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We develop a discrete theory of vector bundles with connection that is natural with respect to appropriate mappings of the base space. The main objects are discrete vector bundle valued cochains. The central operators are a discrete exterior covariant derivative and a combinatorial wedge product. We demonstrate the key properties of these operators and show that they are natural with respect to the mappings referred to above. We give a new interpretation in terms of a double averaging of anti-symmetrized cup product which serves as our discrete wedge product. We also formulate a well-behaved definition of metric compatible discrete connections. We characterize when a discrete vector bundle with connection is trivializable or has a trivial lower rank subbundle. This machinery is used to define discrete curvature as linear maps and we show that our formulation satisfies a discrete Bianchi identity.
We develop a ready-to-use comprehensive theory for (super) 2-vector bundles over smooth manifolds. It is based on the bicategory of (super) algebras, bimodules, and intertwiners as a model for 2-vector spaces. We show that our 2-vector bundles form a
In this paper, we study a new operation named pushforward on diffeological vector pseudo-bundles, which is left adjoint to the pullback. We show how to pushforward projective diffeological vector pseudo-bundles to get projective diffeological vector
We study vector bundles over Lie groupoids, known as VB-groupoids, and their induced geometric objects over differentiable stacks. We establish a fundamental theorem that characterizes VB-Morita maps in terms of fiber and basic data, and use it to pr
We introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by the kth pow
VB-groupoids are vector bundles in the category of Lie groupoids. They encompass several classical objects, including Lie group representations and 2-vector spaces. Moreover, they provide geometric pictures for 2-term representations up to homotopy o