# Spliced arrow polycategory

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*, $𝓢ₚℂ$, has objects these of $ℂ×ℂ^{op}$ and its polymorphisms are given by
$$
𝓢ₚℂ \left(
\binom{X₀^{+}}{X₀^{-}},
\dots,
\binom{Xₙ^{+}}{Xₙ^{-}};
\binom{Y₀^{+}}{Y₀^{-}},
\dots,
\binom{Yₘ^{+}}{Yₘ^{-}};
\right)
=
[Y₀^{+},X₀^{+}] ×
[Xₙ^{-},Yₘ^{-}] ×
\prod_{i=0}^{n-1} [Xᵢ^{-},X_{i+1}^{+}] ×
\prod_{i=0}^{n-1} [Y_{i+1}^{+},Y_{i}^{-}].
$$
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 $ℂ \dashv ℂ^{op}$ in the monoidal bicategory of profunctors.