Day convolution

Remarks

Equivalently, $F \ast G$ is the left Kan extension of the functor $(a, b) \mapsto F(a) \times G(b)$ on $\mathcal{C} \times \mathcal{C}$ along $\otimes \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$. Natural transformations $F \ast G \to H$ correspond to families of maps $F(a) \times G(b) \to H(a \otimes b)$ natural in $a$ and $b$. Example: let $\mathcal{C}$ be the discrete category on $\N$ with $\otimes = +$ and $I = 0$, so a functor $F$ is a sequence of sets $F(0), F(1), \ldots$. Since $\Hom(a + b, n)$ is a point when $a + b = n$ and empty otherwise, $(F \ast G)(n) = \bigsqcup_{a + b = n} F(a) \times G(b)$, the Cauchy product. Non-example: the pointwise product $(F \times G)(c) = F(c) \times G(c)$ is also a monoidal structure, but it is not the Day convolution. In the example above it pairs elements of equal degree, and its unit is the constant functor with value a point, not $\Hom(0, -)$. The monoidal category $[\mathcal{C}, \mathbf{Set}]$ with Day convolution is closed, and $a \mapsto \Hom_{\mathcal{C}}(a, -)$ is a strong monoidal functor $\mathcal{C}^{\mathrm{op}} \to [\mathcal{C}, \mathbf{Set}]$. The same construction works for functors into a cocomplete closed symmetric monoidal category $\mathcal{V}$ in place of $\mathbf{Set}$.