Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGrowth of nonsymmetric operads
445
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The paper concerns the Gelfand-Kirillov dimension and the generating series of nonsymmetric operads. An analogue of Bergmans gap theorem is proved, namely, no finitely generated locally finite nonsymmetric operad has Gelfand-Kirillov dimension strictly between $1$ and $2$. For every $rin {0}cup {1}cup [2,infty)$ or $r=infty$, we construct a single-element generated nonsymmetric operad with Gelfand-Kirillov dimension $r$. We also provide counterexamples to two expectations of Khoroshkin and Piontkovski about the generating series of operads.
rate research
Read More
Operads are algebraic devices offering a formalization of the concept of operations with several inputs and one output. Such operations can be naturally composed to form bigger and more complex ones. Coming historically from algebraic topology, operads intervene now as important objects in computer science and in combinatorics. The theory of operads, together with the algebraic setting and the tools accompanying it, promises advances in these two areas. On the one hand, operads provide a useful abstraction of formal expressions, and also, provide connections with the theory of rewrite systems. On the other hand, a lot of operads involving combinatorial objects highlight some of their properties and allow to discover new ones. This book presents the theory of nonsymmetric operads under a combinatorial point of view. It portrays the main elements of this theory and the links it maintains with several areas of computer science and combinatorics. A lot of examples of operads appearing in combinatorics are studied and some constructions relating operads with known algebraic structures are presented. The modern treatment of operads consisting in considering the space of formal power series associated with an operad is developed. Enrichments of nonsymmetric operads as colored, cyclic, and symmetric operads are reviewed. This text is addressed to any computer scientist or combinatorist who looks a complete and a modern description of the theory of nonsymmetric operads. Evenly, this book is intended to an audience of algebraists who are looking for an original point of view fitting in the context of combinatorics.
We introduce a category of locally constant $n$-operads which can be considered as the category of higher braided operads. For $n=1,2,infty$ the homotopy category of locally constant $n$-operads is equivalent to the homotopy category of classical nonsymmetric, braided and symmetric operads correspondingly.
System of systems engineering seeks to analyze, design and deploy collections of systems that together can flexibly address an array of complex tasks. In the Complex Adaptive System Composition and Design Environment program, we developed network operads as a tool for designing and tasking systems of systems, and applied them to domains including maritime search and rescue. The network operad formalism offers new ways to handle changing levels of abstraction in system-of-system design and tasking.
Using the description of enriched $infty$-operads as associative algebras in symmetric sequences, we define algebras for enriched $infty$-operads as certain modules in symmetric sequences. For $mathbf{V}$ a nice symmetric monoidal model category, we prove that strict algebras for $Sigma$-cofibrant operads in $mathbf{V}$ are equivalent to algebras in the associated symmetric monoidal $infty$-category in this sense. We also show that $mathcal{O}$-algebras in $mathcal{V}$ can equivalently be described as morphisms of $infty$-operads from $mathcal{O}$ to endomorphism operads of (families of) objects of $mathcal{V}$.
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 we establish the relevant homotopy properties. For instance, the higher bar-cobar construction provides us with a cofibrant replacement functor for operads over any ring. All these constructions are produced conceptually by applying the curved Koszul duality for colored operads. This paper is a first step toward a new Koszul duality theory for operads, where the action of the symmetric groups is properly taken into account.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
