# Schreiber stable infinity-stacks -- homological algebra

previous: principal infinity-bundles

home: sheaves and stacks

• 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

• In the next installment we will then use these tools to describe the stably abelian part of general cohomology theory: abelian sheaf cohomology.

# The Dold-Kan correspondence

Last revised on September 9, 2009 at 01:00:41. See the history of this page for a list of all contributions to it.