Polycategorical splice

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

Definition

Definition. Let ℂ be a category. Its spliced arrow $\star$-polycategory, ${𝓢}_{p}C$, has objects these of $C\times C^{op}$ and its polymorphisms are given by

$${𝓢}_{p}C\left(\left(\genfrac{}{}{0ex}{}{X_0^{+}}{X_0^{-}}\right),\dots ,\left(\genfrac{}{}{0ex}{}{X_n^{+}}{X_n^{-}}\right);\left(\genfrac{}{}{0ex}{}{Y_0^{+}}{Y_0^{-}}\right),\dots ,\left(\genfrac{}{}{0ex}{}{Y_m^{+}}{Y_m^{-}}\right);\right)=[Y_0^{+},X_0^{+}]\times [X_n^{-},Y_m^{-}]\times \prod _{i=0}^{n-1}[X_i^{-},X_{i+1}^{+}]\times \prod _{i=0}^{n-1}[Y_{i+1}^{+},Y_i^{-}].$$

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 $C⊣C^{op}$ in the monoidal bicategory of profunctors.