Diffeological Principal Bundles, Čech Cohomology and Principal Infinity Bundles

Abstract: Thanks to a result of Baez and Hoffnung, the category of diffeological spaces is equivalent to the category of concrete sheaves on the site of cartesian spaces. By thinking of diffeological spaces as kinds of sheaves, we can therefore think of diffeological spaces as kinds of infinity sheaves. We do this by using a model category presentation of the infinity category of infinity sheaves on cartesian spaces, and cofibrantly replacing a diffeological space within it. By doing this, we obtain a new generalized cocycle construction for diffeological principal bundles, a new version of Čech cohomology for diffeological spaces that can be compared very directly with two other versions appearing in the literature, which is precisely infinity sheaf cohomology, and we show that the nerve of the category of diffeological principal G-bundles over a diffeological space X for a diffeological group G is weak equivalent to the nerve of the category of G-principal infinity bundles over X.

mathematical physicsalgebraic topologycategory theory

Audience: researchers in the topic

( paper )

Topology and Geometry Seminar (Texas, Kansas)