Symmetric homotopy theory for operads

@article{Dehling2015SymmetricHT,
  title={Symmetric homotopy theory for operads},
  author={Malte Dehling and B. Vallette},
  journal={arXiv: Algebraic Topology},
  year={2015}
}
The purpose of this foundational paper is to introduce various notions and constructions in order to develop the homotopy theory for differential graded operads over any ring. The main new idea is to consider the action of the symmetric groups as part of the defining structure of an operad and not as the underlying category. We introduce a new dual category of higher cooperads, a new higher bar-cobar adjunction with the category of operads, and a new higher notion of homotopy operads, for which… Expand
Deformation theory of Cohomological Field Theories
We develop the deformation theory of cohomological field theories (CohFTs), which is done as a special case of a general deformation theory of morphisms of modular operads. This leads us to introduceExpand
A Tale of Three Homotopies
TLDR
The main nontrivial ingredient in establishing this relationship is the Homotopy transfer theorem for homotopy cooperads due to Drummond-Cole and Vallette. Expand
Moduli problems for operadic algebras
A theorem of Pridham and Lurie provides an equivalence between formal moduli problems and Lie algebras in characteristic zero. We prove a generalization of this correspondence, relating formal moduliExpand
On weak Lie 3-algebras
In this article, we introduce a category of weak Lie 3-algebras with suitable weak morphisms. The definition is based on the construction of a partial resolution over $\mathbb{Z}$ of the Koszul dualExpand
Categorified Cyclic Operads
TLDR
The coherence theorem has the form “all diagrams of canonical isomorphisms commute” and the proof of coherence in the entries-only style is of syntactic nature and relies on the coherence of categorified non-symmetric operads established by Došen and Petrić. Expand
Coextension of scalars in operad theory
The functor between operadic algebras given by restriction along an operad map generally has a left adjoint. We give a necessary and sufficient condition for the restriction functor to admit a rightExpand
Combinatorial homotopy theory for operads
We introduce an explicit combinatorial characterization of the minimal model ${\cal O}_{\infty}$ of the coloured operad ${\cal O}$ encoding non-symmetric operads. In our description of ${\calExpand
Cyclic operads : syntactic, algebraic and categorified aspects
In this thesis, we examine different frameworks for the general theory of cyclic operads of Getzler and Kapranov. As suggested by the title, we set up theoretical grounds of syntactic, algebraic andExpand
$EL_\infty$-algebras, Generalized Geometry, and Tensor Hierarchies
We define a generalized form of L8-algebras called EL8-algebras. As we show, these provide the natural algebraic framework for generalized geometry and the symmetries of double field theory as wellExpand
Massey Products for Graph Homology
This paper shows that the operad encoding modular operads is Koszul. Using this result we construct higher composition operations on (hairy) graph homology which characterize its rational homotopyExpand
...
1
2
...

References

SHOWING 1-10 OF 41 REFERENCES
The universal Hopf operads of the bar construction
The goal of this memoir is to prove that the bar complex B(A) of an E-infinity algebra A is equipped with the structure of a Hopf E-infinity algebra, functorially in A. We observe in addition thatExpand
Homotopy theory of homotopy algebras
This paper studies the homotopy theory of algebras and homotopy algebras over an operad. It provides an exhaustive description of their higher homotopical properties using the more general notion ofExpand
Curved Koszul duality theory
We extend the bar–cobar adjunction to operads and properads, not necessarily augmented. Due to the default of augmentation, the objects of the dual category are endowed with a curvature. As usual,Expand
Motivic slices and colored operads
Colored operads were introduced in the 1970's for the purpose of studying homotopy invariant algebraic structures on topological spaces. In this paper we introduce colored operads in motivic stableExpand
Operads, Algebras and Modules in General Model Categories
In this paper we develop the theory of operads, algebras and modules in cofibrantly generated symmetric monoidal model categories. We give J-semi model strucures, which are a slightly weaker versionExpand
Combinatorial operad actions on cochains
  • C. Berger, B. Fresse
  • Mathematics
  • Mathematical Proceedings of the Cambridge Philosophical Society
  • 2004
A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order toExpand
Commutative algebras and cohomology
This paper has three purposes: (1) to develop the machinery of a commutative cohomology theory for commutative algebras, (2) to apply this cohomology theory to give a working theory of ringExpand
Koszul duality for Operads
(0.1) The purpose of this paper is to relate two seemingly disparate developments. One is the theory of graph cohomology of Kontsevich [Kon 2 3] which arose out of earlier works of Penner [Pe] andExpand
RESOLUTION OF COLOURED OPERADS AND RECTIFICATION OF HOMOTOPY ALGEBRAS
Dedicated to Ross Street on the occasion of his 60th birthday Abstract. We provide general conditions under which the algebras for a coloured operad in a monoidal model category carry a Quillen modelExpand
On Operads
A operad is essentially an algebraic book keeping tool. Developed out of a need to systematically document complicated associativity and commutivity relations, they now find many uses in algebraicExpand
...
1
2
3
4
5
...