Day convolution

Day convolution is an operation on functors that can be seen as a categorified version of function convolution. It was first introduced by Brian Day in 1970 ^[1] in the general context of enriched functor categories. Day convolution acts as a tensor product for a monoidal category structure on the category of functors $[\mathbf {C} ,V]$ over some monoidal category $V$ .

Definition

Let $\mathbf {C}$ be a symmetric monoidal category enriched over a monoidal category $V$ . Given two functors $F,G\colon \mathbf {C} \to V$ , we define their Day convolution as the following coend.^[2]

F\ast G=\int ^{x,y\in \mathbf {C} }\mathbf {C} (x\otimes y,-)\otimes Fx\otimes Gy

References

^ Day, Brian (1970). "On closed categories of functors". Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics. 139: 1–38.
^ Loregian, Fosco. "This is the (co)end, my only (co)friend". ArXiv preprint: 51.

[1] Day, Brian (1970). "On closed categories of functors". Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics. 139: 1–38.

[2] Loregian, Fosco. "This is the (co)end, my only (co)friend". ArXiv preprint: 51.

[1]

[2]