8 research outputs found
Comparing the non-unital and unital settings for directed homotopy
This note explores the link between the q-model structure of flows and the
Ilias model structure of topologically enriched small categories. Both have
weak equivalences which induce equivalences of fundamental (semi)categories.
The Ilias model structure cannot be left-lifted along the left adjoint adding
identity maps. The minimal model structure on flows having as cofibrations the
left-lifting of the cofibrations of the Ilias model structure has a homotopy
category equal to the -element totally ordered set. The q-model structure of
flows can be right-lifted to a q-model structure of topologically enriched
small categories which is minimal and such that the weak equivalences induce
equivalences of fundamental categories. The identity functor of topologically
enriched small categories is neither a left Quillen adjoint nor a right Quillen
adjoint between the q-model structure and the Ilias model structure.Comment: 15 page
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
Connectivity of spaces of directed paths in geometric models for concurrent computation
Higher Dimensional Automata (HDA) are higher dimensional relatives to
transition systems in concurrency theory taking into account to which degree
various actions commute. Mathematically, they take the form of labelled cubical
complexes. It is important to know, and challenging from a
geometric/topological perspective, whether the space of directed paths
(executions in the model) between two vertices (states) is connected; more
generally, to estimate higher connectedness of these path spaces.
This paper presents an approach for such an estimation for particularly
simple HDA modelling the access of a number of processors to a number of
resources with given limited capacity each. It defines a spare capacity for a
concurrent program with prescribed periods of access of the processors to the
resources. It shows that the connectedness of spaces of directed paths can be
estimated (from above) by spare capacities. Moreover, spare capacities can also
be used to detect deadlocks and critical states in such a HDA.
The key theoretical ingredient is a transition from the calculation of local
connectedness bounds (of the upper links of vertices of an HDA) to global ones
by applying a version of the nerve lemma due to Anders Bj\"orner
Directed Homology and Persistence Modules
In this note, we give a self-contained account on a construction for a
directed homology theory based on modules over algebras, linking it to both
persistence homology and natural homology. We study its first properties, among
which some exact sequences