L' inj´ ´ enuit´ ´ m^ eme d' un regard neuf (celui de la science l'est
toujours) peut parfois ´ ´clairer d' un jour nouveau d' anciens probl`
emes. J.Monod [77, p. 13] his book is intended as a comprehensive
introduction to the theory of T principalsheaves andtheirconnections
inthesettingofAbstractDi?- ential Geometry (ADG), the latter being
initiated by A. Mallios'sGeometry of Vector Sheaves [62]. Based on
sheaf-theoretic methods and sheaf - homology, the presentGeometry of
Principal Sheaves embodies the classical theory of connections on
principal and vector bundles, and connections on vector sheaves, thus
paving the way towards a uni?ed (abstract) gauge t- ory and other
potential applications to theoretical physics. We elaborate on the
aforementioned brief description in the sequel. Abstract (ADG) vs.
Classical Di?erential Geometry (CDG). M- ern di?erential geometry is
built upon the fundamental notions of di?er- tial (smooth) manifolds and
?ber bundles, based, intheir turn, on ordinary di?erential calculus.
However, the theory of smooth manifolds is inadequate to cope, for -
stance, with spaces like orbifolds, spaces with corners, or other spaces
with more complicated singularities. This is a rather unfortunate
situation, since one cannot apply the powerful methods of di?erential
geometry to them or to any spaces that do not admit an ordinary method
of di?erentiation. The ix x Preface same inadequacy manifests in
physics, where many geometrical models of physical phenomena are
non-smooth.
