---
date: 2022-02-23 18:45
modification date: Friday 25th March 2022 20:21:38
title: Day convolution
aliases: [Day convolution, convolution product, convolution]

---

Last modified date: <%+ tp.file.last_modified_date() %>


---

- Tags 
	- #homotopy/stable-homotopy/equivariant 
- Refs:
	- <https://www.math.ias.edu/~lurie/papers/Broken-new.pdf#page=66>
- Links:
	- [coend](coend.md)

---

# Day convolution

![](attachments/Pasted%20image%2020220510150912.png)

- Turns the functor category $\Fun(\cat C\op, \cat D)$ into a monoidal category $\Fun_{\hat \tensor}(\cat C\op, \cat D)$.

For $\cat{C}$ be a [symmetric monoidal category](Unsorted/monoidal%20category.md) over another monoidal category $(\cat{D}, \tensor_D)$, and define a **convolution product**
\[
\hat{\tensor}: \Fun( \opcat{\cat C}, \cat D)\cartpower{2} &\to \Fun(\opcat{\cat C}, \cat D) \\
(F, G) &\mapsto F\hat{\tensor} G
\]
where $F\hat\tensor G$ is the following left [Kan extension](Kan%20extension.md) :

\begin{tikzcd}
	{\cat{C}^{\times 2}} && {\cat{D}^{\times 2}} \\
	\\
	{\cat{C}} && {\cat{D}}
	\arrow["{(F, G)}", from=1-1, to=1-3]
	\arrow["{\wait\tensor_D\wait}", from=1-3, to=3-3]
	\arrow["{\wait \tensor_C \wait}"', from=1-1, to=3-1]
	\arrow["{F\hat\tensor G}"', dashed, from=3-1, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXGNhdHtDfV57XFx0aW1lcyAyfSJdLFsyLDAsIlxcY2F0e0R9XntcXHRpbWVzIDJ9Il0sWzIsMiwiXFxjYXR7RH0iXSxbMCwyLCJcXGNhdHtDfSJdLFswLDEsIihGLCBHKSJdLFsxLDIsIlxcd2FpdFxcdGVuc29yX0RcXHdhaXQiXSxbMCwzLCJcXHRlbnNvcl9DIiwyXSxbMywyLCJGXFxoYXRcXHRlbnNvciBHIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d)


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

\[
\cat{C}\cat{D}(F\hat\tensor G, ?) \cong \cat{C}^2\cat{D}(\tensor_D \circ (F, G), \,\, ? \circ \tensor_C)
.\]

Equivalently, take the 2-category of [cocomplete](cocomplete) [tensor categories](Unsorted/tensor%20category.md) $\Cat_{c\tensor}$, 
\[
\Cat_{c\tensor}( \Fun_{\hat\tensor}(\cat C\op, \cat D), ?) \cong \Cat_{c\tensor}(\cat C, ?) \times \Cat_{\tensor}(\cat D, ?)
.\]



Equivalently, define by the following [coend](coend.md) :
\[
F\hat\tensor G(\wait) \da \int^{x, y\in \cat{C}} 
\cat{C}(x\tensor_C y, \wait) \tensor_D F(x) \tensor_D G(y)
.\]

![](attachments/Pasted%20image%2020220316203634.png)
![](attachments/Pasted%20image%2020220320035436.png)