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.
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Deep Dive into Reparametrizations of Continuous Paths.
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.
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In elementary differential geometry, the most basic objects studied (after points perhaps) are paths, i.e., differentiable maps p : I โ R n defined on the closed interval I = [0, 1]. Such a path is called regular if p โฒ (t) = 0 for all t โ ]0, 1[. A reparametrization of the unit interval I is a surjective differentiable map ฯ : I โ I with ฯ โฒ (t) > 0 for all t โ ]0, 1[, i.e. a (strictly increasing) self-diffeomorphism of the unit interval.
Given a path p : I โ R n and a reparametrization ฯ : I โ I, the paths p and p โข ฯ represent the same geometric object. In differential geometry one investigates equivalence classes (identifying p with p โข ฯ for any reparametrization ฯ) and their invariants, like curvature and torsion.
Motivated by applications in concurrency theory, a branch of theoretical Computer Science trying to model and to understand the coordination between many different processors working on a common task, we are interested in continuous paths p : I โ X in more general topological spaces up to more general reparametrizations ฯ : I โ I. When the state space of a concurrent program is viewed as a topological space (typically a cubical complex; cf. [6]), “directed” paths in that space respecting certain “monotonicity” properties correspond to executions. A nice framework to handle directed topological spaces (with an eye to homotopy properties) is the concept of a d-space proposed and investigated by Marco Grandis in [9]. Essentially, a topological space comes equipped with a subset of preferred d-paths in the set of all paths in X, cf. Definition 4.1. Note in particular, that the reverse of a directed path in general is not directed; the slogan is “breaking symmetries”.
We do not try to capture the quantitative behaviour of executions, corresponding to particular parametrizations of paths, but merely the qualitative behaviour, such as the order of shared resources used, or the result of a computation. Hence the objects of study are paths up to certain reparametrizations which 1. do not alter the image of a path, and 2. do not alter the order of events.
We are thus interested in general paths in topological spaces, up to surjective reparametrizations ฯ : I โ I which are increasing (and thus continuous!-cf. Lemma 2.7), but not necessarily strictly increasing. Two paths are considered to have the same behaviour if they are reparametrization equivalent, cf. Definition 1.2.
To understand this equivalence relation, we have to investigate the space of all reparametrizations which includes strange (e.g. nowhere differentiable) elements. Nevertheless, it enjoys remarkable properties: It is a monoid, in which compositions and factorizations can be completely analysed through an investigation of stop intervals and of stop values. The quotient space after dividing out the selfhomeomorphisms has nice algebraic lattice properties.
A path is called regular if it does not “stop”; and we are able to show that the space of general paths modulo reparametrizations is homeomorphic to the space of regular paths modulo increasing auto-homeomorphisms of the interval. Hence to investigate properties of the former, it suffices to consider the latter. This is one of the starting points in the homotopy theoretical and categorical investigation of invariants of d-spaces in [15]. Further possible areas of application of the results (and of higher-dimensionsal generalisations still to be investigated) include categorical homotopy theory as in [8], categorified gauge theory as in [1] and n-transport theory; cf. the blog “The n-category cafรฉ” at http://golem.ph.utexas.edu/category
. This article does not build on any sophisticated machinery. Most of the concepts and proofs can be understood with an undergraduate mathematical background. There are certain parallels to the elementary theory of distribution functions in probability theory, cf. e.g. [13]. The flavour is nevertheless different, since continuity (no jumps, i.e., surjectivity) is essential for us. For the sake of completeness, we have chosen to include also elementary results and their proofs (some of which may be well-known).
Marco Grandis has studied piecewise linear reparametrizations in [10] for different purposes, but also in the framework of “directed algebraic topology”.
Let always X denote a Hausdorff topological space and I = [0, 1] the unit interval. The set of all (nondegenerate) closed subintervals of I will be denoted by
Let p : I โ X denote a continuous map (a path), and remark that the pre-image p -1 (x) of any element x โ X is a closed set. Definition 1.1.
An interval J โ P [ ] (I) is called a p-stop interval if the restriction p | J is constant and if J is a maximal interval with that property.
The set of all p-stop intervals will be denoted as โ p โ P [ ] (I). Remark that the intervals in โ p are disjoint and that โ p carries a natural total order. We let D p := Jโโp J โ I denote the stop set of p.
