Combinatorics in (2,1)-categories
Krista Zehr
TL;DR
This work extends Lovász's homomorphism-counting paradigm from finite relational structures to (2,1)-categories by developing groupoid cardinality as a robust invariant and establishing a Lovász-type theorem for categories of relatively finite functors. It introduces and analyzes factorization systems in (2,1)-categories, defines relative finiteness (relfin), and constructs the 2-category $\mathrm{relfin}_{\mathcal{B}}$, showing that these combinatorial (2,1)-categories support tractable calculus including coproducts and products compatible with the Cauchy product and derivative. The main combinatorial result expresses hom-categories as coproducts over $\mathcal{E}$-quotients, enabling cancellation-like phenomena in higher categories and connecting to the theory of stuff types and generating functions. The paper also links groupoid cardinality to homotopy cardinality, and outlines pathways to ∞-categorical generalizations via Postnikov systems and higher connected/truncated factorizations, highlighting both concrete results and open directions for extending these ideas to broader higher-categorical contexts.
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 Lovász which characterizes the isomorphism type of relational structures by counting the number of homomorphisms into them.