We clarify in which precise sense the theory of principal bundles and the theory of groupoids are equivalent; and how this equivalence of theories, in the differentiable case, reflects itself in the theory of connections. The method used is that of synthetic differential geometry. Introduction. In this note, we make explicit a sense in which the theory of principal fibre bundles is equivalent t...

Whereas usual Hodge theory concerns mainly the usual or abelian cohomology of an algebraic variety—or eventually the rational homotopy theory or nilpotent completion of π1 which are in some sense obtained by extensions—nonabelian Hodge theory concerns the cohomology of a variety with nonabelian coefficients. Because of the basic fact that homotopy groups in higher dimensions are abelian, and si...

For an arbitrary localic étale groupoid G we provide simple descriptions, in terms of modules over the quantale O(G) of the groupoid, of the continuous actions of G, including actions on open maps and sheaves. The category of G-actions is isomorphic to a corresponding category of O(G)-modules, and as a corollary we obtain a new quantale based representation of étendues.

Suppose G is a second countable, locally compact, Hausdorff, principal groupoid with a fixed left Haar system. We define a notion of integrability for groupoids, and show G is integrable if and only if the groupoid C∗-algebra C∗(G) has bounded trace.

In this paper, we provide two types of boundary path groupoids from a generalized Boolean dynamical system (B,L,θ,Iα). For the first groupoid, associate an inverse semigroup to and use tight spectrum T as unit space groupoid Γ(B,L,θ,Iα) that is isomorphic Gtight. The other one defined Renault-Deaconu Γ(∂E,σE) arising topological correspondence E associated with system. We then prove homeomorphi...

The aim of this paper is to obtain a group-2-groupoid as 2-groupoid object in the category groups and also special kind an internal group-groupoids. Corresponding group-2-groupoids, we some categorical structures related crossed modules group-groupoids prove equivalences between them. These results enable us 2-dimensional notions