15 research outputs found
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
The Directed Homotopy Hypothesis
The homotopy hypothesis was originally stated by Grothendieck: topological spaces should be "equivalent" to (weak) infinite-groupoids, which give algebraic representatives of homotopy types. Much later, several authors developed geometrizations of computational models, e.g., for rewriting, distributed systems, (homotopy) type theory etc.
But an essential feature in the work set up in concurrency theory, is that time should be considered irreversible, giving rise to the field of directed algebraic topology. Following the path proposed by Porter, we state here a directed homotopy hypothesis: Grandis\u27 directed topological spaces should be "equivalent" to a weak form of topologically enriched categories, still very close to (infinite,1)-categories. We develop, as in ordinary algebraic topology, a directed homotopy equivalence and a weak equivalence, and show invariance of a form of directed homology
Trace spaces in a pre-cubical complex
AbstractIn directed algebraic topology, directed irreversible (d)-paths and spaces consisting of d-paths are studied from a topological and from a categorical point of view. Motivated by models for concurrent computation, we study in this paper spaces of d-paths in a pre-cubical complex. Such paths are equipped with a natural arc length which moreover is shown to be invariant under directed homotopies. D-paths up to reparametrization (called traces) can thus be represented by arc length parametrized d-paths. Under weak additional conditions, it is shown that trace spaces in a pre-cubical complex are separable metric spaces which are locally contractible and locally compact. Moreover, they have the homotopy type of a CW-complex
Some collapsing operations for 2-dimensional precubical sets
In this paper, we consider 2-dimensional precubical sets, which can be used
to model systems of two concurrently executing processes. From the point of
view of concurrency theory, two precubical sets can be considered equivalent if
their geometric realizations have the same directed homotopy type relative to
the extremal elements in the sense of P. Bubenik. We give easily verifiable
conditions under which it is possible to reduce a 2-dimensional precubical set
to an equivalent smaller one by collapsing an edge or eliminating a square and
one or two free faces. We also look at some simple standard examples in order
to illustrate how our results can be used to construct small models of
2-dimensional precubical sets.Comment: New title, completely revised version of "Reducing cubical set models
of concurrent systems