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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.
List of contents
Sheaves and all that.- The category of differential triads.- Lie sheaves of groups.- Principal sheaves.- Vector and associated sheaves.- Connections on principal sheaves.- Connections on vector and associated sheaves.- Curvature.- Chern-Weil theory.- Applications and further examples.
Additional text
From the reviews of the first edition:
"The book provides an accessible, well-written monograph devoted to the theory of principal sheaves and their connections in the setting initiated by, Geometry of vector sheaves. … It is designed also as a reference book with detailed expositions and complete and self-contained proofs." (Witold Mozgawa, Zentralblatt MATH, Vol. 1082, 2006)
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From the reviews of the first edition:
"The book provides an accessible, well-written monograph devoted to the theory of principal sheaves and their connections in the setting initiated by, Geometry of vector sheaves. ... It is designed also as a reference book with detailed expositions and complete and self-contained proofs." (Witold Mozgawa, Zentralblatt MATH, Vol. 1082, 2006)
