Title: The Cylinder Simplicial DG Ring 

Authors: Amnon Yekutieli

Publication status: 


Abstract: Given a DG ring B and a number q ≥ 0, we construct the q-th cylinder DG ring Cyl_q(B). For q = 1 this is just Keller’s cylinder DG ring (sometimes called the path object of B), which encodes homotopies between DG ring homomorphisms A → B.

As q changes the cylinder DG rings form a simplicial DG ring Cyl(B). Hence, given another DG ring A, the DG ring homomorphisms A → Cyl(B) form a simplicial set Hom(A, Cyl(B)). Our main theorem states that when A is a semi-free DG ring, the simplicial set Hom(A, Cyl(B)) is a Kan complex.

For the verification of the Kan condition we introduce a new construction, which may be of independent interest. Given a horn Y, we define the DG ring N(Y, B), and we prove that N(Y , B) represents this horn in the simplicial set Hom(A, Cyl(B)). In this way the Kan condition is implemented intrinsically in the category of DG rings, thus facilitating calculations.

Presumably all the above can be extended, with little change, from DG rings to (small) DG categories. That would enable easy constructions and explicit calculations of some simplicial aspects of DG categories.



updated 14 Feb 2026