Envelopes for Algebraic Patterns

TL;DR

This paper extends Lurie's symmetric monoidal envelope construction to algebraic patterns, providing a comparison framework for $ $-operads over different algebraic patterns, including $G$-$ $-operads and fibrous patterns.

Contribution

It generalizes the symmetric monoidal envelope to algebraic patterns and characterizes its essential image, enabling comparisons of $ $-operads across various algebraic patterns.

Findings

The envelope becomes fully faithful over the envelope of the terminal object.

$ $-operads over different algebraic patterns are comparable via the new framework.

$G$-$ $-operads are equivalent to fibrous patterns over spans of finite $G$-sets.

Abstract

We generalize Lurie's construction of the symmetric monoidal envelope of an $\infty$-operad to the setting of algebraic patterns. This envelope becomes fully faithful when sliced over the envelope of the terminal object, and we characterize its essential image. Using this, we prove a comparison result that allows us to compare analogues of $\infty$-operads over various algebraic patterns. In particular, we show that the $G$-$\infty$-operads of Nardin-Shah are equivalent to "fibrous patterns" over the $(2, 1)$-category $\mathrm{Span}(\mathbb{F}_G)$ of spans of finite $G$-sets. When $G$ is trivial this means that Lurie's $\infty$-operads can equivalently be defined over $\mathrm{Span}(\mathbb{F})$ instead of $\mathbb{F}_*$.