We compare two models for ∞-operads: the complete Segal operads of Barwick and the complete dendroidal Segal spaces of Cisinski and Moerdijk. Combining this with comparison results already in the literature, this implies that all known models for ∞-operads are equivalent — for instance, it follows that the homotopy theory of Lurie’s ∞-operads is equivalent to that of dendroidal sets and that of...

The structure of a k-fold monoidal category as introduced by Balteanu, Fiedorowicz, Schwänzl and Vogt in [2] can be seen as a weaker structure than a symmetric or even braided monoidal category. In this paper we show that it is still sufficient to permit a good definition of (n-fold) operads in a k-fold monoidal category which generalizes the definition of operads in a braided category. Further...

We establish model category structures on algebras and modules over operads in symmetric spectra, and study when a morphism of operads induces a Quillen equivalence between corresponding categories of algebras (resp. modules) over operads.

We define a new monoidal category on collections (shuffle composition). Monoids in this category (shuffle operads) turn out to bring a new insight in the theory of symmetric operads. For this category, we develop the machinery of Gröbner bases for operads, and present operadic versions of Bergman’s Diamond Lemma and Buchberger’s algorithm. This machinery can be applied to study symmetric operad...

The purpose of this paper is twofold. First, we review applications of the bar duality of operads to the construction of explicit cofibrant replacements in categories of algebras over an operad. In view toward applications, we check that the constructions of the bar duality work properly for algebras over operads in unbounded differential graded modules over a ring. In a second part, we use the...

The purpose of this foundational paper is to introduce various notions and constructions in order develop the homotopy theory for differential graded operads over any ring. main new idea consider action symmetric groups as part defining structure an operad not underlying category. We a dual category higher cooperads, bar-cobar adjunction with operads, notion which we establish relevant properti...

In this article we give a conceptual definition of Manin products in any category endowed with two coherent monoidal products. This construction can be applied to associative algebras, non-symmetric operads, operads, colored operads, and properads presented by generators and relations. These two products, called black and white, are dual to each other under Koszul duality functor. We study thei...

The theory of operads, defined through categories of labeled graphs, is generalized to suit definitions of higher categories with arbitrary basic shapes. Constructions of cubical, globular and opetopic weak higher categories are obtained as examples.

In this paper it is shown that an assortment of operads near to many topologists’ hearts enjoy (homotopy) universal properties of an expressly combinatorial nature. These include the operads An and En. The main idea that makes this possible is the new notion of an operator category, which controls the homotopy types of these operads in a strong sense.

A new sort of combinatorial generating system, called bud generating system, is introduced. Bud generating systems are devices for specifying sets of various kinds of combinatorial objects, called languages. They can emulate context-free grammars, regular tree grammars, and synchronous grammars, allowing to work with all these generating systems in a unified way. The theory of bud generating sy...