We introduce relative homological and weakly homological categories $(\mathbf{C},\mathbf{E})$, where ``relative'' refers to a distinguished class $\mathbf{E}$ of normal epimorphisms in $\mathbf{C}$. It is a generalization of homological categories, but also protomodular categories can be regarded as examples. We indicate that the relative versions of various homological lemmas can be proved in a relative homological category.
Journal of Homotopy and Related Structures, Vol. 1(2006), No. 1, pp. 185-194 http://jhrs.rmi.acnet.ge/volumes/2006/n1a8/