Operads,quasiorders, and regular languages

We generalize the construction of multi-tildes in the aim to provide double multi-tilde operators for regular languages. We show that the underlying algebraic structure involves the action of some operads. An operad is an algebraic structure that mimics the composition of the functions. The involved operads are described in terms of combinatorial objects. These operads are obtained from more primitive objects, namely precompositions, whose algebraic counter-parts are investigated. One of these operads acts faithfully on languages in the sense that two different operators act in two different ways.     

Pluriassociative algebras II: The polydendriform operad and related operads

Dendriform algebras form a category of algebras recently introduced by Loday. A dendriform algebra is a vector space endowed with two nonassociative binary operations satisfying some relations. Any dendriform algebra is an algebra over the dendriform operad, the Koszul dual of the diassociative operad. We introduce here, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter γ of dendriform algebras, called γ-polydendriform algebras, so that 1-polydendriform algebras are dendriform algebras. For that, we consider the operads obtained as the Koszul duals of the γ-pluriassociative operads introduced by the author in a previous work. In the same manner as dendriform algebras are suitable devices to split associative operations into two parts, γ-polydendriform algebras seem adapted structures to split associative operations into 2γ operation so that some partial sums of these operations are associative. We provide a complete study of the γ-polydendriform operads, the underlying operads of the category of γ-polydendriform algebras. We exhibit several presentations by generators and relations, compute their Hilbert series, and construct free objects in the corresponding categories. We also provide consistent generalizations on a nonnegative integer parameter of the duplicial, triassociative and tridendriform operads, and of some operads of the operadic butterfly.     

Differential-geometric structures in operads

We describe certain structures of formal differential geometry in terms of the theory of operads and introduce group structures, Lie-algebra structures, exponential mappings, and an analog of the de Rham complex.     

Models for operads

We study properties of differential graded (dg) operads modulo weak equivalences, that is, modulo the relation given by the existence of a chain of dg operad maps including a homology isomorphism. This approach, naturally arising in string theory, leads us to consider various versions of models. Some applications in topology (homotopy-everything spaces), algebra (cotangent cohomology) and mathematical physics (closed string-field theory) - are also given     

Axiomatic homotopy theory for operads

We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.     

Functional equations and their related operads

Using functional equations, we define functors that generalize standard examples from calculus of one variable. Examples of such functors are discussed, and their Taylor towers are computed. We also show that these functors factor through objects enriched over the homology of little -cubes operads and discuss the relationship between functors defined via functional equations and operads. In addition, we compute the differentials of the forgetful functor from the category of -Poisson algebras in terms of the homology of configuration spaces.

     

Modular operads

We develop a higher genus analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor F on the category of modular operads, the Feynman transform, which generalizes Kontsevichs graph complexes and also the bar construction for operads. We calculate the Euler characteristic of the Feynman transform, using the theory of symmetric functions: our formula is modelled on Wicks theorem. We give applications to the theory of moduli spaces of pointed algebraic curves.     

Group actions on Segal operads

We give a Quillen equivalence between model structures for simplicial operads, described via the theory of operads, and Segal operads, thought of as certain reduced dendroidal spaces. We then extend this result to give a Quillen equivalence between the model structures for simplicial operads equipped with a group action and the corresponding Segal operads.     

Minimal models for graph-related (hyper)operads

We construct explicit minimal models for the (hyper)operads governing modular, cyclic and ordinary operads, and wheeled properads, respectively. Algebras for these models are homotopy versions of the corresponding structures.     

The Eckmann-Hilton argument and higher operads

The classical Eckmann-Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2-category, then its Hom-set is a commutative monoid. A similar argument due to A. Joyal and R. Street shows that a one object, one arrow tricategory is ‘the same’ as a braided monoidal category.In this paper we begin to investigate how one can extend this argument to arbitrary dimension. We provide a simple categorical scheme which allows us to formalise the Eckmann-Hilton type argument in terms of the calculation of left Kan extensions in an appropriate 2-category. Then we apply this scheme to the case of n-operads in the author's sense and classical symmetric operads. We demonstrate that there exists a functor of symmetrisation Symn from a certain subcategory of n-operads to the category of symmetric operads such that the category of one object, one arrow, … , one (n−1)-arrow algebras of A is isomorphic to the category of algebras of Symn(A). Under some mild conditions, we present an explicit formula for Symn(A) which involves taking the colimit over a remarkable categorical symmetric operad.We will consider some applications of the methods developed to the theory of n-fold loop spaces in the second paper of this series.     

Traces,dependency graphs and DNLC grammars

We point out the use of graph grammars for specifying (generating) languages of dependency graphs that arise in theoretical studies of concurrent systems.     

Combinatorial properties of boundary NLC graph languages

Node label controlled (NLC) grammars are graph grammars (operating on node labeled undirected graphs)which rewrite single nodes only and establish connections between the embedded graph and the neighbors of the rewritten node on the basis of the labels of the involved nodes only. They define (possibly infinite) languages of undirected node labeled graphs (or, if we just omit the labels, languages of unlabeled graphs). Boundary NLC (BNLC) grammars are NLC grammars with the property that whenever - in a graph already generated - two nodes may be rewritten, then these nodes are not adjacent. The graph languages generated by this type of grammars are called BNLC languages.In this paper we investigate the behaviour of graph invariants within BNLC languages. First we demonstrate that there is a dependency between the chromatic number and the clique number of graphs in BNLC languages (while this is wellknown not to be true for arbitrary graph languages). Secondly, we introduce a new graph invariant, the so-called index of a graph which seems to be very suitable for describing the adjacency structure of a graph. Then we prove that every BNLC language is of bounded index (which is shown not to be true for arbitrary graph languages). Thus we exhibit properties (concerning graph invariants) of BNLC languages which are intrinsic to this class. We use them to demonstrate that certain graph languages are not BNLC languages.     

The homotopy theory of simplicial props

The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. In this paper, the second in a series on ‘higher props,’ we show that the category of all small colored simplicial props admits a cofibrantly generated model category structure. With this model structure, the forgetful functor from props to operads is a right Quillen functor.     

Pluriassociative algebras I: The pluriassociative operad

Diassociative algebras form a category of algebras recently introduced by Loday. A diassociative algebra is a vector space endowed with two associative binary operations satisfying some very natural relations. Any diassociative algebra is an algebra over the diassociative operad, and, among its most notable properties, this operad is the Koszul dual of the dendriform operad. We introduce here, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter γ of diassociative algebras, called γ-pluriassociative algebras, so that 1-pluriassociative algebras are diassociative algebras. Pluriassociative algebras are vector spaces endowed with 2γ associative binary operations satisfying some relations. We provide a complete study of the γ-pluriassociative operads, the underlying operads of the category of γ-pluriassociative algebras. We exhibit a realization of these operads, establish several presentations by generators and relations, compute their Hilbert series, show that they are Koszul, and construct the free objects in the corresponding categories. We also study several notions of units in γ-pluriassociative algebras and propose a general way to construct such algebras. This paper ends with the introduction of an analogous generalization of the triassociative operad of Loday and Ronco.     

A model structure for coloured operads in symmetric spectra

We describe a model structure for coloured operads with values in the category of symmetric spectra (with the positive model structure), in which fibrations and weak equivalences are defined at the level of the underlying collections. This allows us to treat R-module spectra (where R is a cofibrant ring spectrum) as algebras over a cofibrant spectrum-valued operad with R as its first term. Using this model structure, we give sufficient conditions for homotopical localizations in the category of symmetric spectra to preserve module structures.     

The homotopy theory of bialgebras over pairs of operads

We endow the category of bialgebras over a pair of operads in distribution with a cofibrantly generated model category structure. We work in the category of chain complexes over a field of characteristic zero. We split our construction in two steps. In the first step, we equip coalgebras over an operad with a cofibrantly generated model category structure. In the second step we use the adjunction between bialgebras and coalgebras via the free algebra functor. This result allows us to do classical homotopical algebra in various categories such as associative bialgebras, Lie bialgebras or Poisson bialgebras in chain complexes.     

ECO:a methodology for the enumeration of combinatorial objects

In this Paper, we illustrate a method (called the ECO method) for enumerating some classes of combinatorial objects. The basic idea of this method is the following: by means of an operator that performs a "local expansion" on the objects, we give some recursive constructions of these classes. We use these constructions to deduce some new funtional equations verified by classes' generating functions. By solving the functional equations, we enumerate the combinatorial objects according to various parameters. We show some applications of the method referring to some classical combinatorial objects, such as: trees, paths, polyminoes and permutations     

Curved Koszul duality of algebras over unital versions of binary operads

We develop a curved Koszul duality theory for algebras presented by quadratic-linear-constant relations over unital versions of binary quadratic operads. As an application, we study Poisson n-algebras given by polynomial functions on a standard shifted symplectic space. We compute explicit resolutions of these algebras using curved Koszul duality. We use these resolutions to compute derived enveloping algebras and factorization homology on parallelized simply connected closed manifolds with coefficients in these Poisson n-algebras.     

On products and duality of binary, quadratic, regular operads

Since its introduction by Loday in 1995, with motivation from algebraic K-theory, dendriform dialgebras have been studied quite extensively with connections to several areas in Mathematics and Physics. A few more similar structures have been found recently, such as the tri-, quadri-, ennea- and octo-algebras, with increasing complexity in their constructions and properties. We consider these constructions as operads and their products and duals, in terms of generators and relations, with the goal to clarify and simplify the process of obtaining new algebra structures from known structures and from linear operators.