@AndréHenriques Thank you! The notion of "ribbon braided operad" seems very close to what I am looking for. Is there a relationship between this "untwisted" notion of planar algebra and E2 algebras or (Koszul dually) Hopf algebras?

Thank you for the reference! Can you give a rough idea of how these imply the result? Am I correct in understanding that you observe that simplicial operads and sOper-algebras give two model category structures on the same category and the free object transferability result of Corollary 9.4.1 implies they are Quillen equivalent?

Thanks! I'm aware of Efimov's construction but didn't know about the Hoyois paper, which is great. I actually have a paper that I need to update that incorrectly computes the HH_* of an almost category, so this is particularly helpful

Thank you @AgustíRoig! This looks like exactly the kind of statement I need.

@ranicky it is.

Thank you! From your references I also found this survey article, which was illuminating: arxiv.org/pdf/quant-ph/0303034.pdf