Cosimplicial versus DG-rings: a version of the Dold–Kan correspondence

Abstract

The (dual) Dold-Kan correspondence says that there is an equivalence of categories K : Ch ¿0 → Ab between nonnegatively graded cochain complexes and cosimplicial abelian groups, which is inverse to the normalization functor. We show that the restriction of K to DG-rings can be equipped with an associative product and that the resulting functor DGR * → Rings , although not itself an equivalence, does induce one at the level of homotopy categories. In other words both DGR * and Rings are Quillen closed model categories and the total left derived functor of K is an equivalence:LK : Ho DGR * ∼ → Ho Rings :The dual of this result for chain DG and simplicial rings was obtained independently by Schwede and Shipley, Algebraic and Geometric Topology 3 (2003) 287, through di erent methods. Our proof is based on a functor Q : DGR * → Rings , naturally homotopy equivalent to K, and which preserves the closed model structure. It also has other interesting applications. For example, we use Q to prove a noncommutative version of the Hochschild-Kostant-Rosenberg and Loday-Quillen theorems. Our version applies to the cyclic module [n] → n R S that arises from a homomorphism R → S of not necessarily commutative rings, using the coproduct R of associative R-algebras. As another application of the properties of Q, we obtain a simple, braid-free description of a product on the tensor power S ⊗ n R originally deÿned by Nuss K-theory 12 (1997) 23, using braids.

Homotopical Adjoint Lifting Theorem

“…Dual Dold-Kan Equivalence. The dual Dold-Kan equivalence is studied in [CC04], where the authors study the classical Quillen pair K ∶ Ch ≥0 ⇆ Ab ∆ ∶ N between non-negatively graded cochain complexes Ch ≥0 and cosimplicial abelian groups, where N is the normalized complex functor (also known as the Moore complex), and K is its left adjoint. They replace K ⊣ N with the following Quillen pair, which we will refer to as the monoidal dual Dold-Kan equivalence:…”

Peiffer elements in simplicial groups and algebras

“…This description is in the spirit of that given in [8] for the inverse of the conormalization functor.…”

Homotopy Colimits and Global Observables in Abelian Gauge Theory

“…Restricting to infinitesimal gauge transformations, this picture reduces nicely to the well-known BRST formalism, see [FR12] for a presentation of this topic in the context of the algebraic approach to field theory. By the dual Dold-Kan correspondence (see Appendix A) we can regard our cosimplicial algebra as a differential graded algebra (dg-algebra), see also [CC04] for more details. This dg-algebra can be 'linearized' via a procedure called the van-Est map (here we require the smooth structure mentioned above) to yield the BRST algebra (Chevalley-Eilenberg dg-algebra) of gauge theory, see e.g.…”