The Leibniz PROP is a Crossed Presimplicial Algebra - Askarogullari, Murat Can; Kaygun, Atabey (Springer Science and Business Media LLC, 2025)
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.
Keywords: FOS: Mathematics; Mathematics - Category Theory; Category Theory (math.CT); Category theory; homological algebra; A32, 18m85, 18m32