promonoidal category of spliced arrows
Let $ℂ$ be a category. The multicategory of spliced arrows has objects pairs of objects in $ℂ$, and the multimorphisms are sequences of arrows in $ℂ$ separated by $n$ gaps; the sequence of arrows goes from $X$ to $Y$, with holes typed by ${ X_i, Y_i }_{i \in [1,\dots,n]}$. In other words,
$$\mathcal{S}ℂ \left( \binom{X_1}{Y_1}, \dots, \binom{X_n}{Y_n}; \binom{X}{Y} \right) = ℂ(X;X_1) × \left( \prod_{k=1}^{n-1} ℂ(Y_k, X_{k+1})\right) × ℂ(Y_n, Y).$$
Composition in the multicategory is defined by substitution of a spliced arrow into one of the gaps of the second; the identity is just $id_A - id_B$, the spliced arrow with a single gap typed by $(A,B)$.
Proposition. The multicategory of spliced arrows, $\mathcal{S}ℂ$, is precisely the promonoidal category induced by the duality $ℂ^{op} \dashv ℂ$ in the monoidal bicategory of profunctors: a promonoidal category over $ℂ \times ℂ^{op}$.
$$\mathcal{S}ℂ_2 \left( \binom{X_1}{Y_1}, \binom{X_2}{Y_2}; \binom{X}{Y} \right) = ℂ(X;X_1) × ℂ(Y_1;X_2) × ℂ(Y_2,Y). $$ $$\mathcal{S}ℂ \left( \binom{X_1}{Y_1}; \binom{X}{Y} \right) = ℂ(X;X_1) × ℂ(Y_1,Y). $$ $$\mathcal{S}ℂ_0 \binom{X}{Y} = ℂ(X;Y). $$
Tags: multicategory - promonoidal category, multicategory.
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