Related papers: Parallel Transport and Functors

Kendall shape spaces are a widely used framework for the statistical analysis of shape data arising from many domains, often requiring the parallel transport as a tool to normalise time series data or transport gradient in optimisation…

A recently-developed theory of quantum general relativity provides a propagator for free-falling particles in curved spacetimes. These propagators are constructed by parallel-transporting quantum states within a quantum bundle associated to…

We point out that double categories provide a natural setting for modular functors obtained by a (bicategorical) string-net construction: The source of the modular functor -- which is now a double functor -- is a symmetric monoidal double…

We consider one possible definition of a diffeological connection on a diffeological vector pseudo-bundle. It is different from the one proposed in [7] and is in fact simpler, since it is obtained by a straightforward adaption of the…

In a fibre bundle, natural derivatives of a section are defined as tangent vector fields on the image of a section of the fibre bundle. A local extension to vector fields in the tangent bundle leads to a direct proof of the formula…

In this article we define and investigate a notion of parallel transport on finite projective modules over finite matrix algebras. Given a derivation-based differential calculus on the algebra and a connection on the module, we construct…

The general problem for consistency between arbitrary transports along paths in fibre bundles and bundle morphisms between them is formulated and investigated. The special case of one fibre bundle, its morphism and transport along paths…

We define a notion of connection in a fibre bundle that is compatible with a singular foliation of the base. Fibre bundles equipped with such connections are in plentiful supply, arising naturally for any Lie groupoid-equivariant bundle,…

We develop an alternative view on the concept of connections over a vector bundle map, which consists of a horizontal lift procedure to a prolonged bundle. We further focus on prolongations to an affine bundle and introduce the concept of…

We study a type of connection forms, given by Chen integrals, over pathspaces by placing such forms within a category-theoretic framework of principal bundles and connections. We introduce a notion of 'decorated' principal bundles, develop…

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…

We establish a relation between smooth 2-functors defined on the path 2-groupoid of a smooth manifold and differential forms on this manifold. This relation can be understood as a part of a dictionary between fundamental notions from…

We remark that forcing on fiber bundles of structures of first order languages is not a compatible semantics with the pullback (of fiber bundles) and we describe a semantics which behaves well with respect to it. This new semantics uses…

We develop a theory of \'etale parallel transport for vector bundles with numerically flat reduction on a $p$-adic variety. This construction is compatible with natural operations on vector bundles, Galois equivariant and functorial with…

We develop parallel transport on path spaces from a differential geometric approach, whose integral version connects with the category theoretic approach. In the framework of 2-connections, our approach leads to further development of…

In the framework of Category Theory, we study the association between finite--dimensional representations of a compact quantum group and quantum vector bundles with linear connections for a given quantum principal bundle with a principal…

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…

In quantum mechanics, the momentum space and position space wave functions are related by the Fourier transform. We investigate how the Fourier transform arises in the context of geometric quantization. We consider a Hilbert space bundle H…

Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A 2-bundle…

Connections and curvings on gerbes are beginning to play a vital role in differential geometry and mathematical physics -- first abelian gerbes, and more recently nonabelian gerbes. These concepts can be elegantly understood using the…