Related papers: Syntactic approaches to opetopes
Rectangulations are decompositions of a square into finitely many axis-aligned rectangles. We describe realizations of $(n-1)$-dimensional polytopes associated with two combinatorial families of rectangulations composed of $n$ rectangles.…
There is a growing need for abstractions in logic specification languages such as FO(.) and ASP. One technique to achieve these abstractions are templates (sometimes called macros). While the semantics of templates are virtually always…
Starting from any unital colored PROP $P$, we define a category $P(P)$ of shapes called $P$-propertopes. Presheaves on $P(P)$ are called $P$-propertopic sets. For $0 \leq n \leq \infty$ we define and study $n$-time categorified $P$-algebras…
Techniques from higher categories and higher-dimensional rewriting are becoming increasingly important for understanding the finer, computational properties of higher algebraic theories that arise, among other fields, in quantum…
A natural next step in the evolution of constraint-based grammar formalisms from rewriting formalisms is to abstract fully away from the details of the grammar mechanism---to express syntactic theories purely in terms of the properties of…
Developing an idea of Kapranov and Voevodsky, we introduce a model of weak omega-categories based on directed complexes, combinatorial presentations of pasting diagrams. We propose this as a convenient framework for higher-dimensional…
Complex systems of systems (SoS) are characterized by multiple interconnected subsystems. Typically, each subsystem is designed and analyzed using methodologies and formalisms that are specific to the particular subsystem model of…
This is an expository article about operads in homotopy theory written as a chapter for an upcoming book. It concentrates on what the author views as the basic topics in the homotopy theory of operadic algebras: the definition of operads,…
After introducing some motivations for this survey, we describe a formalism to parametrize a wide class of algebraic structures occurring naturally in various problems of topology, geometry and mathematical physics. This allows us to define…
We propose a functional description of rewriting systems on topological vector spaces. We introduce the topological confluence property as an approximation of the confluence property. Using a representation of linear topological rewriting…
In this paper we suggest a typed compositional seman-tics for nominal compounds of the form [Adj Noun] that models adjectives as higher-order polymorphic functions, and where types are assumed to represent concepts in an ontology that…
We revisit Kapranov and Voevodsky's idea of spaces modelled on combinatorial pasting diagrams, now as a framework for higher-dimensional rewriting and the basis of a model of weak omega-categories. In the first part, we elaborate on…
Many interesting and useful symbolic computation algorithms manipulate mathematical expressions in mathematically meaningful ways. Although these algorithms are commonplace in computer algebra systems, they can be surprisingly difficult to…
Abstract axiomatic formulation of mathematical structures are extensively used to describe our physical world. We take here the reverse way. By making basic assumptions as starting point, we reconstruct some features of both geometry and…
We introduce higher dimensional hypergraphs, which is a generalization of Baez-Dolans's opetopic sets and Hermida-Makkai-Power's multigraphs. This is based on a simple combinatorial structure called shells and the formal composites of…
We introduce OpSets, an executable framework for specifying and reasoning about the semantics of replicated datatypes that provide eventual consistency in a distributed system, and for mechanically verifying algorithms that implement these…
We introduce stratified toposes, which are toposes that are stratified by a suitable hierarchy of universes. The term `stratified topos' recalls the notion of stratified pseudotopos of Moerdijk and Palmgren (2002). However, the details of…
Copatterns give functional programs a flexible mechanism for responding to their context, and composition can greatly enhance their expressiveness. However, that same expressive power makes it harder to precisely specify the behavior of…
The determination of block-entropies is a well established method for the investigation of discrete data, also called symbols (7). There is a large variety of such symbolic sequences, ranging from texts written in natural languages,…
In this report, we introduce observation algebras, constructed by considering the downclosed subsets of a coherence space ordered by reverse inclusion. These may be interpreted as specifications of sets of events via some predicates with…