La théorie de l'homotopie de Grothendieck
The aim of the book by G. Maltsiniotis is to give an introduction to some of the key ideas of A. Grothendieck's unpublished treatise on abstract homotopy theory via category theory (“Pursuing Stacks”). The fundamental concept is that of a “test category” A, whose main property is that the localization of the presheaf category Ã with respect to weak equivalences is naturally equivalent to the usual homotopy category (a typical example is the category of simplices/cubes, for which Ã is the category of simplicial/cubical sets). Furthermore, one can consider more general homotopy categories by replacing the class of weak equivalences by an arbitrary “fundamental localize” (a class of functors that satisfies, among other things, Quillen's Theorem A). The exposition of these concepts and of their basic properties is very clear.