This chapter will formulate the special properties which hold in categories such as Ab, R-Mod, Mod-R, and R-Mod-S: They are all Ab-categories (the hom-sets are abelian groups and composition is bilinear), all finite limits and colimits exist, and these limits — especially kernel and cokernel — are well behaved. This leads to a set of axioms describing an “abelian” category. The axioms suffice to prove all the facts about commuting diagrams and connecting morphisms which are proved in Ab by methods of chasing elements. We carry the subject exactly to this point, leaving the subsequent development of homological algebra to more specialized treatments.
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