Sheaves, principal bundles, and Čech cohomology for diffeological spaces

Abstract

The purpose of this note is to define sheaves for diffeological spaces and give a construction of their Čech cohomology. As an application, we prove that the first degree Čech cohomology classes for the sheaf of smooth functions to an abelian diffeological group G classify diffeological principal G-bundles.

Diffeological Principal Bundles and Principal Infinity Bundles

“…(⇒) Proposition 5.3. (4) of [KWW21] proves that q * P → U is a diffeological principal G-bundle over a cartesian space, thus it is trivial. (⇐) If q * P U × G, then the differentiably good cover {U } shows that π is a subduction, namely the following diagram commutes:…”

“…In this section we discuss different definitions of Čech cohomology of a diffeological space in the literature, that of [KWW21] and [Igl20], and offer the perspective of higher topos theory.…”

“…Comparing Diffeological Čech cohomologies. In the literature there are two definitions [Igl20] and [KWW21], of diffeological Čech cohomology that we wish to compare. We admit outright that we are unable to show using our framework that these cohomologies coincide, although we document some interesting relationships between their constructions.…”

“…We've proven that the cohomology of the PIZ complex is in fact given by ∞-sheaf cohomology. Now we shall discuss the Čech cohomology of [KWW21] with coefficients in R δ . Given a diffeological space X, we obtain its nebula B = p∈Plot(X) U p , which is equipped with a map B π − → X defined by π(p, u) = p(u).…”

“…This is of course not the case for the category of finite dimensional smooth manifolds, and thus provides a "better" category in which to work. Various notions of classical differential geometry have been defined for diffeological spaces, like differential forms, deRham cohomology, fiber bundles, tangent spaces [CW15], and recently Čech cohomology [Igl20] [KWW21].…”