Composite Cotriples and Derived Functors

The main result of [Barr (1967)] is that the cohomology of an algebra with respect to the free associate algebra cotriple can be described by the resolution given by U. Shukla in [Shukla (1961)]. That looks like a composite resolution; first an algebra is resolved by means of free modules (over the ground ring) and then this resolution is given the structure of a DG-algebra and resolved by the categorical bar resolution. This suggests that similar results might be obtained for all categories of objects with “two structures”. Not surprisingly this turns out to involve a coherence condition between the structures which, for ordinary algebras, turns out to reduce to the distributive law. It was suggested in this connection by J. Beck and H. Appelgate. If α and β are two morphisms in some category whose composite is defined we let α ·β denote that composite. If S and T are two functors whose composite is defined we let ST denote that composite; we let αβ = αT ′ · Sβ = S ′β · αT : ST // S ′T ′ denote the natural transformation induced by α: S // S ′ and β: T // T ′. We let αX: SX // S ′X denote the X component of α. We let the symbol used for an object, category or functor denote also its identity morphism, functor or natural transformation, respectively. Throughout we let M denote a fixed category and A a fixed abelian category. N will denote the category of simplicial M objects (see 1.3. below) and B the category of cochain complexes over A.