Combinatorics in (2,1)-categories

TL;DR

This paper explores the concept of groupoid cardinality as an extension of set cardinality, establishing new properties and generalizing Lovász's theorem to (2,1)-categories, linking combinatorics and higher category theory.

Contribution

It introduces new properties of groupoid cardinality and extends Lovász's theorem to a broad class of (2,1)-categories, bridging combinatorics with higher category theory.

Findings

Groupoid cardinality generalizes finite set cardinality.

Properties of groupoid cardinality analogous to set functions.

Extension of Lovász's theorem to (2,1)-categories.

Abstract

Groupoid cardinality is an invariant of locally finite groupoids which has many of the properties of the cardinality of finite sets, but which takes values in all non-negative real numbers, and accounts for the morphisms of a groupoid. Several results on groupoid cardinality are proved, analogous to the relationship between cardinality of finite sets and i.e. injective or surjective functions. We also generalize to a broad class of (2,1)-categories a famous theorem of Lov\'asz which characterizes the isomorphism type of relational structures by counting the number of homomorphisms into them.