In Section 2.3.2 of Higher Algebra, Lurie introduces the notion of generalized $\infty$-operads. This is a functor $p:\mathcal{O}^\otimes \to \mathcal{F}\mathrm{in}_\ast$ of $\infty$-categories, where $\mathcal{F}\mathrm{in}_\ast$ denotes the category of pointed sets $\langle n\rangle=(\{\ast,1,\dots,n \},\ast)$, such that:
- $p$ admits cocartesian morphisms over inert morphisms.
- For each diagram of inert maps $\sigma: \Delta ^1 \times \Delta ^1\to \mathsf{Fin}_\ast$ depicted as $\require{AMScd}$ \begin{CD} \langle m\rangle @>>> \langle n\rangle\\ @VVV @VVV\\ \langle m' \rangle @>>> \langle n'\rangle, \end{CD} if the map $\langle m'\rangle^\circ \amalg _{\langle n'\rangle^\circ }\langle n\rangle^\circ \to\langle m \rangle^\circ $ is bijective, then the pullback of $p$ along $\sigma$ classifies a pullback square $\Delta^1\times \Delta ^1\to \mathcal{C}\mathrm{at}_\infty.$ (Here $\langle n \rangle ^\circ = \{1,\dots ,n\}.$)
- Every square $\Delta^1\times \Delta ^1 \to \mathcal{O}^{\otimes}$ of $p$-cocartesian morphism lying over a square as in the previous point is a $p$-limit diagram.
In the case where the fiber of $p$ over $\langle 0\rangle \in \mathcal{F}\mathrm{in}_\ast$ is a contractible $\infty$-groupoid, $p$ is just an $\infty$-operad, which is an $\infty$-categorical version of a symmetric multicategory. This makes me wonder: Do generalized $\infty$-operads also have an antecedent in classical category theory?
Remark: In the non-symmetric case (where the base category is replaced by $\mathbf{\Delta}^{\mathrm{op}}$), there is a clean answer: Generalized non-symmetric $\infty$-operads are an analog of virtual double categories. This is explained in Gepner and Haugseng's article [GH15].
[GH15] David Gepner, Rune Haugseng, Enriched ∞-categories via non-symmetric ∞-operads, Advances in Mathematics, Volume 279, 2015, Pages 575-716.
