Related papers: Opetopes and chain complexes
The intersection graph of a collection of trapezoids with corner points lying on two parallel lines is called a trapezoid graph. Using binary indexed tree data structure, we improve algorithms for calculating the size and the number of…
The category of strict omega-categories has an important full subcategory whose objects are the simple omega-categories freely generated by planar trees or by globular cardinals. We give a simple description of this subcategory in terms of…
We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration…
Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology, singularity theory, and higher algebra. The theory synthesizes elements of classical combinatorial…
We answer a question of Laszlo Babai concerning the abelian sandpile model. Given a graph, the model yields a finite abelian group of recurrent configurations which is closely related to the combinatorial Laplacian of the graph. We…
Simplicial complexes capture the underlying network topology and geometry of complex systems ranging from the brain to social networks. Here we show that algebraic topology is a fundamental tool to capture the higher-order dynamics of…
We generalize some homotopy calculation techniques such as splittings and matching trees that are introduced for the computations in the case of the independence complexes of graphs to arbitrary simplicial complexes, and exemplify their…
Higher-order interactions provide a nuanced understanding of the relational structure of complex systems beyond traditional pairwise interactions. However, higher-order network analyses also incur more cumbersome interpretations and greater…
We propose elementary definitions of opetopes and opetopic sets. We directly define opetopic sets by a simple structure and several axioms. Opetopes are then opetopic sets satisfying one more axiom. We show that our definition is equivalent…
Graphs are fundamental data structures which concisely capture the relational structure in many important real-world domains, such as knowledge graphs, physical and social interactions, language, and chemistry. Here we introduce a powerful…
We show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of…
Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension. As observed…
Complex numbers define the relationship between entities in many situations. A canonical example would be the off-diagonal terms in a Hamiltonian matrix in quantum physics. Recent years have seen an increasing interest to extend the tools…
We define a graph structure associated in a natural way to finite fields that nevertheless distinguishes between different models of isomorphic fields.
Our work is concerned with simplicial complexes that describe higher-order interactions in real complex systems. This description allows to go beyond the pairwise node-to-node representation that simple networks provide and to capture a…
Graph classification plays an important role is data mining, and various methods have been developed recently for classifying graphs. In this paper, we propose a novel method for graph classification that is based on homotopy equivalence of…
We survey interactions between the topology and the combinatorics of complex hyperplane arrangements. Without claiming to be exhaustive, we examine in this setting combinatorial aspects of fundamental groups, associated graded Lie algebras,…
We define treetopes, a generalization of the three-dimensional roofless polyhedra (Halin graphs) to arbitrary dimensions. Like roofless polyhedra, treetopes have a designated base facet such that every face of dimension greater than one…
We give an account of the basic combinatorial structure underlying the notion of type dependency. We do so by considering the category of all dependent sequent calculi, and exhibiting it as the category of algebras for a monad on a presheaf…
Tree-like tableaux are objects in bijection with alternative or permutation tableaux. They have been the subject of a fruitful combinatorial study for the past few years. In the present work, we define and study a new subclass of tree-like…