Polyadic sets and homomorphism counting

A classical result due to Lovász (1967) shows that the isomorphism type of a graph is determined by homomorphism counts. That is, graphs G and H are isomorphic whenever the number of homomorphisms K→G is the same as the number of homomorphisms K→H for all graphs K. Variants of this result, for various classes of finite structures, have been exploited in a wide range of research fields, including graph theory and finite model theory. We provide a categorical approach to homomorphism counting based on the concept of polyadic (finite) set. The latter is a special case of the notion of polyadic space introduced by Joyal (1971) and related, via duality, to Boolean hyperdoctrines in categorical logic. We also obtain new homomorphism counting results applicable to a number of infinite structures, such as trees and profinite algebras.