Joyal's cylinder conjecture

Abstract

For each pair of simplicial sets $A$ and $B$, the category $\mathbf{Cyl}(A,B)$ of cylinders (also called correspondences) from $A$ to $B$ admits a model structure induced from Joyal's model structure for quasi-categories. In this paper, we prove Joyal's conjecture that a cylinder $X \in \mathbf{Cyl}(A,B)$ is fibrant if and only if the canonical morphism $X \longrightarrow A \star B$ is an inner fibration, and that a morphism between fibrant cylinders in $\mathbf{Cyl}(A,B)$ is a fibration if and only if it is an inner fibration. We use this result to give a new proof of a characterisation of covariant equivalences due to Lurie, which avoids the use of the straightening theorem. In an appendix, we introduce a new family of model structures on the slice categories $\mathbf{sSet}/B$, whose fibrant objects are the inner fibrations with codomain $B$, which we use to prove some new results about inner anodyne extensions and inner fibrations.