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