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Free algebras through Day convolution

### Hongyi Chu and Rune Haugseng

Algebraic & Geometric Topology 22 (2022) 3401–3458
##### 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.

##### Keywords
free algebras, Day convolution, infinity-operads, cartesian patterns, Segal conditions
Primary: 18N70
##### Publication
Received: 20 August 2020
Revised: 20 June 2021
Accepted: 31 August 2021
Published: 30 January 2023
##### Authors
 Hongyi Chu Max Planck Institute for Mathematics Bonn Germany Rune Haugseng Department of Mathematical Sciences Norwegian University of Science and Technology Trondheim Norway