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 Rune, 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)

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)

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 Rune, Chu, Gepner, Kock (and many others i'm probably 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 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)

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 Rune, 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)

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 Rune, Gepner, Kock (and many others i'm probably missing and apologize for omitting), we can use symmetric sequences to model $\infty$-operads in the sense of lurie 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 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)

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 Rune, Chu, Gepner, Kock (and many others i'm probably 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 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)