The Leibniz PROP is a crossed presimplicial algebra

TL;DR

The paper shows that the Leibniz $\mathbb{K}$-algebra is isomorphic, at the level of $\Bbbk$-linear categories, to the twisted product $\text{Sym} \otimes_{\omega} \text{Simp}$, i.e., a crossed presimplicial algebra on the pair $((\Delta^+)^{op}, \mathbb{S})$, with a nonstandard distributive law $\omega$ between $(\Delta^+)^{op}$ and the symmetric groups. It builds this through explicit generators and relations for the free algebras Mag, Simp, Braid, and Sym, then constructs the fundamental distributive law $\zeta$ on $\partial$ and $\chi$, and verifies distributive laws that propagate to Mag, Braid, Simp, and Sym. The main result identifies Leib as a crossed presimplicial structure and clarifies its relation to PROPs/operads: Leib is isomorphic to $\text{Sym} \otimes_{\omega} \text{Simp}$ as $\Bbbk$-algebras, but this is not a monoidal isomorphism, hence not an operadic/PRO(P) isomorphism. The paper also highlights homological implications, expressing Leibniz (co)homology in terms of Tor/Ext over the constructed categorical algebras and outlining connections and potential future work comparing Leibniz and Lie homologies via functorial bridges. Overall, it provides a concrete combinatorial realization of Leibniz-type operations as a crossed presimplicial algebra and situates this within the broader operadic/PRO(P) landscape with explicit distributive laws.

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.

Theorems & Definitions (42)

• Remark 1.1
• Proposition 1.2
• proof
• Proposition 1.3
• proof
• Remark 1.4
• Proposition 1.5
• proof
• Proposition 1.6
• proof