Ivancevic_Applied-Diff-Geom

114 Applied Differential Geometry: A Modern Introductionis a short exact sequence when it is exact at A, at B, and at C.Since 0 → a is the zero arrow, exactness at A means just that f ismonic (i.e., 1–1, or injective map); dually, exactness at C means that g isepic (i.e., onto, or surjective map). Therefore, (2.63) is equivalent tof = Ker g, g = Coker f.Similarly, the statement that h = Coker f becomes the statement that thesequenceAf ✲ Bg ✲ C ✲ 0is exact at B and at C. Classically, such a sequence was called a shortright exact sequence. Similarly, k = Ker f is expressed by a short left exactsequence0 ✲ Af ✲ Bg ✲ C.If A and A ′ are Abelian categories, an additive functor F : A → A ′ isa functor from A to A ′ withF(f + f ′ ) = Ff + Ff ′ ,for any parallel pair of arrows f, f ′ : b → c in A. It follows that F0 = 0.A functor F : A → A ′ between Abelian categories A and A ′ is, bydefinition, exact when it preserves all finite limits and all finite colimits. Inparticular, an exact functor preserves kernels and cokernels, which meansthatKer(Ff) = F(Ker f) and Coker(Ff) = F(Coker f);then F also preserves images, coimages, and carries exact sequences toexact sequences. By construction of limits from products and equalizersand dual constructions, F : A → A ′ is exact iff it is additive and preserveskernels and cokernels.A functor F is left exact when it preserves all finite limits. In otherwords, F is left exact iff it is additive and Ker(Ff) = F(Ker f) for all f:the last condition is equivalent to the requirement that F preserves shortleft exact sequences.Similarly, a functor F is right exact when it preserves all finite colimits.In other words, F is right exact iff it is additive and Coker(Ff) =