An infinity operad of normalized cacti

Abstract: Normalized cacti are a graphical model for the moduli space of genus 0 oriented surfaces. They are endowed with a composition that corresponds to glueing surfaces along their boundaries, but this composition is not associative. By introducing a new topological operad of bracketed trees, we show that this operation is associative up-to all higher homotopies and that normalized cacti form an $\infty$-operad in the form of a dendroidal space satisfying a weak Segal condition. In particular, this provides one of the few examples in the literature of infinity operads that are not a nerve of an actual operad.

algebraic topologycategory theoryquantum algebra

Audience: learners

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