Monoidal Envelopes of Families of $\infty$-Operads and $\infty$-Operadic Kan Extensions
Category Theory
2025-07-22 v2 Algebraic Topology
Abstract
We provide details of the proof of Lurie's theorem on operadic Kan extensions. Along the way, we generalize the construction of monoidal envelopes of -operads to families of -operads and use it to construct the fiberwise direct sum functor, both of which we characterize by certain universal properties. Aside from their uses in the proof of Lurie's theorem, these results and constructions have their independent interest.
Keywords
Cite
@article{arxiv.2303.10813,
title = {Monoidal Envelopes of Families of $\infty$-Operads and $\infty$-Operadic Kan Extensions},
author = {Kensuke Arakawa},
journal= {arXiv preprint arXiv:2303.10813},
year = {2025}
}
Comments
Major revision. Proof of 1.1 simplified (now 2.1). Clarified relation to HA (Section 1). Gave a satisfactory universal property of monoidal envelopes of families (3.5). Improved exposition. 32 pages, identical to the version accepted in Applied Categorical Structures except for formatting