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

Criteria for homotopic maps to be so along monotone homotopies

The state spaces of machines admit the structure of time. A homotopy theory respecting this additional structure can detect machine behavior unseen by classical homotopy theory. In an attempt to bootstrap classical tools into the world of abstract spacetime, we identify criteria for classically homotopic, monotone maps of pospaces to future homotope, or homotope along homotopies monotone in both coordinates, to a common map. We show that consequently, a hypercontinuous lattice equipped with its Lawson topology is future contractible, or contractible along a future homotopy, if its underlying space has connected CW type.Comment: 7 pages, 5 figures, partially presented at GETCO 2006. title change; strengthened Cor. 3.3. -> Prop. 3.7, Prop. 3.2 -> Lem. 3.2; corrected def of category of continuous lattices in sec. 2; added 5 figures, 8 eg's, Def. 3.4, Lemmas 2.8, 3.5, refs [1],[4],[5]; rewording throughout; conclusion and abstract rewritte

Directed Homotopy in Non-Positively Curved Spaces

A semantics of concurrent programs can be given using precubical sets, in order to study (higher) commutations between the actions, thus encoding the "geometry" of the space of possible executions of the program. Here, we study the particular case of programs using only mutexes, which are the most widely used synchronization primitive. We show that in this case, the resulting programs have non-positive curvature, a notion that we introduce and study here for precubical sets, and can be thought of as an algebraic analogue of the well-known one for metric spaces. Using this it, as well as categorical rewriting techniques, we are then able to show that directed and non-directed homotopy coincide for directed paths in these precubical sets. Finally, we study the geometric realization of precubical sets in metric spaces, to show that our conditions on precubical sets actually coincide with those for metric spaces. Since the category of metric spaces is not cocomplete, we are lead to work with generalized metric spaces and study some of their properties

Towards Directed Collapsibility

In the directed setting, the spaces of directed paths between fixed initial and terminal points are the defining feature for distinguishing different directed spaces. The simplest case is when the space of directed paths is homotopy equivalent to that of a single path; we call this the trivial space of directed paths. Directed spaces that are topologically trivial may have non-trivial spaces of directed paths, which means that information is lost when the direction of these topological spaces is ignored. We define a notion of directed collapsibility in the setting of a directed Euclidean cubical complex using the spaces of directed paths of the underlying directed topological space relative to an initial or a final vertex. In addition, we give sufficient conditions for a directed Euclidean cubical complex to have a contractible or a connected space of directed paths from a fixed initial vertex. We also give sufficient conditions for the path space between two vertices in a Euclidean cubical complex to be disconnected. Our results have applications to speeding up the verification process of concurrent programming and to understanding partial executions in concurrent programs

06341 Abstracts Collection -- Computational Structures for Modelling Space, Time and Causality

From 20.08.06 to 25.08.06, the Dagstuhl Seminar 06341 ``Computational Structures for Modelling Space, Time and Causality\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

A Hurewicz Model Structure for Directed Topology

This paper constructs an h-model structure for diagrams of streams, locally preordered spaces. Along the way, the paper extends some classical characterizations of Hurewicz fibrations and closed Hurewicz cofibrations. The usual characterization of classical closed Hurewicz cofibrations as inclusions of neighborhood deformation retracts extends. A characterization of classical Hurewicz fibrations as algebras over a pointed Moore cocylinder endofunctor also extends. An immediate consequence is a long exact sequence for directed homotopy monoids, with applications to safety verifications for database protocols

Directed Homotopy in Non-Positively Curved Spaces

A semantics of concurrent programs can be given using precubical sets, in order to study (higher) commutations between the actions, thus encoding the "geometry" of the space of possible executions of the program. Here, we study the particular case of programs using only mutexes, which are the most widely used synchronization primitive. We show that in this case, the resulting programs have non-positive curvature, a notion that we introduce and study here for precubical sets, and can be thought of as an algebraic analogue of the well-known one for metric spaces. Using this it, as well as categorical rewriting techniques, we are then able to show that directed and non-directed homotopy coincide for directed paths in these precubical sets. Finally, we study the geometric realization of precubical sets in metric spaces, to show that our conditions on precubical sets actually coincide with those for metric spaces. Since the category of metric spaces is not cocomplete, we are lead to work with generalized metric spaces and study some of their properties

Directed topological complexity

International audienceIt has been observed that the very important motion planning problem of robotics mathematically speaking boils down to the problem of finding a section to the path-space fibration, raising the notion of topological complexity, as introduced by M. Farber. The above notion fits the motion planning problem of robotics when there are no constraints on the actual control that can be applied to the physical apparatus. In many applications, however, a physical apparatus may have constrained controls, leading to constraints on its potential future dynamics. In this paper we adapt the notion of topological complexity to the case of directed topological spaces, which encompass such controlled systems, and also systems which appear in concurrency theory. We study its first properties, make calculations for some interesting classes of spaces, and show applications to a form of directed homotopy equivalence