Lov\`{a}sz's hom-counting theorem by inclusion-exclusion principle

TL;DR

This paper presents a new, shorter proof of Lovász's hom-counting theorem in finite graph categories, extending its applicability beyond previous assumptions and providing new examples where the theorem holds.

Contribution

The paper introduces a novel generalization of Lovász's theorem without the finiteness assumption, along with a more concise proof and new applicable categories.

Findings

Shorter proof of the generalized theorem

Broader applicability to new categories

Extension beyond previous finiteness assumptions

Abstract

Let ${\mathcal C}$ be the category of finite graphs. Lov\`{a}sz (1967) shows that if $|\mathrm{Hom}(X,A)|=|\mathrm{Hom}(X,B)|$ holds for any $X$, then $A$ is isomorphic to $B$. Pultr (1973) gives a categorical generalization using a similar argument. Both proofs assume that each object has a finite number of isomorphism classes of subobjects. Generalizations without this assumption are given by Dawar, Jakl, and Reggio (2021) and Regio (2021). Here another generalization without this assumption is given, with a shorter proof. Examples of categories are given, for which our theorem is applicable, but the existing theorems are not.