Diffeological Coarse Moduli Spaces of Stacks over Manifolds

TL;DR

This paper develops a framework for associating diffeological coarse moduli spaces to stacks over manifolds, linking the concepts with Lie groupoids, principal bundles, and differential forms to enhance understanding of differentiable stacks.

Contribution

It introduces a left adjoint functor from stacks to diffeological coarse moduli spaces and explores their geometric and differential properties in the context of Lie groupoids.

Findings

The functor from stacks to diffeological spaces is left adjoint to the Grothendieck construction.

Basic differential forms on stacks coincide with forms on their Lie groupoid representatives.

Diffeological forms on the orbit space match basic forms on the stack under certain conditions.

Abstract

In this paper, we consider diffeological spaces as stacks over the site of smooth manifolds, as well as the "underlying" diffeological space of any stack. More precisely, we consider diffeological spaces as so-called concrete sheaves and show that the Grothendieck construction sending these sheaves to stacks has a left adjoint: the functor sending any stack to its diffeological coarse moduli space. As an application, we restrict our attention to differentiable stacks and examine the geometry behind the coarse moduli space construction in terms of Lie groupoids and their principal bundles. Within this context, we define a "gerbe", and show when a Lie groupoid is such a gerbe (or when a stack is represented by one). Additionally, we define basic differential forms for stacks and confirm in the differentiable case that these agree (under certain conditions) with basic differential forms on a representative Lie groupoid. These basic differentiable forms in turn match the diffeological forms on the orbit space.