The Simplicial Cylinder DG Ring

2026/02/28 by Amnon Yekutieli
Mathematics · #math.RA #math.CT #math.KT #msc:16E45 #msc:18N50

paper · pdf

this version: 30 pages, some improvements

arxiv created 2026/08/02 · arxiv updated 2026/08/04

Abstract

The Keller cylinder DG ring encodes homotopies between DG ring homomorphisms f₀, f₁ : A → 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 → 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_{≤ 1}(A,B), including invariance under quasi-isomorphism B' → B, and that the automorphism groups are abelian. We also indicate some applications of this work.

Citations

• The Squaring Operation for Commutative DG Rings
• A-infinity functors and homotopy theory of dg-categories
• Differential graded versus Simplicial categories
• A-infinity algebras, modules and functor categories