The Leibniz PROP is a Crossed Presimplicial Algebra

Abstract

We prove that the Leibniz PROP is isomorphic (as $\Bbbk$-linear categories) to the symmetric crossed presimplicial algebra $\Bbbk[(Δ^+)^{op} \mathbb{S}]$ where $Δ^+$ is the skeletal category of finite well-ordered sets with surjections, but the distributive law between $(Δ^+)^{op}$ and the symmetric groups $\mathbb{S} = \bigsqcup_{n\geq 1} S_n$ is not the standard one.