A-infinity monads and completion

TL;DR

This paper explores A-infinity monads within topological categories, defining a completion functor that preserves the A-infinity structure, thus advancing the understanding of homotopy-coherent monads.

Contribution

It introduces a completion functor for A-infinity monads and proves that this functor retains the A-infinity structure, providing new tools for homotopy-coherent algebraic structures.

Findings

The completion functor is itself an A-infinity-monad.

A-infinity monads can be coherently completed within topological categories.

The framework enhances understanding of homotopy-coherent algebraic structures.

Abstract

Given an operad A of topological spaces, we consider A-monads in a topological category C . When A is an A-infinity-operad, any A-monad K : C -> C can be thought of as a monad up to coherent homotopies. We define the completion functor with respect to an A-infinity-monad and prove that it is an A-infinity-monad itself.