Séminaires : Séminaire d'Algèbre

The Keller cylinder DG ring encodes homotopies between DG ring homomorphisms f_0, f_1 : 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 \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, including to DG categories.

This talk will take place on Zoom only.