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

The goal of this paper is to prove a Koszul duality result for En-operads in differential graded modules over a ring. The case of an E1-operad, which is equivalent to the associative operad, is classical. For n > 1, the homology of an En-operad is identified with the n-Gerstenhaber operad and forms another well known Koszul operad. Our main theorem asserts that an operadic cobar construction on...

Inspired by recent work of Batanin and Berger on the homotopy theory of operads, a general monad-theoretic context for speaking about structures within structures is presented, and the problem of constructing the universal ambient structure containing the prescribed internal structure is studied. Following the work of Lack, these universal objects must be constructed from simplicial objects ari...

Working in the context of symmetric spectra, we describe and study a homotopy completion tower for algebras and left modules over operads in the category of modules over a commutative ring spectrum (e.g., structured ring spectra). We prove a strong convergence theorem that for 0-connected algebras and modules over a (−1)-connected operad, the homotopy completion tower interpolates (in a strong ...

We present a definition of homotopy algebra for an operad, and explore its consequences. The paper should be accessible to topologists, category theorists, and anyone acquainted with operads. After a review of operads and monoidal categories, the definition of homotopy algebra is given. Specifically, suppose that M is a monoidal category in which it makes sense to talk about algebras for some o...

We consider partitions of a set with r elements ordered by refinement. We consider the simplicial complex K̄(r) formed by chains of partitions which starts at the smallest element and ends at the largest element of the partition poset. A classical theorem asserts that K̄(r) is equivalent to a wedge of r− 1-dimensional spheres. In addition, the poset of partitions is equipped with a natural action...

This paper explicitly constructs cofree coalgebras over operads in the category of DG-modules. It is shown that the existence of an operadaction on a coalgebra implies a “generalized coassociativity” that facilitates the construction. Special cases are considered in which the general expression simplifies (such as the pointed, irreducible case). In the pointed irreducible case, this constructio...

According to a result of H. Cartan (cf. [5]), the homotopy of a simplicial commutative algebra is equipped with divided power operations. In this paper, we provide a general approach to the construction of such operations in the context of simplicial algebras over an operad. To be precise, we work over a fixed field F, and we consider operads in the category of F-modules. An operad is an algebr...

For a connected pasting scheme G, under reasonable assumptions on the underlying category, the category of C-colored G-props admits a cofibrantly generated model category structure. In this paper, we show that, if G is closed under shrinking internal edges, then this model structure on G-props satisfies a (weaker version) of left properness. Connected pasting schemes satisfying this property in...

There are many different types of algebra: associative, associative and commutative, Lie, Poisson, etc., etc. Each comes with an appropriate notion of a module. As is becoming more and more important in a variety of fields, it is often necessary to deal with algebras and modules of these sorts “up to homotopy”. I shall give a very partial overview, concentrating on algebra, but saying a little ...