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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.

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When you think of an non-symmetric operad as special kind of monoidal in the first paragraph, the objects of the monoidal category are not the objects of the corresponding operad but rather strings of these objects. But for both Lurie'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. –  Omar Antolín-Camarena Mar 27 '13 at 1:50
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For what you want, given an operad $p : C \to Fin_{\ast}$, you would need to construct a $q : D \to Fin_{\ast}$ where the objects of $q^{-1}\langle 1 \rangle$ come from all of $C$, not just from $p^{-1} \langle 1 \rangle$. –  Omar Antolín-Camarena Mar 27 '13 at 1:53
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I think the construction you are looking for, that given an operad produces a symmetric monoidal category, is the monoidal envelope described in Higher Algebra, section 2.2.4. –  Omar Antolín-Camarena Mar 27 '13 at 17:28
    
If you feel like it, you could compile these comments into an answer. –  David Spivak Mar 27 '13 at 23:31
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up vote 6 down vote accepted

Given a symmetric monoidal category one can construct its underlying operad (well, symmetric colored operad, but I won't keep mentioning this). This operad has the same objects as the SMC. The operads arising this way can be characterized, so that we can regard a SMC as a special kind of operad.

Conversely, given an operad, one can construct its symmetric monoidal envelope, a symmetric monoidal category whose objects are not just the objects of the operad but rather formal (commutative) products of them. The SMCs arising this way can be characterized, as David mentions in the question, so that we can also regard operads as special SMCs.

Now, when operads and SMC are implemented by Lurie's definitions, the direction in which it is easy to go is that of regarding a SMC as an operad. If the SMC is given by a functor $p : C \to Fin_{\ast}$, then regarded as an operad it is the same functor, and whether we view it as a SMC or as an operad its objects are the objects of the category $p^{-1}\langle 1 \rangle$.

To go in the other direction, we need to change the functor: given an operad $p : C \to Fin_{\ast}$, its symmetric monoidal envelope will be a different functor $q : D \to Fin_{\ast}$. The objects of the envelope, which are the objects of $q^{-1}\langle 1 \rangle$, will be all the objects of $C$, not just the objects of the subcategory $p^{-1}\langle 1 \rangle$.

Lurie constructs the symmetric monoidal envelope in section 2.2.4 of Higher Algebra.

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And ${\bf Oprd}\to {\bf SMC}$ is left adjoint to ${\bf SMC}\to{\bf Oprd}$, correct? –  David Spivak Mar 29 '13 at 22:17
    
Yes, that's right. –  Omar Antolín-Camarena Mar 31 '13 at 17:10
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