Composition along a channel

Last updated Feb 3, 2024

Let $𝔸, 𝔹, ℂ$ be categories and let $(𝔹,⊗,I)$ be a monoidal structure. Two profunctors $P ﹕ 𝔸^{op} × 𝔹 → \mathbf{Set}$ and $Q﹕ 𝔹^{op} × ℂ → \mathbf{Set}$ compose along a $𝔹$-channel into the profunctor $(P \odot Q) ﹕ 𝔸^{op} × 𝔹 × 𝔹^{op} × ℂ → \mathbf{Set}$ defined by $$(P \odot Q)(A,B;B’,C) = ∫^{M ∈ 𝔹} P(A,B ⊗ M) × Q(M ⊗ B’, C).$$

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