Ivancevic_Applied-Diff-Geom

Technical Preliminaries: Tensors, Actions and Functors 115F(Coker f) for all f: the last condition is equivalent to the requirementthat F preserves short right exact sequences.In an Abelian category A, a chain complex is a sequence... ✲ cn+1∂ n+1✲ cn∂ n ✲ cn−1✲ ...of composable arrows, with ∂ n ∂ n+1 = 0 for all n. The sequence need not beexact at c n ; the deviation from exactness is measured by the nth homologyobjectH n c = Ker(∂ n : c n✲ cn−1 )/ Im(∂ n+1 : c n+1✲ cn ).Similarly, a cochain complex in an Abelian category A is a sequence... ✲ wn+1d n+1✲ wnd n ✲ wn−1✲ ...of composable arrows, with d n d n+1 = 0 for all n. The sequence neednot be exact at w n ; the deviation from exactness is measured by the nthcohomology objectH n w = Ker(d n+1 : w n✲ wn+1 )/ Im(d n : w n−1✲ wn ).A cycle is a chain C such that ∂C = 0. A boundary is a chain C suchthat C = ∂B, for any other chain B.A cocycle (a closed form) is a cochain ω such that dω = 0. A coboundary(an exact form) is a cochain ω such that ω = dθ, for any other cochain θ.