An infinity Operad of Normalized Cacti

Gluing surfaces along their boundaries allows us to define composition laws that have been used to define cobordism categories, as well as operads and props associated to surfaces. These have played an important role in recent years, for example in constructing topological field theory or computing the homology of the moduli space of Riemann surfaces.

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 particular, this provides one of the few examples in the literature of infinity operads that are not a nerve of an actual operad.