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We had found a concrete cocycle description of hom-sets in the homotopy category of infinity-stacks and had announced that these hom-sets describe very general notions of cohomology.
As a first application of this, we looked at the special case of Cech cohomology. This provides a concrete component formula for cocycles in nonabelian cohomology.
More simplifications occur when we restrict attention to special coefficient objects, i.e. to special simplicial sheaves. Namely it turns out that whenever the infinity-groupoids that we deal with have a “maximally nice” abelian group structure on them – in which case they are called stable infinity-groupoids, great simplifications kick in and we obtain strong algebraic models for computing with these beasts.
At the heart of this convenient reformulation of stably abelian $\infty$-groupoids is the Dold-Kan correspondence. This says that and how precisely certain particularly nice stable $\infty$-groupoids are modeled by the methods of homological algebra.
Indeed, conversely, it is helpful to understand the origin and purpose of homological algebra as such as being the study of stably abelian $\infty$-groupoids. This clarifies many structures and constructions in this old subject.
Indeed, the very notion of chain complex is seen, by the classical Dold-Kan correspondence, to be precisely equivalent to nothing but a more efficient encoding of the structure given by a Kan complex with strict abelian group structure.
Therefore we now
look at some concepts in homological algebra
and then discuss the Dold-Kan correspondence.
In the next installment we will then use these tools to describe the stably abelian part of general cohomology theory: abelian sheaf cohomology.
Last revised on September 9, 2009 at 01:00:41. See the history of this page for a list of all contributions to it.