Related papers: Path space connections and categorical geometry
A general noncommutative-geometric theory of principal bundles is presented. Quantum groups play the role of structure groups. General quantum spaces play the role of base manifolds. A differential calculus on quantum principal bundles is…
A vector bundle with connection over a supermanifold leads naturally to a notion of parallel transport along superpaths. In this note we show that {\it every} such parallel transport along superpaths comes form a vector bundle with…
It is shown that the principle of locality and noncommutative geometry can be connnected by a sheaf theoretical method. In this framework quantum spaces are introduced and examples in mathematical physics are given. With the language of…
Using tangent bundle geometry we construct an equivalent reformulation of classical field theory on flat spacetimes which simultaneously encodes the perspectives of multiple observers. Its generalization to curved spacetimes realizes a new…
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…
The motivation for this paper stems \cite{CR} from the need to construct explicit isomorphisms of (possibly nontrivial) principal $G$-bundles on the space of loops or, more generally, of paths in some manifold $M$, over which I consider a…
This note addresses the construction of a notion of parallel transport along superpaths arising from the concept of a superconnection on a vector bundle over a manifold $M$. A superpath in $M$ is, loosely speaking, a path in $M$ together…
A noncommutative-geometric formalism of framed principal bundles is sketched, in a special case of quantum bundles (over quantum spaces) possessing classical structure groups. Quantum counterparts of torsion operators and Levi-Civita type…
In generalization of knot quandles we introduce similar algebraic structures associated with arbitrary pairs consisting of a path-connected topological space and its path-connected subspace.
In this paper, we introduce the concept of principal bundles on statistical manifolds. After necessary preliminaries on information geometry and principal bundles on manifolds, we study the $\alpha$-structure of frame bundles over…
In this paper we introduce a notion of parallel transport for principal bundles with connections over differentiable stacks. We show that principal bundles with connections over stacks can be recovered from their parallel transport thereby…
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…
In this paper we study cobordism categories consisting of manifolds which are endowed with geometric structure. Examples of such geometric structures include symplectic structures, flat connections on principal bundles, and complex…
The definitions and some basic properties of the linear transports along paths in vector bundles and the normal frames for them are recalled. The formalism is specified on line bundles and applied to a geometrical description of the…
We define functorial isomorphisms of parallel transport along etale paths for a class of G-principal bundles on a p-adic curve where G is a connected reductive algebraic group of finite presentation. This class consists of all principal…
We introduce a general definition of higher-form connections on principal $\infty$-bundles in differential geometry. This is achieved by developing the formal differentiation and integration of maps from smooth manifolds to derived stacks…
We review the basic elements of the geometrical formalism for description of gauge fields and the theory of invariant connections, and their applications to the coset space dimensional reduction of Yang-Mills theories. We also discuss the…
The characteristic forms in the bundle of connections of a principal bundle P over M determine the characteristic classes of P for degree less or equal to the dimension of M, and differential forms on the space of connections for higher…
We investigate relative holomorphic connections on a principal bundle over a family of compact complex manifolds. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic principal bundle over…
Tangent categories provide an axiomatic approach to key structural aspects of differential geometry that exist not only in the classical category of smooth manifolds but also in algebraic geometry, homological algebra, computer science, and…