The Language of Categories

Abstract

A chain complex is a sequence of abelian groups and homomorphisms $$ {C_.}:...{\text{ }}{C_n}{\text{ }}{\text{ }}{C_{n - 1}}{\text{ }}{\text{ }}...$$ with the property d n ∘ d n+1 = 0 for all n. Homomorphisms d n are called boundary operators or differentials. A cochain complex is a sequence of abelian groups and homomorphisms $${C^.}:...{\text{ }}{C^n}{\text{ }}{\text{ }}{C_{n + 1}}{\text{ }}{\text{ }}...$$ with the property d n ∘ d n−1 = 0. A chain complex can be considered as a cochain complex by reversing the enumeration: C n = C −n , d n = d −n . This is why we will usually consider only cochain complexes. A complex of A- modules is a complex for which C n (respectively C n ) are modules over a ring A and d n (resp. d n ) are homomorphisms of modules.