This work demonstrates that homomorphism-counting functions reveal isomorphism types in combinatorial categories, suggesting new insights in group and graph theory.
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.
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Ceres et al. (2025) studied this question.
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