Singularity structures for noncommutative spaces

TL;DR

This paper introduces a categorical framework for noncommutative spaces that assigns distribution spaces with intrinsic regularity and singularity notions, linking propagation of singularities in pseudo-differential operators on groupoids to those on manifolds.

Contribution

It develops a new (bi)category framework for noncommutative spaces with distributional structures and applies it to analyze singularity propagation in pseudo-differential operators.

Findings

Categories can be derived from manifolds, noncommutative spaces, or Lie groupoids.

Spaces of distributions have intrinsic regularity and singularity notions.

Application to propagation of singularities in pseudo-differential operators on groupoids.

Abstract

We introduce a (bi)category $\mathfrak{Sing}$ whose objects can be functorially assigned spaces of distributions and generalized functions. In addition, these spaces of distributions and generalized functions possess intrinsic notions of regularity and singularity analogous to usual Schwartz distributions on manifolds. The objects in this category can be obtained from smooth manifolds, noncommutative spaces, or Lie groupoids. An application of these structures relates the longitudinal propagation of singularities for pseudo-differential operators on a groupoid with propagation of singularities on the base manifold.