The Quotient and its Right Adjoint in Day Convolution

Definition. The Quotient and its Right Adjoint in Day Convolution

Let $𝐶$ be a small monoidal category. Recall that for presheaves $𝐴, 𝐵 : 𝐶^{op} \to 𝐒𝐞𝐭$ , the Day convolution provides a closed monoidal structure:

(𝐴 \otimes 𝐵) (𝑐) = \int^{𝑢, 𝑣} 𝐶 [𝑐, 𝑢 \otimes 𝑣] \times 𝐴 (𝑢) \times 𝐵 (𝑣)

(𝐴 ⊸ 𝐵) (𝑐) = \int_{𝑢} 𝐴 (𝑢) \to 𝐵 (𝑢 \otimes 𝑐)

which forms an adjunction $𝐴 \otimes - ⊣ 𝐴 ⊸ -$ .

For covariant functors $𝐴 : 𝐶 \to 𝐒𝐞𝐭$ and presheaves $𝐵 : 𝐶^{op} \to 𝐒𝐞𝐭$ , we can define the quotient and its right adjoint, each of which is a presheaf on $𝐶$ :

(𝐷_{𝐴} 𝐵) (𝑐) = \int^{𝑢} 𝐴 (𝑢) \times 𝐵 (𝑢 \otimes 𝑐)

(\sqrt[𝐴]{𝐵}) (𝑐) = \int_{𝑢, 𝑣} 𝐶 [𝑢 \otimes 𝑣, 𝑐] \to (𝐴 (𝑢) \to 𝐵 (𝑣))

These form the adjunction $𝐷_{𝐴} ⊣ \sqrt[𝐴]{-}$ .