Free algebras via monoidal envelopes

TL;DR

This paper provides an explicit formula for free algebras in the context of $ $-operads using monoidal envelopes, simplifying the proof of their existence.

Contribution

It introduces a new approach to constructing free $ $-algebras via monoidal envelopes, offering a clearer and more direct proof.

Findings

Explicit colimit formula for free $ $-algebras

Simplified proof of existence of relative free $ $-algebras

Application to $ $-operad morphisms

Abstract

For any morphism of $\infty$-operads $\mathcal{P} \to \mathcal{O}$, we show that the free $\mathcal{O}$-algebra on a $\mathcal{P}$-algebra admits an explicit formula as the colimit over the $\mathcal{O}$-monoidal envelope of $\mathcal{P}$, providing a new and simple proof of the existence of relative free $\mathcal{O}$-algebras.