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• Tags
  • #homotopy/stable-homotopy/equivariant
• Refs:
  • https://www.math.ias.edu/~lurie/papers/Broken-new.pdf#page=66
• Links:
  • coend

Day convolution

• 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 : $$\begin{align*} F\widehat{\otimes}G({-}) \coloneqq\int^{x, y\in \mathsf{C}} \mathsf{C}(x\otimes_C y, {-}) \otimes_D F(x) \otimes_D G(y) .\end{align*}$$