2007/06/25 by Martin Raußen, Martin Raussen, Raussen, Martin +2 · 2 citations
Computer Science · Mathematics · #Algebraic structures and combinatorial models #FOS: Mathematics #General Topology (math.GN) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.GN
paper · pdf · doi:10.48550/arxiv.0706.3560
arxiv created 2007/06/25 · openalex publication_date 2007/06/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 to investigate compositions and factorizations, and we compare it to the distributive lattice of countable subsets of the unit interval. The results obtained are used to analyse the space of traces in a topological space, i.e., the space of continuous paths up to reparametrization equivalence. This space is shown to be homeomorphic to the space of regular paths (without stops) up to increasing reparametrizations. Directed versions of the results are important in directed homotopy theory.