Mario RomΓ‘n

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Spliced arrow polycategory

Last updated Sep 8, 2023

spliced-arrow-polycategory The spliced arrow polycategory generalizes the promonoidal category of spliced arrows. It is right adjoint to polycategorical contour.

# Definition

Definition. Let β„‚ be a category. Its spliced arrow $\star$-polycategory, $π“’β‚šβ„‚$, has objects these of $β„‚Γ—β„‚^{op}$ and its polymorphisms are given by $$ π“’β‚šβ„‚ \left( \binom{Xβ‚€^{+}}{Xβ‚€^{-}}, \dots, \binom{Xβ‚™^{+}}{Xβ‚™^{-}}; \binom{Yβ‚€^{+}}{Yβ‚€^{-}}, \dots, \binom{Yβ‚˜^{+}}{Yβ‚˜^{-}}; \right) = [Yβ‚€^{+},Xβ‚€^{+}] Γ— [Xβ‚™^{-},Yβ‚˜^{-}] Γ— \prod_{i=0}^{n-1} [Xα΅’^{-},X_{i+1}^{+}] Γ— \prod_{i=0}^{n-1} [Y_{i+1}^{+},Y_{i}^{-}]. $$ An intuition is that polymorphisms are spliced circles of morphisms. Composition glues together two of these circles along a typed hole. Duals are given by interchanging the top and bottom objects.

Remark. This polycategory underlies the Frobenius pseudomonoid generated by the pseudoduality $β„‚ \dashv β„‚^{op}$ in the monoidal bicategory of profunctors.