Relative homological algebra. Cohomology of categories, posets and coalgebras

Abstract

This chapter discusses relative homological algebra and presents the main ideas of the theory and some recent advances in it. Generalizations of purity in the category of abelian groups and in module categories have many applications and are tools of homological algebra. The ideas of relative homological algebra are contained internally in the homological algebra of Cartan and Eilenberg. The chapter introduces the notion of a proper class of cokernels in a pre-abelian category. This generalization of the usual notion of a proper class of short exact sequences is used for defining relative derived categories. Relative homological algebra in module categories is presented in the chapter. In addition, the chapter discusses recent results on the classification of inductively closed proper classes, which are closely related with algebraically compact modules. The language of relative homological algebra is useful in defining the cohomology of small categories. The notion of a proper class is more familiar for abelian categories.