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?
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asked Jan 5, 2015 at 4:10
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1$\begingroup$ 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? $\endgroup$– Jonathan BeardsleyCommented Jan 5, 2015 at 5:15
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1$\begingroup$ But that seems to depend on stability and presentability. $\endgroup$– Jonathan BeardsleyCommented Jan 5, 2015 at 5:19
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1$\begingroup$ I think that infinity operads are Quillen equivalent to colored operads of simplicial sets. Does this maybe help? $\endgroup$– Fernando MuroCommented Jan 5, 2015 at 9:22
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6$\begingroup$ 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. $\endgroup$– Rune HaugsengCommented Jan 5, 2015 at 13:29
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1$\begingroup$ 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? $\endgroup$– Rune HaugsengCommented Jan 5, 2015 at 17:40
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1 Answer
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This question was answered in the affirmative by Rune Haugseng in Section 4 of https://arxiv.org/pdf/1708.09632.
answered May 15 at 21:30