The cohomology groups of an epimorphism

Abstract

If p:S→T is a U-split epimorphism in a monadic category ℂ (such as groups or algebras) and ℂ denotes the induced cotriple on ℂ then the cohomology groups H n (p,A) of the epimorphism with coefficients in a abelian group object A in ℂ have been defined by van Osdol ([18]) and interpreted in dimension 1 by Aznar and Cegarra ([1]), as isomorphism classes of 2-torsors which have p as their augmentation. In this paper it is shown that these groups are themselves “cotriple cohomology groups” for the cotriple on the category of simplicial objects of ℂ induced by ℂ and applied to the complex COSK o (p) with coefficients in the abelian group object K(A,1). Van Osdol's long exact sequence in the first variable associated with p is shown to be isomorphic to the standard second variable cotriple cohomology sequence, provided by the short exact sequence 0→K(A,0)→L(A,0)→K(A,1)→0 in this category. The interpretation of H1(p,A) in terms of 2-torsors is shown to be a consequence of the standard interpretation of H1 as isomorphism classes of 1-torsors, combined with the properties of the functor Open image in new window .