Related papers: Twisted-Product Categorical Bundles
A categorical principal bundle is a structure comprised of categories that is analogous to a classical principal bundle; examples arise from geometric contexts involving bundles over path spaces. We show how a categorical principal bundle…
Principal bundles have at least three different definitions, depending on the category of geometric objects studied. In Differential Geometry, they are defined as locally trivial projection map of smooth manifolds with an atlas whose…
We introduce the notion of continuous twisted partial actions of a locally compact group on a C*-algebra. With such, we construct an associated C*-algebraic bundle called the semidirect product bundle. Our main theorem shows that, given any…
Tangent categories are categories equipped with a tangent functor: an endofunctor with certain natural transformations which make it behave like the tangent bundle functor on the category of smooth manifolds. They provide an abstract…
We introduce a geometric construction of a gauge field theory of a complex adaptive system. It is based on a suitable simplicial formulation of a discrete geometry that manifests relevant properties valid in the classical differentiable…
We review the concept of a graded bundle as a natural generalisation of a vector bundle. Such geometries are particularly nice examples of more general graded manifolds. With hindsight there are many examples of graded bundles that appear…
We propose a generalisation of the notion of associated bundles to a principal bundle constructed via group action cocycles rather than via mere representations of the structure group. We devise a notion of connection generalising Ehresmann…
Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we…
We offer here a more direct approach to twisted K-theory, based on the notion of twisted vector bundles (of finite or infinite dimension) and of twisted principal bundles. This is closeely related to the classical notion ot torsors and…
We introduce the concept of a graded bundle which is a natural generalization of the concept of a vector bundle and whose standard examples are higher tangent bundles T^nQ playing a fundamental role in higher order Lagrangian formalisms.…
Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard…
Tangent category theory is a well-established categorical framework for differential geometry. A long list of fundamental geometric constructions, such as the tangent bundle functor, vector fields, Euclidean spaces, and vector bundles have…
We construct the categories of standard vector bundles over schemes and define direct sum and tensor product. These categories are equivalent to the usual categories of vector bundles with additional properties. The tensor product is…
A gauge theory is associated with a principal bundle endowed with a connection permitting to define horizontal lifts of paths. The horizontal lifts of surfaces cannot be defined into a principal bundle structure. An higher gauge theory is…
We define locally trivial quantum vector bundles (QVB) and QVB associated to locally trivial quantum principal fibre bundles. There exists a differential structure on the associated vector bundle coming from the differential structure on…
We introduce the notion of locally trivial quantum principal bundles. The base space and total space are compact quantum spaces (unital $C^{\star}$-algebras), the structure group is a compact matrix quantum group. We prove that a quantum…
We present the notion of a filtered bundle as a generalisation of a graded bundle. In particular, we weaken the necessity of the transformation laws for local coordinates to exactly respect the weight of the coordinates by allowing more…
A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category of fibers. Well known results on fibre bundles are shown to be true for stratified bundles;…
K. S. S. Nambooripad introduced an interesting class of categories known as normal categories, which are categories with subobjects, morphisms admitting factorization and having sufficiently many cones. These normal categories plays…
A tangent category is a category equipped with an endofunctor that satisfies certain axioms which capture the abstract properties of the tangent bundle functor from classical differential geometry. Cockett and Cruttwell introduced…