In *Higher Operads, Higher Categories*, Leinster nicely characterizes operads among monoidal categories (as PROs). Roughly speaking, a monoidal category comes from an operad if its set of objects is freely generated as a monoid by some generating set, and if every morphism is a monoidal product of morphisms with one output. But for the purposes of this question, all I need to say is that Leinster characterizes operads as special monoidal categories. Symmetric operads are again special symmetric monoidal categories.

In *Higher Algebra*, Lurie defines the notion of symmetric monoidal category (SMC) in a really interesting way: it is a coCartesian fibration $p\colon C\to Fin_{\ast}$, where $Fin_{\ast}$ is the category of pointed finite sets, such that the $n$ different "inert" maps $\langle n\rangle\to\langle 1\rangle\ $ induce an isomorphism $p^{-1}\langle n\rangle\cong (p^{-1}\langle 1\rangle)~^n$. But instead of defining a (symmetric colored) operad as a special kind of SMC, Lurie seems to relax the above coCartesian condition. An operad to Lurie is a functor $p\colon C\to Fin_{\ast}$, with coCarteisan lifts guaranteed only for certain arrows downstairs, e.g. over "inert" morphisms. While he does include other conditions not present in the SMC definition, we don't see directly how an operad is a special kind of SMC, even though both are kinds of functors $p\colon C\to Fin_{\ast}$.

While I really like Lurie's discussion (2.1.1) of operads, I find his definition (2.1.1.10) a bit opaque. I was expecting to see that an operad is a monoidal category with a certain extra (PRO-like) condition, but I don't see that in his definition.

**Question:** Is there a nice way to characterize Lurie's operads as special SMCs, i.e. as coCartesian fibrations $p\colon C\to Fin_{\ast}$ satisfying a PRO-like condition? How does that condition manifest in the 1-truncated case?

PS. While Lurie works with $\infty$-categories, I'm only interested in the 1-truncated case. I just appreciate the parsimony in his definition of SMC, and was surprised to see it evaporate in his definition of operad.

notthe objects of the corresponding operad but rather strings of these objects. But forbothLurie's way of encoding operads and SMC the objects are the objects of $p^{-1} \langle 1 \rangle$. The two definitions Lurie gives are much more suited to seeing symmetric monoidal category as special kinds of operads than the other way around. $\endgroup$ – Omar Antolín-Camarena Mar 27 '13 at 1:50Higher Algebra, section 2.2.4. $\endgroup$ – Omar Antolín-Camarena Mar 27 '13 at 17:28