Poincar\'e/Koszul Duality for General Operads

TL;DR

This paper explores the Koszul dual of operads' arity filtration and establishes conditions under which the Poincaré/Koszul duality is an equivalence, connecting to the little n-disks operad and its self-duality.

Contribution

It provides new criteria for the Poincaré/Koszul duality to be an equivalence for general operads, extending previous work on specific cases like the little n-disks operad.

Findings

Identifies the Koszul dual of the arity filtration on operads.

Establishes conditions for the Poincaré/Koszul duality arrow to be an equivalence.

Relates duality results to the self-Koszul duality of the little n-disks operad.

Abstract

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 Poincar\'e/Koszul duality arrow of Ayala-Francis is an equivalence. We discuss how the Poincar\'e/Koszul duality arrow for the little $n$-disks operad $\mathcal{E}_n$ relates to their work when combined with the self-Koszul duality of $\mathcal{E}_n$.