The tower of n-groupoids and the long cohomology sequence

Abstract

Given an exact category E , we associate to it a fibration c above E such that, for each object X of E , the fiber c[ x] is again exact. If, moreover, A is an internal abelian group in E , it determines a family of abelian groups A x in the fibres c[ x], such that the group H 1( E,A) is the colimit of the H 0( c[ x], A x ). This remark allows us to define iteratively H n+1( E,A) as the colimit of the H n ( c[ x], A x ). These groups are shown to have the property of the long cohomology sequence. When E=Ab , the construction coincides, up to isomorphism, with Yoneda's classical description of Ext n . When E=Grp , it coincides with the cohomology groups of a group in the sense of Eilenberg-Mac Lane.