  We develop a generalisation of the path homology theory introduced by
Grigor'yan, Lin, Muranov and Yau (GLMY-theory) in a general simplicial setting.
The new theory includes as particular cases the GLMY-theory for path complexes
and new homology theories: path homology of categories with a chosen set of
morphisms (marked categories) groups with a chosen subset (marked groups) and
path Hochschild homology of algebras with chosen vector subspaces (marked
algebras). Using our general machinery, we also introduce a new homology theory
for quivers that we call square-commutative homology of quivers and compare it
with the theory developed by Grigor'yan, Muranov, Vershinin and Yau.
