An exact sequence in the first variable for torsor cohomology: The 2-dimensional theory of obstructions

Abstract

In [9] J.W. Duskin gets an inner interpretation for the ‘cotriple’ cohomology of ia monadic category in terms of n-dimensional torsors, generalizing to any dimenision Beck’s interpretation of dimension 1. Later, in [15], P. Glenn, using a slightly ,different n-torsor concept, developes a cohomology theory for an exact category in the sense of Barr [4] without requiring free resolutions, which is applicable in both algebraic and topological settings and coincides with the most important cohomology theories in the known examples. Clearly this general context is an appropriate one in which to formulate and solve the classical problems considered in these theories of relating them in low dimensions to ‘obstructions and extensions’. This is the object of our paper. More precisely, with each of the classical cohomology theories in Algebra has been associated a theory relating Hi (Zf2 as classically numbered) to the classification of non-singular extensions and W2 (H3 as classically numbered) to obstructions to the existence of such extensions. This problem has been studied using cocycle calculations in several algebraic contexts (e.g. [5], [12], [163, [17], [22], [23], [24]) and without these ([13], [26]). In this paper, we interpret the two-dimensional torsors E. under an abelian group A (whose connected components [E.] E Tors’(R, A) define H2(R,A) in the Glenn theory) as obstructions to the existence of (l-dimensional) torsors under the groupoid G(E.) associated to E.. Specifically, we have