Homotopy Theories and Model Categories

摘要: This chapter explains homotopy theories and model categories. A category is just an ordinary with three specified classes of morphisms—fibrations, cofibrations, weak equivalences—which satisfy a few simple axioms that are deliberately reminiscent the properties topological spaces. Surprisingly enough, these give reasonably general context in which it possible to set up basic machinery theory. The can then be used immediately large number different settings, as long checked each case. Although many settings geometric, some them not. provides background material, principally discussion categorical constructions (limits colimits), come almost any attempt build new objects abstract out old ones. also conceptual interpretation category. Surprisingly, this depends only on class equivalences. suggests category, equivalences carry fundamental theoretic information, while fibrations, they function mostly tools make various constructions.