The Cylinder Simplicial DG Ring
Amnon Yekutieli
TL;DR
This work constructs the q-th cylinder DG ring $Cyl_q(B)$ and assembles them into a simplicial DG ring $Cyl(B)$ to encode homotopies between DG ring maps. The central result is that for a semi-free DG ring $A$, the simplicial set $Hom_{ ext{DGRng}}(A, Cyl(B))$ is a Kan complex, furnishing an explicit intrinsic horn-filling mechanism via the new DG ring $N(Y,B)$ representing horns. A key technical development shows $N(oldsymbol{ abla}^{q}_{i}, B)$ represents horns in $Hom(A, Cyl(B))$, enabling fillers through a lifting property for semi-free resolutions. The paper also discusses extending these constructions to (small) DG categories, suggesting a path toward a concrete simplicial enrichment of DG categories, with potential implications for explicit computations in derived categories and higher morphisms.
Abstract
Given a DG ring $B$ and an integer $q \geq 0$, 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 $A \to B$. As $q$ changes the cylinder DG rings form a simplicial DG ring $Cyl(B)$. Hence, given another DG ring $A$, the DG ring homomorphisms $A \to Cyl(B)$ 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.