The Simplicial Cylinder DG Ring
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917436005023744 |
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| author | Yekutieli, Amnon |
| author_facet | Yekutieli, Amnon |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11943 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Simplicial Cylinder DG Ring Yekutieli, Amnon Rings and Algebras Category Theory K-Theory and Homology 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. |
| title | The Simplicial Cylinder DG Ring |
| topic | Rings and Algebras Category Theory K-Theory and Homology |
| url | https://arxiv.org/abs/2602.11943 |