Inverting operations in operads

TL;DR

This paper develops a method to invert certain operations in operads using Dwyer-Kan localization, creating a new operad where specified one-ary operations become homotopy invertible, with applications to algebra categories.

Contribution

It introduces a localization technique for operads with respect to one-ary operations, extending the Dwyer-Kan hammock localization to operads and establishing a universal functor between algebra categories.

Findings

Constructed a localization for operads with respect to one-ary operations.

Established a universal property for the functor between algebra categories.

Provided a canonical map from the original operad to the localized operad.

Abstract

We construct a localization for operads with respect to one-ary operations based on the Dwyer-Kan hammock localization. For an operad O and a sub-monoid of one-ary operations W we associate an operad LO and a canonical map O to LO which takes elements in W to homotopy invertible operations. Furthermore, we give a functor from the category of O-algebras to the category of LO-algebras satisfying an appropriate universal property.