19 research outputs found
Flow does not model flows up to weak dihomotopy
We prove that the category of flows cannot be the underlying category of a
model category whose corresponding homotopy types are the flows up to weak
dihomotopy. Some hints are given to overcome this problem. In particular, a new
approach of dihomotopy involving simplicial presheaves over an appropriate
small category is proposed. This small category is obtained by taking a full
subcategory of a locally presentable version of the category of flows.Comment: v2 16 pages, 3 figures ; updated bibliography + slight improvements
and corrections of typos ; v3 only the Journal-ref fiel
Inverting weak dihomotopy equivalence using homotopy continuous flow
A flow is homotopy continuous if it is indefinitely divisible up to
S-homotopy. The full subcategory of cofibrant homotopy continuous flows has
nice features. Not only it is big enough to contain all dihomotopy types, but
also a morphism between them is a weak dihomotopy equivalence if and only if it
is invertible up to dihomotopy. Thus, the category of cofibrant homotopy
continuous flows provides an implementation of Whitehead's theorem for the full
dihomotopy relation, and not only for S-homotopy as in previous works of the
author. This fact is not the consequence of the existence of a model structure
on the category of flows because it is known that there does not exist any
model structure on it whose weak equivalences are exactly the weak dihomotopy
equivalences. This fact is an application of a general result for the
localization of a model category with respect to a weak factorization system.Comment: 22 pages; LaTeX2e ; v2 : corrected bibliography + improvement of the
statement of the main theorems ; v3 final version published in
http://www.tac.mta.ca/tac
The homotopy branching space of a flow
In this talk, I will explain the importance of the homotopy branching space
functor (and of the homotopy merging space functor) in dihomotopy theory. The
paper is a detailed abstract of math.AT/0304112 and math.AT/0305169.Comment: Expository paper ; 11 pages ; to appear in GETCO'03 proceedin
Relative directed homotopy theory of partially ordered spaces
Algebraic topological methods have been used successfully in concurrency
theory, the domain of theoretical computer science that deals with distributed
computing. L. Fajstrup, E. Goubault, and M. Raussen have introduced partially
ordered spaces (pospaces) as a model for concurrent systems. In this paper it
is shown that the category of pospaces under a fixed pospace is both a
fibration and a cofibration category in the sense of H. Baues. The homotopy
notion in this fibration and cofibration category is relative directed
homotopy. It is also shown that the category of pospaces is a closed model
category such that the homotopy notion is directed homotopy.Comment: 20 page
T-homotopy and refinement of observation (II) : Adding new T-homotopy equivalences
This paper is the second part of a series of papers about a new notion of
T-homotopy of flows. It is proved that the old definition of T-homotopy
equivalence does not allow the identification of the directed segment with the
3-dimensional cube. This contradicts a paradigm of dihomotopy theory. A new
definition of T-homotopy equivalence is proposed, following the intuition of
refinement of observation. And it is proved that up to weak S-homotopy, a old
T-homotopy equivalence is a new T-homotopy equivalence. The left-properness of
the weak S-homotopy model category of flows is also established in this second
part. The latter fact is used several times in the next papers of this series.Comment: 20 pages, 3 figure
T-homotopy and refinement of observation (III) : Invariance of the branching and merging homologies
This series explores a new notion of T-homotopy equivalence of flows. The new
definition involves embeddings of finite bounded posets preserving the bottom
and the top elements and the associated cofibrations of flows. In this third
part, it is proved that the generalized T-homotopy equivalences preserve the
branching and merging homology theories of a flow. These homology theories are
of interest in computer science since they detect the non-deterministic
branching and merging areas of execution paths in the time flow of a higher
dimensional automaton. The proof is based on Reedy model category techniques.Comment: 30 pages ; final preprint version before publication ; see
http://nyjm.albany.edu:8000/j/2006/Vol12.ht
Comparing globular complex and flow
A functor is constructed from the category of globular CW-complexes to that
of flows. It allows the comparison of the S-homotopy equivalences (resp. the
T-homotopy equivalences) of globular complexes with the S-homotopy equivalences
(resp. the T-homotopy equivalences) of flows. Moreover, it is proved that this
functor induces an equivalence of categories from the localization of the
category of globular CW-complexes with respect to S-homotopy equivalences to
the localization of the category of flows with respect to weak S-homotopy
equivalences. As an application, we construct the underlying homotopy type of a
flow.Comment: 54 pages ; 3 figures ; Second paper corresponding to the content of
math.AT/0201252 ; v2 : very minor changes ; v3 : final versio