Related papers: Path space connections and categorical geometry
The theory of principal $G$-bundles over a Lie groupoid is an important one, unifying the various types of principal $G$-bundles, including those over manifolds, those over orbifolds, as well as equivariant principal $G$-bundles. In this…
We prove that isomorphism classes of principal bundles over a diffeological space are in bijection to certain maps on its free loop space, both in a setup with and without connections on the bundles. The maps on the loop space are smooth…
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.
Curvature and torsion of linear transports along paths in, respectively, vector bundles and the tangent bundle to a differentiable manifold are defined and certain their properties are derived.
Following the approach of Budzy\'nski and Kondracki, we define covariant differential algebras and connections on locally trivial quantum principal fibre bundles. We also consider covariant derivatives, connection forms and curvatures and…
In the main part of this paper a connection is just a fiber projection onto a (not necessarily integrable) distribution or sub vector bundle of the tangent bundle. Here curvature is computed via the Froelicher-Nijenhuis bracket, and it is…
We construct a new category of vector spaces which contains both the standard category of vector spaces and Grassmannians. Its space of objects classifies vector bundles, its space of morphisms classifies bundle isomorphisms, and it can be…
Directed Algebraic Topology is beginning to emerge from various applications. The basic structure we shall use for such a theory, a 'd-space', is a topological space equipped with a family of 'directed paths', closed under some operations.…
The theory of frames normal for general connections on differentiable bundles is developed. Links with the existing theory of frames normal for covariant derivative operators (linear connections) in vector bundles are revealed. The…
What remains of a geometrical notion like that of a principal bundle when the base space is not a manifold but a coarse graining of it, like the poset formed by a base for the topology ordered under inclusion? Motivated by finding a…
Distributions, i.e., subsets of tangent bundles formed by piecing together subspaces of tangent spaces, are commonly encountered in the theory and application of differential geometry. Indeed, the theory of distributions is a fundamental…
For finite coverings we elucidate the interaction between transferred Chern classes and Chern classes of transferred bundles. This involves computing the ring structure for the complex oriented cohomology of various homotopy orbit spaces.…
An introduction to the theory of bundle gerbes and their relationship to Hitchin-Chatterjee gerbes is presented. Topics covered are connective structures, triviality and stable isomorphism as well as examples and applications.
A bounded curvature path is a continuously differentiable piece-wise $C^2$ path with bounded absolute curvature connecting two points in the tangent bundle of a surface. These paths have been widely considered in computer science and…
We prove that the category of abelian gerbes with connection over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These bundles are equipped with a connection and with a "fusion" product…
Starting categorically, we give simple and precise models of equivariant classifying spaces. We need these models for work in progress in equivariant infinite loop space theory and equivariant algebraic K-theory, but the models are of…
In the geometric version of the Langlands correspondence, irregular singular point connections play the role of Galois representations with wild ramification. In this paper, we develop a geometric theory of fundamental strata to study…
We argue that the quantum-theoretical structures studied in several recent lines of research cannot be adequately described within the standard framework of quantum circuits. This is in particular the case whenever the combination of…
This paper presents a research program aimed at establishing relational foundations for relativistic quantum physics. Although the formalism is still under development, we believe it has matured enough to be shared with the broader…
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…