Directed homotopy theory, I. The fundamental category

Marco Grandis

AMS Classification numbers: 55Pxx, 18G55.
Key words: homotopy theory, homotopical algebra, directed homotopy, fundamental category.

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

e-mail: grandis@dima.unige.it
http://www.dima.unige.it/~grandis/

Included files:
Grandis.Dht1.pdf

Notes: to appear in: Cahiers Topologie Geometrie Differentielle Categoriques
Preprint: Dip. Mat. Univ. Genova, Preprint 443 (2001), 26 p.
Revised version: 5 Nov 2001.

Abstract. Directed Algebraic Topology is beginning to emerge from
various applications.

The basic structure we shall use for the foundations of 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.