Day Convolution

The Day convolution of two Presheaves F , G : 𝒱 m ⁢ a ⁢ t ⁢ h ⁢ r ⁢ m ⁢ o ⁢ p → Sets . : 𝐹 𝐺 → superscript 𝒱 𝑚 𝑎 𝑡 ℎ 𝑟 𝑚 𝑜 𝑝 Sets F,G:\mathcal{V}^{mathrm{op}}\to\mathrm{Sets}{.} on a Monoidal Category 𝒱 𝒱 \mathcal{V} is given by the following Coend:

(F⊗G)⁢(X)=∫Y,Z𝒱⁢(X,Y⊗Z)×F⁢(Y)×G⁢(Z)tensor-product𝐹𝐺𝑋superscript𝑌𝑍𝒱𝑋tensor-product𝑌𝑍𝐹𝑌𝐺𝑍\par(F\otimes G)(X)=\int^{Y,Z}\mathcal{V}(X,Y\otimes Z)\times F(Y)\times G(Z)

This gives rise to a Monoidal structure on the Presheaf Category 𝒱 ^ ^ 𝒱 \widehat{\mathcal{V}} , where the unit is the functor − → I {{-}\to{I}} ; EG: the Yoneda Embedding y ⁢ ( I ) 𝑦 𝐼 y(I) .