Related papers: Path homology of directed hypergraphs

A directed graph is set-homogeneous if, whenever U and V are isomorphic finite subdigraphs, there is an automorphism g of the digraph with U^g=V. Here, extending work of Lachlan on finite homogeneous digraphs, we classify finite…

In this paper we introduce a novel family of attributed graphs for the purpose of shape discrimination. Our graphs typically arise from variations on the Mapper graph construction, which is an approximation of the Reeb graph for point cloud…

In this note, we give a self-contained account on a construction for a directed homology theory based on modules over algebras, linking it to both persistence homology and natural homology. We study its first properties, among which some…

We prove analogues of classical results for higher homotopy groups and singular homology groups of pseudotopological spaces. Pseudotopological spaces are a generalization of (\v{C}ech) closure spaces which are in turn a generalization of…

Discrete homotopy theory or A-homotopy theory is a combinatorial homotopy theory defined on graphs, simplicial complexes, and metric spaces, reflecting information about their connectivity. The present paper aims to further understand the…

We discuss the homological algebra of representation theory of finite dimensional algebras and finite groups. We present various methods for the construction and the study of equivalences of derived categories: local group theory, geometry…

This paper gives a first step towards developing synthetic differential geometry within homotopy type theory. Its model theory will be discussed in a subsequent paper.

We classify the finite connected-homogeneous digraphs, as well as the infinite such digraphs with precisely one end. This completes the classification of all the locally finite connected-homogeneous digraphs.

This is the second in a series of papers. Here we develop here an intersection theory for manifolds equipped with an action of a finite group. As in our previous paper, our approach will be homotopy theoretic, enabling us to circumvent the…

We present an exact formula for the ordinary generating series of the simple paths between any two vertices of a graph. Our formula involves the adjacency matrix of the connected induced subgraphs and remains valid on weighted and directed…

There has been a great deal of research on graphs defined on algebraic structures in the last two decades. In this paper we begin an exploration of hypergraphs defined on algebraic structures, especially groups, to investigate whether this…

We give an overview of differential cohomology from the point of view of algebraic topology. This includes a survey of several different definitions of differential cohomology groups, a discussion of differential characteristic classes, an…

One of the prime motivation for topology was Homotopy theory, which captures the general idea of a continuous transformation between two entities, which may be spaces or maps. In later decades, an algebraic formulation of topology was…

A directed space is a topological space $X$ together with a subspace $\vec{P}(X)\subset X^I$ of \emph{directed} paths on $X$. A symmetry of a directed space should therefore respect both the topology of the underlying space and the topology…

Our purpose is to explore, in the context of loop ensembles on finite graphs, the relations between combinatorial group theory, loops topology, loop measures, and signatures of discrete paths. We determine the distributions of the loop…

We consider two categories related to symplectic manifolds: 1. Objects are symplectic manifolds and morphisms are symplectic embeddings. 2. Objects are symplectic manifolds endowed with compatible almost complex structure and morphisms are…

We want to investigate 'spaces' where paths have a 'weight', or 'cost', expressing length, duration, price, energy, etc. The weight function is not assumed to be invariant up to path-reversion. Thus, 'weighted algebraic topology' can be…

We construct cobordism maps on link Floer homology associated to decorated link cobordisms. The maps are defined on a curved chain homotopy type invariant. We describe the construction, and prove invariance. We also make a comparison with…

We define the notions of Reidemeister torsion and analytic torsion for directed graphs by means of the path homology theory introduced by the authors in \cite{Grigoryan-Lin-Muranov-Yau2013, Grigoryan-Lin-Muranov-Yau2014,…

We show that the category of graphs has the structure of a 2-category with homotopy as the 2-cells. We then develop an explicit description of homotopies for finite graphs, in terms of what we call `spider moves'. We then create a category…