categories: preprint: 'Directed homotopy theory, I'

[grandis@dima.unige.it (Marco Grandis)]

The following preprint is available on line:

Marco Grandis,
'Directed homotopy theory, I. The fundamental category'

Abstract.

   Directed Algebraic Topology is beginning to emerge from various applications.

   The basic structure we shall use for such a theory, a 'd-space', is a
topological space equipped with a family of 'directed paths', closed under some
operations. This allows for 'directed homotopies', generally non reversible,
represented by a cylinder and cocylinder functors. The existence of 'pastings'
(colimits) yields a geometric realisation of cubical sets as d-spaces, together
with homotopy constructs which will be developed in a sequel. Here, the
'fundamental category' of a d-space is introduced and a 'Seifert - van Kampen'
theorem proved; its homotopy invariance rests on 'directed homotopy' of
categories. In the process, new shapes appear, for d-spaces but also for small
categories, their elementary algebraic model.

   Applications of such tools are briefly considered or suggested, for objects
which model a directed image, or a portion of space-time, or a concurrent
process.

Dip. Mat. Univ. Genova, Preprint 443 (October 2001).
26 pages.

Available at:

http://www.dima.unige.it/~grandis/
ftp://www.dima.unige.it/Home/grandis/public/Dht1.ps

http://arXiv.org/abs/math.AT/0111048

_____________

Marco Grandis

Dipartimento di Matematica
Universita' di Genova
via Dodecaneso 35
16146 GENOVA, Italy

e-mail: grandis@dima.unige.it
tel: +39.010.353 6805   fax: +39.010.353 6752

http://www.dima.unige.it/~grandis/
ftp://www.dima.unige.it/Home/grandis/public/