Related papers: Inessential directed maps and directed homotopy eq…
Extending the results of reconstruction of compact metric spaces by inverse limits, we show that if $(X, d), (Y, d)$ are compact metric spaces, then the mapping space $Y^X$ is homotopy equivalent to the inverse limit of an inverse system of…
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…
We study the homotopy types of certain spaces closely related to the spaces of algebraic (rational) maps from the $m$ dimensional real projective space into the $n$ dimensional complex projective space for $2\leq m\leq 2n$ (we conjecture…
We identify Grandis' directed spaces as a full reflective subcategory of the category of multipointed $d$-spaces. When the multipointed $d$-space realizes a precubical set, its reflection coincides with the standard realization of the…
Given pointed cellular spaces $X$ and $Y$, $X$ compact, and an integer $r\ge0$, we define a relation $\overset r\approx$ on $[X,Y]$ and argue for the conjecture that it always coincides with the $r$-similarity $\overset r\sim$.
We introduce several homotopy equivalence relations for proper holomorphic mappings between balls. We provide examples showing that the degree of a rational proper mapping between balls (in positive codimension) is not a homotopy invariant.…
Given a smooth manifold $M$ and a totally nonholonomic distribution $\Delta\subset TM$ of rank $d$, we study the effect of singular curves on the topology of the space of horizontal paths joining two points on $M$. Singular curves are…
Two important invariants of directed graphs, namely magnitude homology and path homology, have recently been shown to be intimately connected: there is a 'magnitude-path spectral sequence' or 'MPSS' in which magnitude homology appears as…
Motivated by the recent work of Algom-Kfir and Bestinva introducing the mapping class group of an infinite graph via proper homotopy equivalences, we give a necessary and sufficient condition for a surface to be properly homotopy equivalent…
A flow is a directed space structure on a homotopy type. It is already known that the underlying homotopy type of the realization of a precubical set as a flow is homotopy equivalent to the realization of the precubical set as a topological…
We introduce the persistent homotopy type distance dHT to compare real valued functions defined on possibly different homotopy equivalent topological spaces. The underlying idea in the definition of dHT is to measure the minimal shift that…
We develop a homotopy theory of directed graphs based on cubical homotopy groups, also referred to as A-groups or reduced GLMY homotopy groups. Localizing the category of directed graphs at morphisms that induce isomorphisms on these groups…
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 that any globular subdivision of multipointed $d$-spaces gives rise to a dihomotopy equivalence between the associated flows. As a straightforward application, the flows associated to two multipointed $d$-spaces related by a finite…
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…
In this paper, we present a directed homotopy type theory for reasoning synthetically about (higher) categories, directed homotopy theory, and its applications to concurrency. We specify a new `homomorphism' type former for Martin-L\"of…
We begin the study the algebraic topology of semi-coarse spaces, which are generalizations of coarse spaces that enable one to endow non-trivial `coarse-like' structures to compact metric spaces, something which is impossible in coarse…
We introduce a homotopy theory of digraphs (directed graphs) and prove its basic properties, including the relations to the homology theory of digraphs constructed by the authors in previous papers. In particular, we prove the homotopy…
Let $\mathcal{E}(X)$ be the group of homotopy classes of self homotopy equivalences for a connected CW complex $X$. We observe two classes of maps $\mathcal{E}$-maps and co-$\mathcal{E}$-maps. They are defined as the maps $X\to Y$ that…
On one hand, together with Pelle Steffens, we recently characterized the infinity category of derived manifolds up to equivalence by a universal property. On the other hand, it is shown in recent work of Behrend-Liao-Xu that the category of…