cogdef.html

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<TITLE>cogeometry - definitions</TITLE>

<H1>Definitions</H1>

 <P>The aim of this file is to provide some definitions for the
theoretical explanation of the <A HREF="cogeometry.html">cogeometry
concept</A>.

<H3><A NAME="geometry">Geometry</A></H3>

 <P>If we have a space X (usually R<sup>2</sup>, R<sup>3</sup>), a
<B>geometry</B> is a subdivision of this space into parts called <B><A
HREF="#region">regions</A></B>. That means, each point has to be part
of one region, or to be part of the boundary of two or more regions.

<H3><A NAME="region">Region</A></H3>

 <P>A closed subset of the space X of codimension 0. Every point of
the space is in the inner part of a region (inner point) or on the
boundary of two or more regions (boundary point).  The set of all
boundary points is the <A HREF="#skeleton">1-skeleton</A>.

<H3><A NAME="Face">Face</A></H3>

 <P>A closed subset of X of codimension 1, consisting only of boundary
points.  In X, it has no inner points. But in the subset topology of
the 1-skeleton X<sup>1</sup> inner points can be defined. Such points are
named "boundary face points" (codimension 1). A boundary face point is
part of two regions and only one boundary face.  The points which are
not inner points of any face form the 2-skeleton (codimension &gt;1).

<H3><A NAME="edge">Edges</A>, <A NAME="vertex">Vertices</A>, ...</H3>

The same scheme can be continued for higher codimension. Edges have
codimension 2, vertices codimension 3. Higher codimensions are not
possible in 3D, that's why we don't need a name for them.

<H3><A NAME="segment">Segments</A></H3>

 <P>A d-segment is a common notion for regions (0-segments), faces
(1-segments), edges (2-segments), vertices (3-segments).  It can be
generalized also into higher dimension. 

 <P>A d-segment is a closed subset of X with codimension d. It is part
of the set of all d-boundary-points.  It has - in the topology of all
d-boundary-points - inner points (codimension = d) and boundary points
(codimension &gt; d - (d+1)-boundary-points).  Every inner
d-boundary-point lies only in one d-segment, every
(d+1)-boundary-point in two or more d-segments.

<H3><A NAME="skeleton">Skeleton</A></H3>

 <P>The d-skeleton X<sup>d</sup> is the union of all d-segments. Thus, the
0-skeleton of X is X. The 1-skeleton is the set of all boundary
points.

<H3><A NAME="covariance">Covariance</A></H3>

 <P>A class of objects which may be defined on spaces X is
<B>covariant</B>, if for certain, appropriate functions (morphisms)
<B>f: X-&gt;Y</B> for every object on X there is a uniquely defined
<B>image</B> on Y.

 <P>Examples of covariant objects are points, functions Z-&gt;X into
the space, a tangential vector, probability distributions, homology
and homotopy groups.

<H3><A NAME="contravariance">Contravariance</A></H3>

 <P>A class of objects which may be defined on spaces X is
<B>contravariant</B>, if for certain, appropriate functions
(morphisms) <B>f: X-&gt;Y</B> for every object on Y there is a
uniquely defined <B>preimage</B> on X.

 <P>Examples of contravariant objects are functions X-&gt;Z on the
space, a covector, a metric, differential forms, cohomology groups,
fibre bundles, connections (gauge fields), open and close subsets.