Composition along a channel

Let $\mathbb{A},\mathbb{B},C$ be categories and let $(\mathbb{B},\otimes ,I)$ be a monoidal structure. Two profunctors $P﹕{\mathbb{A}}^{op}\times \mathbb{B}\to \mathbf{Set}$ and $Q﹕{\mathbb{B}}^{op}\times C\to \mathbf{Set}$ compose along a $\mathbb{B}$-channel into the profunctor $(P\odot Q)﹕{\mathbb{A}}^{op}\times \mathbb{B}\times {\mathbb{B}}^{op}\times C\to \mathbf{Set}$ defined by $(P\odot Q)(A,B;B^{\prime },C)={\int }^{M\in \mathbb{B}}P(A,B\otimes M)\times Q(M\otimes B^{\prime },C).$

Tags: profunctor References

• Open Diagrams via Coend Calculus (Román, 2020)