A pr 2 00 3 Axiomatic homotopy theory for operads (original) (raw)

Algebraic operads up to homotopy

arXiv: Algebraic Topology, 2017

This paper deals with the homotopy theory of differential graded operads. We endow the Koszul dual category of curved conilpotent cooperads, where the notion of quasi-isomorphism barely makes sense, with a model category structure Quillen equivalent to that of operads. This allows us to describe the homotopy properties of differential graded operads in a simpler and richer way, using obstruction methods.

From homotopy operads to infinity-operads

2014

The goal of the present paper is to compare, in a precise way, two notions of operads up to homotopy which appear in the literature. Namely, we construct a functor from the category of strict unital homotopy colored operads to the category of infinity-operads. The former notion, that we make precise, is the operadic generalization of the notion of A-infinity-categories and the latter notion was defined by Moerdijk-Weiss in order to generalize the simplicial notion of infinity-category of Joyal-Lurie. This functor extends in two directions the simplicial nerve of Faonte-Lurie for A-infinity-categories and the homotopy coherent nerve of Moerdijk-Weiss for differential graded operads; it is also shown to be equivalent to a big nerve à la Lurie for differential graded operads. We prove that it satisfies some homotopy properties with respect to weak equivalences and fibrations; for instance, it is shown to be a right Quillen functor.

Algebraic structures associated to operads

arXiv: Rings and Algebras, 2017

We study different algebraic structures associated to an operad and their relations: to any operad mathbfP\mathbf{P}mathbfP is attached a bialgebra,the monoid of characters of this bialgebra, the underlying pre-Lie algebra and its enveloping algebra; all of them can be explicitely describedwith the help of the operadic composition. non-commutative versions are also given. We denote by mathbfbinfty\mathbf{b\_\infty}mathbfbinfty the operad of mathbfbinfty\mathbf{b\_\infty}mathbfbinfty algebras, describing all Hopf algebra structures on a symmetric coalgebra.If there exists an operad morphism from mathbfbinfty\mathbf{b\_\infty}mathbfbinfty to mathbfP\mathbf{P}mathbfP, a pair (A,B)(A,B)(A,B) of cointeracting bialgebras is also constructed, that it to say:$B$ is a bialgebra, and AAA is a graded Hopf algebra in the category of BBB-comodules. Most examples of such pairs (on oriented graphs, posets$\ldots$) known in the literature are shown to be obtained from an operad; colored versions of these examples andother ones, based on Feynman graphs, are introduced and compared.

From operator categories to higher operads

Geometry & Topology

In this paper we introduce the notion of an operator category and two different models for homotopy theory of ∞-operads over an operator category-one of which extends Lurie's theory of ∞-operads, the other of which is completely new, even in the commutative setting. We define perfect operator categories, and we describe a category Λ(Φ) attached to a perfect operator category Φ that provides Segal maps. We define a wreath product of operator categories and a form of the Boardman-Vogt tensor product that lies over it. We then give examples of operator categories that provide universal properties for the operads An and En (1 ≤ n ≤ +∞), as well as a collection of new examples. Contents 0. Introduction 1. Operator categories 2. Complete Segal operads 3. Wreath products 4. Perfect operator categories 5. The canonical monad on a perfect operator category 6. Leinster categories 7. Quasioperads and their algebras 8. Boardman-Vogt tensor products and weak algebras 9. Boardman-Vogt tensor products and ∞-algebras 10. Complete Segal operads versus quasioperads 11. Some examples of complete Segal Φ-monoids Appendix A. A proof of Th. 8.12 Appendix B. A proof of Th. 5.10 Appendix C. A proof of Th. 5.18 References

Monoidal Functors, Acyclic Models and Chain Operads

Canadian Journal of Mathematics, 2008

We prove that for a topological operad P the operad of oriented cubical chains, C ord * (P), and the operad of singular chains, S * (P), are weakly equivalent. As a consequence, C ord * (P ; Q) is formal if and only if S * (P ; Q) is formal, thus linking together some formality results that are spread out in the literature. The proof is based on an acyclic models theorem for monoidal functors. We give different variants of the acyclic models theorem and apply the contravariant case to study the cohomology theories for simplicial sets defined by R-simplicial differential graded algebras. 2 F. GUILLÉN, V. NAVARRO, P.PASCUAL, AND A. ROIG 5. Application: comparison of singular and cubical chains 17 5.1. 17 5.2. A monoidal cotriple in Top 17 5.3. 18 6. Symmetric monoidal functors 19 6.1. Acyclic models for symmetric monoidal functors 19 6.2. The Kleisli cotriple 20 6.3. Ordered cubical chains 20 7. Application to operads 22 7.1. Operads 22 7.2. 23 7.3. Formality 24 7.4. Modular operads 25 8. Contravariant functors 25 8.1. The standard construction 26 8.2. Acyclic models for contravariant monoidal functors 26 8.3. Application to singular and cubical cochains 26 9. Application to cohomology theories 27 9.1. Cohomology theories 27 9.2. 30 References 31

The Boardman–Vogt resolution of operads in monoidal model categories

Topology, 2006

We extend the W-construction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for well-pointed Σ -cofibrant operads. The standard simplicial resolution of Godement as well as the cobar-bar chain resolution are shown to be particular instances of this generalised W-construction.