Related papers: The Directed Van Kampen Theorem in Lean
This paper proves that labelled flows are expressive enough to contain all process algebras which are a standard model for concurrency. More precisely, we construct the space of execution paths and of higher dimensional homotopies between…
Let $S$ be a complete flat surface, such as the Euclidean plane. We determine the homeomorphism class of the space of all curves on $S$ which start and end at given points in given directions and whose curvatures are constrained to lie in a…
Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms. We survey results in computational topology that are concerned with graphs drawn…
We give new lower bounds for the (higher) topological complexity of a space, in terms of the Lusternik-Schnirelmann category of a certain auxiliary space. We also give new lower bounds for the rational topological complexity of a space, and…
In condensed matter physics and related areas, topological defects play important roles in phase transitions and critical phenomena. Homotopy theory facilitates the classification of such topological defects. After a pedagogic introduction…
In this work, we explore links between natural homology and persistent homology for the classification of directed spaces. The former is an algebraic invariant of directed spaces, a semantic model of concurrent programs. The latter was…
We define the notion of {\em classifying space} of a topological stack and show that every topological stack \X has a classifying space X which is a topological space well-defined up to weak homotopy equivalence. Under a certain…
This work presents a new path classification criterion to distinguish paths geometrically and topologically from the workspace, which is divided through cell decomposition, generating a medial-axis-like skeleton structure. We use this…
We develop a discrete-time version of the blended dynamics theorem for the use of designing distributed computation algorithms. The blended dynamics theorem enables to predict the behavior of heterogeneous multi-agent systems. Therefore,…
We introduce the weighted path homology on the category of weigh\-ted directed hypergraphs and describe conditions of homotopy invariance of weighted path homology groups. We give several examples that explain the nontriviality of the…
We prove the validity over $\mathbb{R}$ of a commutative differential graded algebra model of configuration spaces for simply connected closed smooth manifolds, answering a conjecture of Lambrechts--Stanley. We get as a result that the real…
A reparametrization (of a continuous path) is given by a surjective weakly increasing self-map of the unit interval. We show that the monoid of reparametrizations (with respect to compositions) can be understood via ``stop-maps'' that allow…
This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space. We show that this functor satisfies a version of the van Kampen theorem, and so is a…
A duality of $\kappa$-normed topological vector spaces is defined and investigated. For such spaces the analog of the Mackey-Arens theorem is proved. There are investigated cases, when $\kappa$-normability of a topological vector space…
We develop a homology theory for directed spaces, based on the semi-abelian category of (non-unital) associative algebras. The major ingredient is a simplicial algebra constructed from convolution algebras of certain trace categories of a…
We present and analyze a topologically induced transition from ordered, synchronized to disordered dynamics in directed networks of oscillators. The analysis reveals where in the space of networks this transition occurs and its underlying…
Trajectory prediction in a cluttered environment is key to many important robotics tasks such as autonomous navigation. However, there are an infinite number of possible trajectories to consider. To simplify the space of trajectories under…
Despite the attention that the problem of path planning for tethered robots has garnered in the past few decades, the approaches proposed to solve it typically rely on a discrete representation of the configuration space and do not exploit…
In the vector-field guided path-following problem, a sufficiently smooth vector field is designed such that its integral curves converge to and move along a one-dimensional geometric desired path. The existence of singular points where the…
The spaces of directed paths on the geometric realizations of pre-cubical sets, called also $\square$--sets, can be interpreted as the spaces of possible executions of Higher Dimensional Automata, which are models for concurrent…