Hom-counting functions, combinatorial categories and related problems

TL;DR

This paper explores hom-counting functions in combinatorial categories, demonstrating their ability to recover object types and applying these insights to isomorphism problems in algebraic structures.

Contribution

It establishes that hom-counting functions satisfy a strong Yoneda Lemma property for small categories and applies this to isomorphism problems in various mathematical domains.

Findings

Hom-counting functions can recover object isomorphism types.

The algebra of hom-counting functions is studied for small categories.

Applications to isomorphism problems in group, graph, and ring theory.

Abstract

Combinatorial categories satisfy a stronger form of Yoneda Lemma, namely, the isomorphism type of an object can be recovered by counting the number of homomorphisms from all other objects into it. In this work, we show that this property holds for sufficiently small categories by studying the algebra of homomorphism-counting functions. We present applications of the results to the isomorphism problem in group, graph and ring theory.