Stacks, costacks and axiomatic homology

Abstract

Let p: $ -+ £ be a sheaf (espace etale) of abelian groups. Applying singular functor S one obtains a simplicial map tt: A-> X with A=5(«f), X=S(X)and tt — S(p). The fibers 77_1(x), x e X, form a system of over X which will be called a costack of abelian groups over the simplicial set X. In general, a costack is defined as a functor on X, regarded as a category. This is a generalized dual of the notion of a stack defined by Spanier [5]. The main objects of this note are (1) to develop a general theory of stacks and costacks over simplicial sets, (2) to construct a semisimplicial homology theory with variable coefficients which is unique in the sense of Eilenberg-Steenrod. The coefficients of the homology are a costack in an abelian category. In particular, when the coefficient costack is a locally constant costack the homology becomes the usual homology with local coefficients. There are three chapters in this note. Chapter I is devoted to a study of stacks and costacks. It is partially a preparatory chapter. In Chapter II we define homology of costacks via usual chain complexes and prove that the homology so defined can be computed by projective resolutions by introducing a generalized torsion product functor. Under the equivalence of costacks and modules, this generalized functor is essentially the genuine torsion product functor of modules. The rest of Chapter II is a preparation for Chapter III, in which a homology theory of pairs of simplicial sets over a fixed simplicial set Ais defined. Results of Chapter II ensure the existence of such a theory. Chapter III concludes with a proof of the uniqueness of this homology theory. This is a generalization of Eilenberg-Steenrod uniqueness theorem [1]. The results presented in this note are a part of the author's Ph.D. thesis at the City University of New York written under the direction of Professor Alex Heller.