Simplicial Methods in Homological Algebra

Abstract

By now, the reader has seen several examples of chain complexes in which the boundary maps C n → C n −1 are alternating sums d 0 − d 1 + … ± d n . The primordial example is the singular chain complex of a topological space X ; elements of C n (X) are formal sums of maps f from the n -simplex Δ n into X , and d i ( f ) is the composition of f with the inclusion Δ n−1 ⊂ Δ n of the i th face of the simplex (1.1.4). Other examples of this phenomenon include Koszul complexes (4.5.1), the bar resolution of a group (6.5.1), and the Chevalley-Eilenberg complex of a Lie algebra (7.7.1). Complexes of this form arise from simplicial modules, which are the subject of this chapter. Simplicial Objects Let Δ be the category whose objects are the finite ordered sets [ n ] = {0 n } for integers n ≥ 0, and whose morphisms are nondecreasing monotone functions. If A is any category, a simplicial object A in A is a contravariant functor from Δ to A , that is, A : Δ op → A . For simplicity, we write A n for A ([ n ]). Similarly, a cosimplicial object C in A is a covariant functor C : Δ → A , and we write A n for A ([ n ]). A morphism of simplicial objects is a natural transformation, and the category S A of all simplicial objects in A is just the functor category A Δ op .