We build new algebraic structures, which we call genuine equivariant operads, can be thought of as a hybrid between operads and coefficient systems. then prove an Elmendorf-Piacenza type theorem stating that with their graph model structure, are equivalent to projective structure. As application, explicit models for the $N_{\infty}$-operads Blumberg Hill.

Abstract This paper shows that the operad encoding modular operads is Koszul. Using this result, we construct higher composition operations on (hairy) graph homology, which characterize its rational homotopy type.

We give a framework for comparing on the one hand theories of ncategories that are weakly enriched operadically, and on the other hand ncategories given as algebras for a contractible globular operad. Examples of the former are the definition by Trimble and variants (Cheng-Gurski) and examples of the latter are the definition by Batanin and variants (Leinster). We will show how to take a theory...

Let M g,l be the moduli space of stable algebraic curves of genus g with l marked points. With the operations which relate the different moduli spaces identifying marked points, the family (M g,l) g,l is a modular operad of projective smooth Deligne-Mumford stacks, M. In this paper we prove that the modular operad of singular chains C * (M; Q) is formal; so it is weakly equivalent to the modula...

— The goal of this article is to make explicit a structured complex computing the cohomology of a profinite completion of the n-fold loop space of a sphere of dimension d < n. Our construction involves: the free complete algebra in one variable associated to any fixed En-operad; an element in this free complete algebra determined by a morphism from the operad of L∞-algebras to an operadic suspe...

In the theory of operads we consider generalized symmetric power functors defined by sums of coinvariant modules. One observes classically that the symmetric functor construction provides an isomorphism from the category of symmetric modules to a split subcategory of the category of functors on dgmodules (if dg-modules form our ground category). The purpose of this article is to obtain a simila...

Let M g,l be the moduli space of stable algebraic curves of genus g with l marked points. With the operations which relate the different moduli spaces identifying marked points, the family (M g,l) g,l is a modular operad of projective smooth Deligne-Mumford stacks, M. In this paper we prove that the modular operad of singular chains C * (M; Q) is formal; so it is weakly equivalent to the modula...

Let Mg,l be the moduli space of stable algebraic curves of genus g with l marked points. With the operations that relate the different moduli spaces identifying marked points, the family (Mg,l)g,l is a modular operad of projective smooth Deligne-Mumford stacksM. In this paper, we prove that the modular operad of singular chains S∗(M;Q) is formal, so it is weakly equivalent to the modular operad...

I exhibit a pair of non-symmetric operads that, while not themselves isomorphic, induce isomorphic monads. The existence of such a pair implies that if ‘algebraic theory’ is understood as meaning ‘monad’ then operads cannot be regarded as algebraic theories of a special kind.

In this article we give a construction of a polynomial 2-monad from an operad and describe the algebras of the 2-monads which then arise. This construction is different from the standard construction of a monad from an operad in that the algebras of our associated 2-monad are the categorified algebras of the original operad. Moreover it enables us to characterise operads as categorical polynomi...