d-homology and extended d-homology functors are defined for categorical and conceptual homology generalization. Singular (extended) d-homology is introduced by composition of this functor with other certain functors. Dimension axiom, Homotopy axiom and in certain conditions Exactness axiom have been investigated in (extended) d-homology theory. Poincare - Hurewicz Theorem which is an important relation between homotpy and homology theory is explained and established in relation to (extended) d-homology in this paper.
