Poincaré/Koszul duality for general operads

We record a result concerning the Koszul dual of the arity filtration on an operad. This result is then used to give conditions under which, for a general operad, the Poincaré/Koszul duality arrow of Ayala and Francis is an equivalence, using a proof similar to theirs. We discuss how the Poincaré/Koszul duality arrow for the little disks operad $\mathcal{E}_n$ relates to the work of Ayala and Francis when combined with the self-Koszul duality of $\mathcal{E}_n$.

Keywords

operad, Koszul duality, Goodwillie calculus, factorization homology

2010 Mathematics Subject Classification

55Pxx

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The author was supported by NSF Grant No. 1122374 while completing this work.

Received 25 March 2021

Received revised 7 August 2021

Accepted 7 August 2021

Published 10 August 2022