Coend

A coend of a Functor F : 𝒞 op × 𝒞 → 𝒟 : 𝐹 → superscript 𝒞 op 𝒞 𝒟 F:\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\to\mathcal{D} is the Weighted Colimit of F 𝐹 F weighted by the Hom Functor of 𝒞 op superscript 𝒞 op \mathcal{C}^{\mathrm{op}} .

Elementary Definition

A coend is the universal Cowedge of F 𝐹 F ; for any other cowedge w : F ⁢ ( c , c ) → W : 𝑤 → 𝐹 𝑐 𝑐 𝑊 w:F(c,c)\to W , we have a unique map ∫ c F ⁢ ( c , c ) → W → superscript 𝑐 𝐹 𝑐 𝑐 𝑊 \int^{c}F(c,c)\to W .