An operadic proof of Baez-Dolan stabilization hypothesis

We prove a stabilization theorem for algebras of n-operads in a monoidal model category. It implies a version of Baez-Dolan stabilization hypothesis for Rezk's weak n-categories and some other stabilization results.Comment: 14 pages, the paper is now in its final form accepted for publication in Proceedings of AM

Operadic categories and Duoidal Deligne's conjecture

The purpose of this paper is two-fold. In Part 1 we introduce a new theory of operadic categories and their operads. This theory is, in our opinion, of an independent value. In Part 2 we use this new theory together with our previous results to prove that multiplicative 1-operads in duoidal categories admit, under some mild conditions on the underlying monoidal category, natural actions of contractible 2-operads. The result of D. Tamarkin on the structure of dg-categories, as well as the classical Deligne conjecture for the Hochschild cohomology, is a particular case of this statement.Comment: 54 pages, to appear in Advances in Mathematic

Homotopy theory for algebras over polynomial monads

We study the existence and left properness of transferred model structures for "monoid-like" objects in monoidal model categories. These include genuine monoids, but also all kinds of operads as for instance symmetric, cyclic, modular, higher operads, properads and PROP's. All these structures can be realised as algebras over polynomial monads. We give a general condition for a polynomial monad which ensures the existence and (relative) left properness of a transferred model structure for its algebras. This condition is of a combinatorial nature and singles out a special class of polynomial monads which we call tame polynomial. Many important monads are shown to be tame polynomial.Comment: Final version. Remark 5.16 extended. Bibliography complete

Koszul duality in operadic categories

Our aim is to set up the cornerstones of Koszul duality in general operadic categories. In particular, we will prove that operads (in our generalized sense) governing the most important operad- and/or PROP-like structures as classical operads, their variants as cyclic, modular or wheeled operads, and also diverse versions of PROPs such as properads, dioperads, 1/2-PROPs, and still more exotic stuff as permutads and pre-permutads are quadratic, and describe their Koszul duals. To this end we single out some additional properties of operadic categories ensuring that free operads admit a nice explicit description, and investigate how these properties interact with discrete operadic (op)fibrations which we use as a mighty tool to construct new operadic categories from the old ones. Particular attention is payed to the operadic category of graphs and to its clones, but several other examples are given as well. Our present work provides an answer to the questions "What does encode a type of operads?" and "How to construct Koszul duals to these objects?" formulated in the last Loday's 2012 talk in Lille.Comment: 108 pages; a section about the +-construction adde