Seminar

In this talk, I will discuss my recent preprint (arXiv:1911.02631) in which I proved Joyal's cylinder conjecture.

In quasi-category theory, "cylinders" (a.k.a. "correspondences") are a model for the collages of ∞-categorical profunctors. For each pair of simplicial sets A and B, the category Cyl(A,B) of cylinders from A to B admits a model structure induced from Joyal's model structure for quasi-categories. Joyal conjectured that a cylinder X in Cyl(A,B) is fibrant in this model structure iff the canonical morphism from X to the join of A and B is an inner fibration, and moreover that a morphism between fibrant cylinders in Cyl(A,B) is a fibration in this model structure iff it is an inner fibration.

This talk will consist of three parts: in the first, I will review the necessary background to an understanding of the statement of Joyal's conjecture; in the second, I will give an overview of my proof of Joyal's conjecture; in the third, I will use this result to give a new proof of Lurie's characterisation of covariant equivalences, which proof avoids the use of the straightening theorem.

Notes 189 KB application/pdf

Joyal's Cylinder Conjecture