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abelian category anticommutes B+NA C+NB+NA called canonical factorization category of pairs cokernel commutative diagram commutative square connected sequence connecting homomorphism Consider the commutative D+NA defined denoted diagram of l.e.s.'s dual E+NP Eckmann and Hilton embedding epic epimorphism exact functor exact rows exact sequence Ext2n extended Extn(W gives h-functor Homological Algebra identity maps induced maps injective class injective homotopy groups isomorphism ker f ker g Ker-Coker sequence kernel of f left excision left R-modules lemma map f map induced mapping cylinder modules monic morphisms natural transformations Notice obtain the l.e.s. obtains the commutative Peter Hilton proof prove pushout R(cok replace result right excision s.e.s. of Q sequence of functors short exact sequence shown Similarily small category Sn(f Sn(g Sn(gf split s.e.s. square of monomorphisms Theorem Tn(A Tn(b Tn(c Tn(f Tn(g Tn(p Tr(B two-sided excision property Vi<A zero zero object