Koszul duality for operadic categories

Abstract

The aim of this sequel to [10] is to set up the cornerstones of Koszul duality and Koszulity in the context of a large class of operadic categories. In particular, we will prove that operads, in the generalized sense of [7], governing important operad-and/or PROP-like structures such as the classical operads, their variants such as cyclic, modular or wheeled operads, and also diverse versions of PROPs such as properads, dioperads, 1 2 PROPs, and still more exotic objects such as permutads and pre-permutads are binary quadratic, and describe their Koszul duals. We then prove, using the results of [11], that operads describing the most common structures are Koszul.

Modular operads as modules over the Brauer properad

“…This is different to some other approaches where a difference between the Koszul dual and the bar construction does yield a different notion of homotopy (or ∞-) modular operads, see work of Ward [War,§3] (which includes a homotopy transfer theorem in [War,Theorem 2.58]) and Batanin-Markl [BM21]. Also note that a definition of ∞-modular operads more along the lines of ∞-category theory is given by Hackney-Robertson-Yau in [HRYa,§3.2].…”

“…Various other treatments of modular operads and the Feynman transform exist in the literature. For example they appear as a special case of the theory of Feynman categories due to Kaufmann-Ward [KW], of the theory of groupoid-colored operads by work of Ward [War] (see also Dotsenko-Shadrin-Vaintrob-Vallette [DSVV]), and of the theory of operadic categories due to BM18;BM21]. The latter two are similar to our approach in the sense that, there too, modular operads appear as algebras over an operad-like object, which is shown to be Koszul (though in a more complicated fashion than for us).…”