Free algebras through Day convolution

Chu, Hongyi; Haugseng, Rune

doi:10.2140/agt.2022.22.3401

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Free algebras through Day convolution

Hongyi Chu and Rune Haugseng

Algebraic & Geometric Topology 22 (2022) 3401–3458

DOI: 10.2140/agt.2022.22.3401

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

Mathematical Subject Classification

Primary: 18N70

References

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

Volume 22, issue 7 (2022)