symmetric monoidal $\mathcal{V}$-functors and symmetric monoidal $\mathcal{V}$-natural transformations. Here $\mathcal{B}$ is any symmetric monoidal $\mathcal{V}$-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 $\mathcal{B}=\mathbf{1}$, the "unit" symmetric monoidal $\mathcal{V}$-category. In this case the universal property reduces to an equivalence between the category of commutative monoids in $[\mathcal{A},\mathcal{V}]$ and the category of symmetric monoidal functors $\mathcal{A}\to\mathcal{V}$. This [MathOverflow question](http://mathoverflow.net/questions/130616/reference-for-lax-monoidal-functors-monoids-under-day-convolution) identifies a couple of references to this fact.

   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 $\mathcal{V}$. All symmetric monoidal functors are taken to be lax, not strong and not strict.

   The universal property: Let $\mathcal{A}$ be a small symmetric monoidal $\mathcal{V}$-category. The $\mathcal{V}$-category $[\mathcal{A},\mathcal{V}]$ of all $\mathcal{V}$-functors $\mathcal{A}\to\mathcal{V}$ can be made symmetric monoidal under the Day Convolution:

   $$(\Phi\ast\Psi)(a) = \int^{b_1,b_2} \Phi(b_1) \otimes \Psi(b_2) \otimes A(b_1\otimes b_2,a)$$

   In this situation the evaluation functor

   $$[\mathcal{A},\mathcal{V}]\times \mathcal{A}\longrightarrow \mathcal{V}$$

   is symmetric monoidal, and so induces a functor

   $$e\colon SymMon(\mathcal{B},[\mathcal{A},\mathcal{V}]) \longrightarrow SymMon(\mathcal{B}\times\mathcal{A},\mathcal{V})$$

   between categories of symmetric monoidal $\mathcal{V}$-functors and symmetric monoidal $\mathcal{V}$-natural transformations. Here $\mathcal{B}$ is any symmetric monoidal $\mathcal{V}$-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 $\mathcal{B}=\mathbf{1}$, the “unit” symmetric monoidal $\mathcal{V}$-category. In this case the universal property reduces to an equivalence between the category of commutative monoids in $[\mathcal{A},\mathcal{V}]$ and the category of symmetric monoidal functors $\mathcal{A}\to\mathcal{V}$. This MathOverflow question identifies a couple of references to this fact.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeOct 23rd 2013
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexSurprisingly, I don't recall having seen this before.

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

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeDec 27th 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThis came up again on [mathoverflow](https://mathoverflow.net/questions/319580/monoidal-functors-mathcal-c-to-mathcal-d-mathcal-v-are-monoidal-functors) and I suggested a somewhat more abstract way to prove it (though I still don't know a reference).

   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).