Subject: categories: Preprint available

From: [email-masked] (Marco Grandis)
Date: Thu, 12 Nov 1998 15:26:28 +0100

The following preprint:

M. Grandis,
"An intrinsic homotopy theory for simplicial complexes
with applications to image processing"

is available at:

ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/

as:   Lnk.Nov98.ps

***

Abstract. A simplicial complex is a set equipped with a down-closed family
of distinguished finite subsets; this structure is mostly viewed as
codifying a triangulated space. Here, this structure is used directly to
describe "spaces" of interest in various applications, where the associated
triangulated space would be misleading. An intrinsic homotopy theory, not
based on topological realisation, is introduced.
        The applications considered here are aimed at metric spaces and
digital topology; concretely, at image processing and computer graphics. A
metric space  X  has a structure  t_e(X)  of simplicial complex at each
"resolution"  e > 0;  the resulting n-homotopy group  \pi_n(t_e(X))  detects
those singularities which can be captured by an n-dimensional grid, with
edges bound by  e;  this works equally well for continuous or discrete
regions of euclidean spaces.

***

Comments would be appreciated.

In particular, I am uneasy about a question of terminology.

In my opinion, the term "simplicial complex", quite appropriate when the
structure is viewed as codifying a triangulated space, is unfit when such
objects are treated as "spaces" in themselves (somewhat close to
bornological spaces, which have similar axioms on objects and maps).

In other words, "simplicial complex" should not refer to the category
itself, say  C,  but to its usual embedding in  Top,  the simplicial
realisation. The two aspects may clash, e.g. with respect to initial or
final structures: the coarse C-object on three points (final structure, all
parts are distinguished) is realised as a euclidean triangle; a C-subobject
is sufficient to produce a topological subspace (a regular subobject in
Top), but a C-subspace (a regular subobject in  C)  is a stronger notion.
Moreover, from a more concrete point of view, the simplicial realisation is
quite inappropriate in most of the applications considered in this work.

The opposition  "C-object / simplicial complex" is in part similar to
"sequence / series": the second term refers to a more specific view & use
of the same data; the clashing of the opposition is particularly evident in
the notions of convergence, for a sequence or a series.

That's why I am calling a C-object a "combinatorial space". (The term
"combinatorial complex" has also been used for simplicial complex; and I
wanted a term of the form "attribute + space", to use freely of topological
terms like discrete, coarse, subspace...)
But of course it is embarassing to propose a new term for a classical notion.

Marco Grandis