Preface This monograph, wThich is aimed at the graduate level and beyond, consists of two parts. In part II we develop the beginnings of a kind of "relative" category theory of what we will call homotopical categories. These are categories with a single distin- guished class of maps (called weak equivalences) containing all the isomorphisms and satisfying one simple two out of six axiom which states that (*) for every three maps r, s and t for which the two compositions sr and ts are defined and are weak equivalences, the four maps r, s, t and tsr are also weak equivalences, which enables one to define "homotopicar versions of such basic categorical no- tions as initial and terminal objects, colimit and limit functors, adjunctions, Kan extensions and universal properties. In part I we use the results of part II to get a better understanding of Quilleir s so useful model categories, which are categories with three distinguished classes of maps (called cofibrations, fibrations and weak equivalences) satisfying a few simple axioms which enable one to "do homotopy theory*'. In particular we show that such model categories are homotopically cocornplete and homotopically complete in a sense which is much stronger than the existence of all small homotopy colimit and limit functors. Both parts are essentially self-contained. A reader of part II is assumed to have some familiarity with the categorical notions mentioned above, wrhile those who read part I (and especially the introductory chapter) should also know something about model categories. In the hope of increasing the local as well as the global readability of this monograph, we not only start each section with some introductory remarks and each chapter with an introductory section, but also each of the two parts with an introductory chapter, with the first chapter of part I serving as motivation for and introduction to the whole monograph and the first chapter of part II summarizing the main results of its other three chapters.

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