'Directed Algebraic Topology'

[Marco Grandis <grandis@dima.unige.it>]

Dear categorists,

My book

    'Directed Algebraic Topology'
    Models of non-reversible worlds

has appeared, at Cambridge University Press. Its aims are mentioned
below.

It is likely well known that the policy of Cambridge UP, with respect
to publication,
is open and liberal.

But I must say I was pleased and surprised, during the preparation of
this volume,
by their way of handling things, which was at the same time effective
and informal,
precise and very flexible.

For the many people in this list that are concerned with the problems
of our libraries,
because of the high prices of scientific books and journals, I will
add that royalties for
this volume have been converted into CUP books for the library of my
Departement.

Marco Grandis
________

FROM THE BEGINNING OF THE INTRODUCTION
Aims

Directed Algebraic Topology is a recent subject which arose in the
1990's, on the one hand in abstract settings for homotopy theory,
and on the other hand in investigations in the theory of concurrent
processes.
Its general aim should be stated as `modelling non-reversible
phenomena'.
The subject has a deep relationship with category theory.

	The domain of Directed Algebraic Topology should be distinguished
from the domain of classical Algebraic Topology by the principle that
{\it directed spaces
have privileged directions and directed paths therein need not be
reversible}.
While the classical domain of Topology and Algebraic Topology is a
reversible world,
where a path in a space can always be travelled backwards, the study
of non-reversible
phenomena requires broader worlds, where a directed space can have
non-reversible paths.

The homotopical tools of Directed Algebraic Topology, corresponding
in the classical case to
ordinary homotopies, the fundamental group and fundamental $n$-
groupoids, should be similarly
`non-reversible': {\it directed homotopies}, the {\it fundamental
monoid} and {\it fundamental
$n$-categories}.
Similarly, its homological theories will take values in `directed'
algebraic structures, like {\it
preordered} abelian groups or abelian {\it monoids}. Homotopy
constructions like mapping cone,
cone and suspension, occur here in a directed version; this gives
rise to new `shapes', like (lower and
upper) directed cones and directed spheres, whose elegance is
strengthened by the fact that such
constructions are determined by universal properties.

Applications will deal with domains where privileged directions
appear, such as concurrent
processes, rewrite systems, traffic networks, space-time models,
biological systems, etc.
At the time of writing, the most developed ones are concerned with
concurrency.


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