## Joyal's cylinder conjecture

### Alexander Campbell – 9 October 2019, 16 October 2019, 30 October 2019, 6 November 2019

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 series of talks, I will prove Joyal's conjecture that a cylinder X in Cyl(A,B) is fibrant iff the canonical morphism from X to the join of A and B is an inner fibration, and that a morphism between fibrant cylinders in Cyl(A,B) is a fibration iff it is an inner fibration. I will 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.

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