Envelopes for Algebraic Patterns

We generalize Lurie’s construction of the symmetric monoidal envelope of an $\mathrm{\infty }$-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 $\mathrm{\infty }$-operads over various algebraic patterns. In particular, we show that the $G$-$\mathrm{\infty }$-operads of Nardin-Shah are equivalent to “fibrous patterns” over the $(2,1)$-category $\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}({\mathbb{F}}_G)$ of spans of finite $G$-sets. When $G$ is trivial this means that Lurie’s $\mathrm{\infty }$-operads can equivalently be defined over $\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}(\mathbb{F})$ instead of ${\mathbb{F}}_{\ast }$.