The Derived Category

Abstract

There are many formal similarities between homological algebra and algebraic topology. The Dold-Kan correspondence, for example, provides a dictionary between positive complexes and simplicial theory. The algebraic notions of chain homotopy, mapping cones, and mapping cylinders have their historical origins in simplicial topology. The derived category D ( A ) of an abelian category is the algebraic analogue of the homotopy category of topological spaces. D ( A ) is obtained from the category Ch ( A ) of (cochain) complexes in two stages. First one constructs a quotient K ( A ) of Ch ( A ) by equating chain homotopy equivalent maps between complexes. Then one “localizes” K ( A ) by inverting quasi-isomorphisms via a calculus of fractions. These steps will be explained below in sections 10.1 and 10.3. The topological analogue is given in section 10.9. The Category K ( A ) Let A be an abelian category, and consider the category Ch = Ch ( A ) of cochain complexes in A . The quotient category K = K ( A ) of Ch is defined as follows: The objects of K are cochain complexes (the objects of Ch ) and the morphisms of K are the chain homotopy equivalence classes of maps in Ch. That is, Hom k ( A, B ) is the set Hom ch ( A B )/ ∼ of equivalence classes of maps in Ch. We saw in exercise 1.4.5 that K is well defined as a category and that K is an additive category in such a way that the quotient Ch ( A ) → K ( A ) is an additive functor.