相关论文: Parallel Transport and Functors

We extend the usual notion of parallel transport along a path to triangulated surfaces. A homotopy of paths is lifted into a fibered category with connection and this defines a functor between the fibers above the boundary paths. These…

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…

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…

A review of the parallel transport (translation) in fibre bundles is presented. The connections between transports along paths and parallel transports in fibre bundles are examined. It is proved that the latter ones are special cases of the…

The axiomatic approach to parallel transport theory is partially discussed. Bijective correspondences between the sets of connections, (axiomatically defined) parallel transports, and transports along paths satisfying some additional…

We prove a Lie 2-group torsor version of the well-known one-one correspondence between fibered categories and pseudofunctors. Consequently, we obtain a weak version of the principal Lie group bundle over a Lie groupoid. The correspondence…

In this note we make use of some properties of vector fields on a manifold to give an alternate proof to [3] for the equivalence between connections and parallel transport on vector bundles over manifolds. Out of the proof will emerge a new…

A nice differential-geometric framework for (non-abelian) higher gauge theory is provided by principal 2-bundles, i.e. categorified principal bundles. Their total spaces are Lie groupoids, local trivializations are kinds of Morita…

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…

Manifolds and fiber bundles, while superficially different, have strong parallels; in particular, they are both defined in terms of equivalence classes of atlases or in terms of maximal atlases, with the atlases treated as mere adjuncts.…

We provide a new perspective on parallel 2-transport and principal 2-group bundles with 2-connection. We define parallel 2-transport as a 2-functor from the thin fundamental 2-groupoid to the 2-category of 2-group torsors. The definition of…

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…

The linear transports along paths in vector bundles introduced in Ref. [1] are applied to the special case of tensor bundles over a given differentiable manifold. Links with the transports along paths generated by derivations of tensor…

Transports along path in fibre bundles are axiomatically introduced. Their general functional form and some their simple properties are investigated. The relationships of the transports along paths and lifting of paths are studied.

The (parallel) linear transports along paths in vector bundles are axiomatically described. Their general form and certain properties are found. It is shown that these transports are locally (i.e. along every fixed path) always Euclidean…

Parallel transport in a fibre bundle with respect to smooth paths in the base space B have recently been extended to representations of the smooth singular simplicial set Sing_{smooth}(B). Inspired by these extensions,I revisit the…

The parallel linear transports defined by flat linear connection are axiomatically described. On this basis a number of properties, some of which are new, of these transports and connections are derived.

The theory of linear transports along paths in vector bundles, generalizing the parallel transports generated by linear connections, is developed. The normal frames for them are defined as ones in which their matrices are the identity…

Categorical bundles provide a natural framework for gauge theories involving multiple gauge groups. Unlike the case of traditional bundles there are distinct notions of triviality, and hence also of local triviality, for categorical…

We associate to a 2-vector bundle over an essentially finite groupoid a 2-vector space of parallel sections, or, in representation theoretic terms, of higher invariants, which can be described as homotopy fixed points. Our main result is…