C-G

Abstract

In ordinary homological algebra, if M is an R-module, the usual way of starting to construct a projective resolution of M is to let F be the free R-module generated by the elements of M and F // M the epimorphism determined by (m) Â // m. One then takes the kernel of F // M and continues the process. But notice that in the construction of F // M a lot of structure is customarily overlooked. F is actually a functor MG of M , F // M is an instance of a natural transformation G // (identity functor); there is also a “comultiplication” G // GG which is a little less evident. The functor G, equipped with these structures, is an example of what is called a standard construction or “cotriple”. In this paper we start with a category C, a cotriple G in C, and show how resolutions and derived functors or homology can be constructed by means of this tool alone. The category C will be non-abelian in general (note that even for modules the cotriple employed fails to respect the additive structure of the category), and the coefficients will consist of an arbitrary functor E:C // A , where A is an abelian category. For ordinary homology and cohomology theories, E will be tensoring, homming or deriving with or into a module of some kind. To summarize the contents of the paper: In Section 1 we define the derived functors and give several examples of categories with cotriples. In Section 2 we study the derived functors Hn( , E)G as functors on C and give several of their properties. In Section 3 we fix a first variable X ∈ C and study Hn(X, )G as a functor of the abelian variable E. As such it admits a simple axiomatic characterization. Section 4 considers the case in which C is additive and shows that the general theory can always, in effect, be reduced to that case. In Section 5 we study the relation between cotriples and projective classes (defined—essentially—by Eilenberg-Moore [Eilenberg & Moore (1965)]) and show that the homology only depends on the projective class defined by the cotriple. Sections 6–9 are concerned largely with various special properties that these derived functors possess in well known algebraic categories (groups, modules, algebras, . . . ). In Section 10 we consider the problem of defining a cotriple to produce a given projective class (in a sense, the converse problem to that studied in Section 5) by means of “models”. We also compare the results with other theories of derived functors based on models. Section 11 is concerned with some technical items on acyclic models.