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Added by
John Baez
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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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.
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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.
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.
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.
We present a homotopy theory for a weak version of modular operads whose compositions and contractions are only defined up to homotopy. This homotopy theory takes the form of a Quillen model structure on the collection of simplicial presheaves for a certain category of undirected graphs. This new category of undirected graphs, denoted $mathbf{U}$, plays a similar role for modular operads that the dendroidal category $Omega$ plays for operads. We carefully study properties of $mathbf{U}$, including the existence of certain factorization systems. Related structures, such as cyclic operads and stable modular operads, can be similarly treated using categories derived from $mathbf{U}$.
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