On analytic groupoid cardinality

TL;DR

This paper introduces a method to assign complex-valued cardinalities to certain groupoids using analytic continuation of generating functions, revealing a new recursive structure called 'nested equivalence' that extends traditional notions of groupoid size.

Contribution

It proposes a novel approach to define complex cardinalities for divergent groupoid series via analytic continuation, uncovering the concept of nested equivalence.

Findings

Complex-valued cardinalities can be assigned to divergent groupoids.

Analytic continuation reveals a recursive 'nested equivalence' structure.

The approach extends traditional groupoid cardinality concepts.

Abstract

Groupoids graded by the groupoid of bijections between finite sets admit generating functions which encode the groupoid cardinalities of their graded components. As suggested in the work of Baez and Dolan, we use analytic continuation of such generating functions to define a complex-valued cardinality for groupoids whose usual groupoid cardinality diverges. The complex nature of such a cardinality invariant is shown to reflect a recursion of structure which we refer to as `nested equivalence'.