Homological algebra i $${varvec{}}$$-abelia categories

In this paper, we study the homological theory in n-abelian categories. First, we prove some useful properties of n-abelian categories, such as ((n+2)times (n+2))-lemma, 5-lemma and n-Horseshoes lemma. Secondly, we introduce the notions of right(left) n-derived functors of left(right) n-exact functors, n-(co)resolutions, and n-homological dimensions of n-abelian categories. For an n-exact sequence, we show that the long n-exact sequence theorem holds as a generalization of the classical long exact sequence theorem. As a generalization of (textsf {Ext}^*(-,-)), we study the n-derived functor (textsf {nExt}^*(-,-)) of hom-functor (mathrm {Hom}(-,-)). We give an isomorphism between the abelian group of equivalent classes of m-fold n-extensions (textsf {nE}^m(A,B)) of A, B and (textsf {nExt}_{mathcal A}^m(A,B)) using n-Baer sum for (m,nge 1).