The Simplicial Cylinder DG Ring

TL;DR

This paper introduces the simplicial cylinder DG ring, generalizing Keller's construction, and proves that the associated simplicial Hom set is a Kan complex for semi-free DG rings.

Contribution

It develops higher cylinder DG rings forming a simplicial structure and establishes the Kan complex property of the simplicial Hom set for semi-free DG rings.

Findings

The simplicial set SHom(A,B) is a Kan complex when A is semi-free.

Automorphism groups in the fundamental groupoid are abelian.

SHom_{ extless=1}(A,B) is invariant under quasi-isomorphisms.

Abstract

The Keller cylinder DG ring encodes homotopies between DG ring homomorphisms $f_0, f_1 : A \to B$. Recently we discovered the higher cylinder DG rings $Cyl_q(B)$, which assemble into the simplicial cylinder DG ring $Cyl(B)$. For $q=1$ this recovers Keller's original construction. The sets $SHom_q(A,B)$ of DG ring homomorphisms $A \to Cyl_q(B)$ form the simplicial Hom set $SHom(A,B)$. Our main result is that when $A$ is a semi-free DG ring, the simplicial set $SHom(A,B)$ is a Kan complex. We prove several results about the fundamental groupoid $SHom_{\leq 1}(A,B)$, including invariance under quasi-isomorphism $B' \to B$, and that the automorphism groups are abelian. We also indicate some applications of this work.