Related papers: Opetopes and chain complexes
In this paper, we review a method for computing and parameterizing the set of homotopy classes of chain maps between two chain complexes. This is then applied to finding topologically meaningful maps between simplicial complexes, which in…
The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasi-isomorphisms. Such examples are called `algebraic' because they originate from abelian…
Two distinct structures of aggregates of atoms connected by anisotropic bonds with a network configuration are discussed from the viewpoint of a point set topology. A specific topological space connects the two types of topological…
We introduce the Insertion Chain Complex, a higher-dimensional extension of insertion graphs, as a new framework for analyzing finite sets of words. We study its topological and combinatorial properties, in particular its homology groups,…
The goal of this article is to describe several presentations of the infinity category of algebras over some monad on the infinity category of chain complexes.
Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets.…
This paper describes a new approach to the problem of the structural research of clusters based on the theory of geodetic and k-geodetic graphs. We firmly believe that this same approach can be used when solving problems of correlation…
The goal of this paper is to generalize some of the existing toolkit of combinatorial algebraic topology in order to study the homology of abstract chain complexes. We define shellability of chain complexes in a similar way as for cell…
We describe the structure of triconnected graph with the help of its decomposition by 3-cutsets. We divide all 3-cutsets of a triconnected graph into rather small groups with a simple structure, named complexes. The detailed description of…
This paper proposes a categorical framework for knowledge graphs linking combinatorial graph structure with topos-theoretic semantics. Knowledge graphs are represented as labelled directed multigraphs and analysed through incidence matrices…
Graph-based signal processing techniques have become essential for handling data in non-Euclidean spaces. However, there is a growing awareness that these graph models might need to be expanded into `higher-order' domains to effectively…
This document develops general concepts useful for extracting knowledge embedded in large graphs or datasets that have pair-wise relationships, such as cause-effect-type relations. Almost no underlying assumptions are made, other than that…
The notion of polytopal map between two polytopal complexes is defined. Surprisingly, this definition is quite simple and extends naturally those of simplicial and cubical maps. It is then possible to define an induced chain map between the…
The development of mathematics has been characterized by the increasing interconnectivity of seemingly separate disciplines. Such interplay has been facilitated by a massive development in formalism; category theory has provided a common…
We give an explicit construction of the category Opetope of opetopes. We prove that the category of opetopic sets is equivalent to the category of presheaves over Opetope.
The theory of complex trees is introduced as a new approach to study a broad class of self-similar sets. Systems of equations encoded by complex trees tip-to-tip equivalence relations are used to obtain one-parameter families of connected…
We construct a category, $\Omega$, of which the objects are pointed categories and the arrows are pointed correspondences. The notion of a "spec datum" is introduced, as a certain relation between categories, of which one has been given a…
In enumerative combinatorics, it is often a goal to enumerate both labeled and unlabeled structures of a given type. The theory of combinatorial species is a novel toolset which provides a rigorous foundation for dealing with the…
When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in…
We study aggregation as a mechanism for the creation of complex networks. In this evolution process vertices merge together, which increases the number of highly connected hubs. We study a range of complex network architectures produced by…