Joyal's cylinder conjecture

Abstract

For each pair of simplicial sets A and B, the category 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∈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. Show more