Schreiber stable infinity-stacks: homological algebra

previous: principal infinity-bundles

home: sheaves and stacks

next: categories of stable objects – additive, abelian and triangulated categories

  • 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.

Abstract nonsense linear algebra

Abstract nonsense homological algebra

The Dold-Kan correspondence

Last revised on September 9, 2009 at 00:55:26. See the history of this page for a list of all contributions to it.