Day convolution

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Turns the functor category \({\mathsf{Fun}}(\mathsf{C}^{\operatorname{op}}, \mathsf{D})\) into a monoidal category \({\mathsf{Fun}}_{\widehat{\otimes}}(\mathsf{C}^{\operatorname{op}}, \mathsf{D})\).

For \(\mathsf{C}\) be a symmetric monoidal category over another monoidal category \((\mathsf{D}, \otimes_D)\), and define a convolution product \begin{align*} \widehat{\otimes}: {\mathsf{Fun}}( { {\mathsf{C}}^{\operatorname{op}}}, \mathsf{D}){ {}^{ \scriptscriptstyle\times^{2} } } &\to {\mathsf{Fun}}({ {\mathsf{C}}^{\operatorname{op}}}, \mathsf{D}) \\ (F, G) &\mapsto F\widehat{\otimes} G \end{align*} where \(F\widehat{\otimes}G\) is the following left Kan extension :

Link to Diagram

Here the diagram is not required to commute, but rather satisfy some universal property: there is an equivalence of categories? #todo

\begin{align*} \mathsf{C}\mathsf{D}(F\widehat{\otimes}G, ?) \cong \mathsf{C}^2\mathsf{D}(\otimes_D \circ (F, G), \,\, ? \circ \otimes_C) .\end{align*}

Equivalently, take the 2-category of cocomplete tensor categories \(\mathsf{Cat}_{c\otimes}\), \begin{align*} \mathsf{Cat}_{c\otimes}( {\mathsf{Fun}}_{\widehat{\otimes}}(\mathsf{C}^{\operatorname{op}}, \mathsf{D}), ?) \cong \mathsf{Cat}_{c\otimes}(\mathsf{C}, ?) \times \mathsf{Cat}_{\otimes}(\mathsf{D}, ?) .\end{align*}

Equivalently, define by the following coend : \begin{align*} F\widehat{\otimes}G({-}) \coloneqq\int^{x, y\in \mathsf{C}} \mathsf{C}(x\otimes_C y, {-}) \otimes_D F(x) \otimes_D G(y) .\end{align*}

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