Primitive path homology

In this paper we introduce a primitive path homology theory in the category of simple digraphs.In the subcategory of asymmetric digraphs, this theory coincides with the path homology theory introduced by Grigor'yan, Lin, Muranov and Yau, but these theories differ in the general case.We study properties of the primitive path homology and describe relations between the primitive path homology and the path homology.Our approach gives a possibility to construct primitive homology theories of paths which have a given tail vertex a or (and) a given head vertex b.We study these theories and describe also relationships between them and the path homology theory.