TITLE: Day convolution intuition
QUESTION [13 upvotes]: In the nLab, Day convolution is introduced as a generalisation of convolution of complex-valued functions, but I'm wondering how exactly to understand this.
I can (just about) parse the definitions, but have absolutely no intuition or geometric insight at all.
Here is my thinking so far (excuse all of the quotation marks) :

A way of thinking about simple examples of presheaves is to imagine an association: to each open set of a topological space associate a set of continuous functions on that open set (really we could obtain a group or ring structure on this set of functions, but let's ignore that for a moment and just look at set-valued presheaves).
Then a presheaf is a 'function' that maps an open set to a set of functions. So a convolution of presheaves is obtained by 'blurring' these 'functions' together.

But what this actually means is rather beyond me.
The issue (for me) is twofold:

What does Day convolution look like in this simple case where we take our starting category to be $\mathsf{Op}(T)$, which is the category of open sets and inclusion maps of some topological space $T$ (which is usually the motivating example for presheaves).
In fact, can we even look at this example?
As far as I can see, $\mathsf{Op}(T)$ doesn't admit a monoidal structure;
What does Day convolution look like generally?
Given some convolution of presheaves, are there any simple examples (if 1. doesn't work) that give a good intuition, where the Day convolution has a reasonably succinct description?

Edit:

How does Day convolution fit in with 'regular' convolution.
That is, can we recover the usual convolution from Day convolution?

I hope this question is rigorous enough, and if not then I'll try to edit it to be more so.

REPLY [19 votes]: Day convolution is a categorification of the monoid algebra construction. There is a formal analogy between the two, but one is not a literal generalisation of the other. So to address your question 3, we should not expect to recover the usual convolution from Day convolution.
Let's develop the following analogy:
\begin{array}{|c|c|} \hline
\textbf{monoid algebra} & \textbf{Day convolution} \\ \hline \hline
\text{set} & \text{category} \\ \hline
\text{monoid} & \text{monoidal category} \\ \hline 
\text{ring } R & \text{monoidally cocomplete category } \mathcal{V} \\ \hline
R\text{-module} & \text{cocomplete } \mathcal{V}\text{-category} \\ \hline
R\text{-algebra} & \text{monoidally cocomplete } \mathcal{V}\text{-category} \\ \hline
\text{free } R\text{-module on a set } X& \text{free cocomplete } \mathcal{V}\text{-category on a category } \mathcal{C}\\
R^{(X)}& [\mathcal{C}^\text{op},\mathcal{V}]\\ \hline
\text{free } R\text{-algebra on a monoid } M & \text{free monoidally cocomplete } \mathcal{V}\text{-category}\\ 
&  \text{on a monoidal category } \mathcal{A} \\
R^{(M)} \text{ with convolution product} & [\mathcal{A}^\text{op},\mathcal{V}] \text{ with Day convolution}\\ \hline
\end{array}
Here a monoidally cocomplete category is a cocomplete monoidal category $\mathcal{V}$ such that $\otimes \colon \mathcal{V} \times \mathcal{V} \to \mathcal{V}$ is cocontinuous in each variable. This condition corresponds in our analogy to the distributivity of multiplication over addition in a ring.
Let $(e_x)_{x\in M}$ be the canonical basis for $R^{(M)}$, so that each element of $R^{(M)}$ can be written $f = \sum_{x} f(x) e_x$. The convolution product on $R^{(M)}$ is then determined by the requirement that $M \to R^{(M)}, x \mapsto e_x$ is a monoid homomorphism. For:
\begin{equation}
\begin{split}
 f \ast g & = \left(\sum_{x} f(x) e_x \right) \ast \left(\sum_{y} g(y) e_y \right) \\
& = \sum_{x,y} f(x)g(y) e_x \ast e_y \\
& = \sum_{x,y} f(x)g(y) e_{xy}.
\end{split}
\end{equation}
An analogous argument gives the formula for Day convolution. The representables $\mathcal{A}(-,A)$ provide a ''basis'' of $[\mathcal{A}^{op},\mathcal{V}]$: each object may be expressed as the canonical colimit $$F \cong \int^{A} FA \otimes \mathcal{A}(-,A).$$ The Day convolution is determined by the requirement that the Yoneda embedding $\mathcal{A} \to [\mathcal{A}^\text{op},\mathcal{V}]$ be strong monoidal. We have:
\begin{equation}
\begin{split}
 F \ast G & \cong \left(\int^{A} FA \otimes \mathcal{A}(-,A)  \right) \ast \left(\int^{B} GB \otimes \mathcal{A}(-,B) \right) \\
& \cong \int^{A,B} F(A)\otimes G(B) \otimes \mathcal{A}(-,A) \ast \mathcal{A}(-,B) \\
& \cong \int^{A,B} F(A)\otimes G(B) \otimes \mathcal{A}(-,A\otimes B).
\end{split}
\end{equation}
Note that we have used the requirement that the Day convolution product must preserve colimits in each variable.
Now, Day convolution can be defined for the more general case of a promonoidal category $\mathcal{A}$. Here we can continue our analogy and think of the promonoidal structure as providing the ''structure coefficients'' of the Day convolution product.