Profinite $\infty$-operads

TL;DR

This paper develops a profinite completion functor for infinity-operads, establishing a model category framework and characterizing lean infinity-operads through homotopical finiteness, with applications to unital and infinity-category cases.

Contribution

It introduces a new profinite completion functor for infinity-operads as a left Quillen functor, based on lean infinity-operads and their homotopical properties.

Findings

Constructed a profinite completion functor as a left Quillen functor.

Characterized lean infinity-operads via homotopical finiteness.

Extended the construction to unital and infinity-category cases.

Abstract

We show that a profinite completion functor for (simplicial or topological) operads with good homotopical properties can be constructed as a left Quillen functor from an appropriate model category of infinity-operads to a certain model category of profinite infinity-operads. The construction is based on a notion of lean infinity-operad, and we characterize those infinity-operads weakly equivalent to lean ones in terms of homotopical finiteness properties. Several variants of the construction are also discussed, such as the cases of unital (or closed) infinity-operads and of infinity-categories.