# Straightening for $\infty$-operads

There is this straightening/unstraightening procedure of Jacob Lurie's which takes a symmetric monoidal $\infty$-category (which is the data of a coCartesian morphism of simplicial sets $C^\otimes\to N(Fin_\ast)$ satisfying a Segal condition) and produces a "stack" object which is a symmetric monoidal functor $N(Fin_\ast)\to Cat_\infty$ whose target is $C^\otimes$ (where the image of $\{1,\ast\}$ is $C$, the underlying $\infty$-category of interest). Is there any kind of "straightening" construction for $\infty$-operads? In particular, is it necessarily impossible in general to produce such a functor corresponding to an $\infty$-operad (which is also a functor $O^\otimes\to N(Fin_\ast)$ satisfying certain properties)? If it's impossible, is there some weaker version of $Cat_\infty$ (perhaps without a full symmetric monoidal structure?) in which such a "stacky" presentation of an $\infty$-operad might land? My real goal here to ask if there is a way in which I can think of $\infty$-operads as (possibly some weakened version of) commutative algebra objects in some $\infty$-category. Does anyone know of a description in this way?

• It seems that, in light of section 6.3 of Lurie's Higher Algebra, we can think of $\infty$-operads as monoid objects in symmetric sequences on a symmetric monoidal category? – Jonathan Beardsley Jan 5 '15 at 5:15
• But that seems to depend on stability and presentability. – Jonathan Beardsley Jan 5 '15 at 5:19
• I think that infinity operads are Quillen equivalent to colored operads of simplicial sets. Does this maybe help? – Fernando Muro Jan 5 '15 at 9:22
• For a fixed set of objects, $\infty$-operads should presumably be the associative algebras in "coloured symmetric sequences" in spaces. As far as I know the monoidal $\infty$-category required for this to make sense has not been constructed, though. – Rune Haugseng Jan 5 '15 at 13:29
• I think that's just a different construction (though related). Certainly you can define the composition product on symmetric sequences in any reasonably nice ordinary symmetric monoidal category (including, say, sets). There's a definition on the nlab that I imagine one could make sense of for $\infty$-categories too... What were you thinking of using this construction for? – Rune Haugseng Jan 5 '15 at 17:40