Free algebras through Day convolution

TL;DR

This paper develops a framework using Day convolution on presheaves to characterize free algebras and left adjoints in higher category theory, providing new proofs and generalizations of existing results.

Contribution

It introduces conditions based on Segal structures and cartesian patterns to compute free algebras via colimits, extending Lurie's work on operadic extensions.

Findings

Established colimit formulas for free algebras

Provided simplified proofs of Lurie's operadic results

Extended the theory to new categorical contexts

Abstract

Building on the foundations in our previous paper, we study Segal conditions that are given by finite products, determined by structures we call cartesian patterns. We set up Day convolution on presheaves in this setting and use it to give conditions under which there is a colimit formula for free algebras and other left adjoints. This specializes to give a simple proof of Lurie's results on operadic left Kan extensions and free algebras for symmetric $\infty$-operads.