Koszulity of a certain properad

TL;DR

This work proves the Koszul contractibility of the properad $\mathscr{Y}^{(n)}_{prop}$ encoding planar genus-zero bialgebra-like structures, by combining planar dioperad Koszul theory with a geometric model built from moduli spaces of cloven Strebel quadratic differentials. The central method identifies the relevant bar/cobar complexes with cellular chains on assocoipahedra and carefully analyzes cloven subcomplexes to ensure vanishing of higher-genus contributions, yielding a quasi-isomorphism between the cobar-bar constructions and the original properad. The authors show that the canonical map $\Omega(\mathscr{Y}^{(n)}_{prop})^{!'} \to \mathscr{Y}^{(n)}_{prop}$ is a quasi-isomorphism, giving a cofibrant resolution and enabling homotopy transfer of $\mathscr{Y}^{(n)}_{prop}$-algebras. By tying Koszul duality for planar dioperads to properadic contractibility, the paper extends Koszulity results to a class of properads defined by genus-zero relations and links these algebraic structures to moduli spaces of meromorphic quadratic differentials, with potential implications for formality phenomena in higher algebraic structures.

Abstract

We establish that the properad $Y^{(n)}$, encoding bialgebras with a product of degree zero, a coproduct of degree $(1-n)$ and a rank three cyclic tensor, which satisfy a deformed version of the balanced infinitesimal bialgebra condition, is Koszul. This result is established by geometric methods, by studying cellular chain complexes of moduli spaces of a certain type of meromorphic quadratic differential on $\mathbb{CP}^1$, which we call cloven Strebel differentials. Using this geometric interpretation we can control the topology of these spaces, establishing vanishing of higher cohomology of the relevant bar complexes.

Theorems & Definitions (35)

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• Definition 1
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