Day Convolution is Closed

Theorem. Day Convolution is Closed

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 and $𝐴, 𝐵$ be $𝒱︀$ -enriched presheaves on $𝒞︀$ . Define

(𝐴 ⊸ 𝐵) 𝑐 = \int_{𝑢} 𝐴 𝑢 ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑐) .

Then, for Day convolution $\otimes_{Day}$ ,

𝐴 \otimes_{Day} - ⊣ 𝐴 ⊸_{Day} - .

Proof. Proof that Day Convolution is Closed day-closed-structure-proof

For $𝑋, 𝐵 : {𝒞︀}^{op} \to 𝒱︀$ , the enriched hom in $[{𝒞︀}^{op}, 𝒱︀]$ is given by the end:

\begin{aligned} \int_{𝑐} ((𝐴 \otimes_{Day} 𝑋) 𝑐 ⊸_{𝒱︀} 𝐵 𝑐) & ≅ \int_{𝑐} ((\int^{𝑢, 𝑣} 𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) \otimes_{𝒱︀} 𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 𝑐) \\ ≅ \int_{𝑐} \int_{𝑢, 𝑣} ((𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) \otimes_{𝒱︀} 𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 𝑐) \\ ≅ \int_{𝑢, 𝑣} ((𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} \int_{𝑐} (𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) ⊸_{𝒱︀} 𝐵 𝑐)) \\ ≅ \int_{𝑢, 𝑣} ((𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑣)) \\ ≅ \int_{𝑣} (𝑋 𝑣 ⊸_{𝒱︀} \int_{𝑢} (𝐴 𝑢 ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑣))) \\ ≅ \int_{𝑣} (𝑋 𝑣 ⊸_{𝒱︀} (𝐴 ⊸ 𝐵) 𝑣) . \end{aligned}

∎

Symmetrically, $(𝐵 ⟜ 𝐴) 𝑐 = \int_{𝑣} 𝐴 𝑣 ⊸_{𝒱︀} 𝐵 (𝑐 \otimes_{𝒞︀} 𝑣)$ and $- \otimes_{Day} 𝐴 ⊣ - ⟜ 𝐴$ , so the enriched presheaf category $[{𝒞︀}^{op}, 𝒱︀]$ is biclosed [1].

References

Construction of biclosed categories ↗