Day Convolution

Definition. Day Convolution

Let $𝒱︀$ be a symmetric monoidal closed category that is complete and cocomplete. Let $(𝒞︀, \otimes_{𝒞︀}, 𝐼)$ be a small monoidal $𝒱︀$ -enriched category. The Day convolution of $𝒱︀$ -enriched presheaves $𝐴, 𝐵 : {𝒞︀}^{op} \to 𝒱︀$ is the enriched presheaf

(𝐴 \otimes_{Day} 𝐵) 𝑐 = \int^{𝑢, 𝑣} 𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) \otimes_{𝒱︀} 𝐴 𝑢 \otimes_{𝒱︀} 𝐵 𝑣,

with unit the representable $𝒞︀ (-, 𝐼)$ .

Day convolution makes the enriched presheaf category $[{𝒞︀}^{op}, 𝒱︀]$ a monoidal category, symmetric when $𝒞︀$ is [1]. It is moreover closed.

Under the enriched Yoneda embedding the convolution of representables is representable, $𝒞︀ (-, 𝑥) \otimes_{Day} 𝒞︀ (-, 𝑦) ≅ 𝒞︀ (-, 𝑥 \otimes_{𝒞︀} 𝑦)$ , so Day convolution is the cocontinuous extension of the tensor of $𝒞︀$ .

As a Kan Extension

Equivalently, $𝐴 \otimes_{Day} 𝐵$ is the left Kan extension of $(𝑢, 𝑣) \mapsto 𝐴 𝑢 \otimes_{𝒱︀} 𝐵 𝑣$ along $\otimes_{𝒞︀}^{op} : {𝒞︀}^{op} \times {𝒞︀}^{op} \to {𝒞︀}^{op}$ .

In $𝐒𝐞𝐭$

When $𝒱︀ = 𝐒𝐞𝐭$ , we recover the ordinary Day convolution of presheaves $𝐴, 𝐵 : {𝒞︀}^{op} \to 𝐒𝐞𝐭$ , where the formula simplifies to:

(𝐴 \otimes_{Day} 𝐵) 𝑐 = \int^{𝑢, 𝑣} 𝒞︀ [𝑐, 𝑢 \otimes_{𝒞︀} 𝑣] \times 𝐴 𝑢 \times 𝐵 𝑣 .

References

Construction of biclosed categories ↗

As a Kan Extension

In 𝐒𝐞𝐭

References

In $𝐒𝐞𝐭$