A pr 2 00 3 Axiomatic homotopy theory for operads (original) (raw)
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.
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.
and Ieke Moerdijk. Resolution of coloured operads and rectification of homotopy algebras
2012
We provide general conditions under which the algebras for a coloured operad in a monoidal model category carry a Quillen model structure, and prove a Comparison Theorem to the effect that a weak equivalence between suitable such operads induces a Quillen equivalence between their categories of algebras. We construct an explicit Boardman-Vogt style cofibrant resolution for coloured operads, thereby giving a uniform approach to algebraic structures up to homotopy over coloured operads. The Comparison Theorem implies that such structures can be rectified.
2003
In this paper we generalize the plus-construction given by M. Livernet for algebras over rational differential graded Koszul operads to the framework of admissible operads (the category of algebras over such operads admits a closed model category structure). We follow the modern approach of J. Berrick and C. Casacuberta defining topological plus-construction as a nullification with respect to a universal acyclic space. Similarly, we construct a universal H ∗ -acyclic algebra U and we define A −→ A+ as the U -nullification of the algebra A. This map induces an isomorphism on Quillen homology and quotients out the maximal perfect ideal of π0(A). As an application, we consider for any associative algebra R the plusconstructions of gl(R) in the categories of Lie and Leibniz algebras up to homotopy. This gives rise to two new homology theories for associative algebras, namely cyclic and Hochschild homologies up to homotopy. In particular, these theories coincide with the classical cyclic...