Categories and Functors

In Chapter I we discussed various algebraic structures (rings, abelian groups, modules) and their appropriate transformations (homomorphisms). We also saw how certain constructions (for example, the formation of Hom_Λ(A, B) for given Λ-modules A, B) produced new structures out of given structures. Over and above this we introduced certain “universal” constructions (direct sum, direct product) and suggested that they constituted special cases of a general, and important, procedure. Our objective in this chapter is to establish the appropriate mathematical language for the general description of mathematical systems and of mappings of systems, insofar as that language is applicable to homological algebra.