Joyal's cylinder conjecture

For each pair of simplicial sets Aand B, the category Cyl(A, B)of cylinders (also called correspondences) from Ato Badmits a model structure induced from Joyal’s model structure for quasi-categories. In this paper, we prove Joyal’s conjecture that a cylinder X ∈ Cyl(A, B)is fibrant if and only if the canonical morphism X → A * B is an inner fibration, and that a morphism between fibrant cylinders in 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 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.