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In the paper Bar-Cobar Duality by Michael Ching, he proves that the category of operads in spectra is equivalent via the Bar-Cobar adjunction to some model category of co-operads defined in the paper. Since this paper was written there have been several advancements in the theory of $\infty$-operads and specifically enriched $\infty$-operads.

In particular, due to work of Haugseng, Chu, Gepner, Kock (and many others i'm missing and apologize for omitting), we can use symmetric sequences to model $\infty$-operads in the sense of Lurie's HA and we can also do so in the enriched setting. I think the model category defined in Ching's paper is compatible with any of the existing models for $\infty$-operads enriched in spectra that exist today (at least I believe this to be true - I haven't checked this too carefully).

What's a lot less clear to me is what $\infty$-categorical notion does the model category of PreCooperads (which are at the other side of the Bar-Cobar equivalence) model?

Question: What is the notion of enriched $\infty$-Cooperad for which the category of Cooperads in Spectra coincides with the underlying $\infty$-category of the model category of $PreCooperads$? (compatibly with the Bar-Cobar adjunction)

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  • $\begingroup$ Is Ching's model category of cooperads obtained by writing down something natural and then Bousfield localizing to force an equivalence of categories? Also, it might be worth mentioning that in Higher Algebra, Lurie uses a kind of strange construction to do some "spectral cooperad"-y things in the chapter on Goodwillie calculus. I think Heuts uses the same model. So there are other things one might like to compare too. $\endgroup$
    – Tim Campion
    Commented Dec 7, 2019 at 0:10

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