The volume of a differentiable stack

TL;DR

This paper generalizes the concept of cardinality from discrete groupoids to Lie groupoids, defining a stack volume invariant using additional geometric data, and introduces an intrinsic Hilbert space structure for stacks.

Contribution

It extends the notion of cardinality to Lie groupoids and defines a new invariant called the volume of a stack, incorporating geometric data.

Findings

Defines the volume of a stack as an invariant under groupoid equivalence.

Introduces an intrinsic Hilbert space structure for stacks based on line bundle sections.

Abstract

We extend the notion of the cardinality of a discrete groupoid (equal to the Euler characteristic of the corresponding discrete orbifold) to the setting of Lie groupoids. Since this quantity is an invariant under equivalence of groupoids, we call it the volume of the associated stack rather than of the groupoid itself. Since there is no natural measure in the smooth case like the counting measure in the discrete case, we need extra data to define the volume. This data has the form of an invariant section of a natural line bundle over the groupoid. Sections of a square root of this line bundle constitute an "intrinsic Hilbert space" of the stack.