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A linear constraint system is specified by linear equations over the group$\ZZ_d$ of integers modulo $d$. Their operator solutions play an important rolein the study of quantum contextuality and non-local games. In this paper, weuse the theory of simplicial sets to develop a framework for studying operatorsolutions of linear systems. Our approach refines the well-knowngroup-theoretical approach based on solution groups by identifying these groupsas algebraic invariants closely related to the fundamental group of a space. Inthis respect, our approach also makes a connection to the earlier homotopicalapproach based on cell complexes. Within our framework, we introduce a newclass of linear systems that come from simplicial sets and show that any linearsystem can be reduced to one of that form. Then we specialize in linear systemsthat are associated with groups. We provide significant evidence for aconjecture stating that for odd $d$ every linear system admitting a solution ina group admits a solution in $\ZZ_d$.

Simplicial techniques for operator solutions of linear constraint systems