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Wednesday Online Seminar: Joyal's cylinder conjecture April 08, 2020 (02:00 PM PDT - 03:00 PM PDT)
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Location: MSRI: Online/Virtual
Speaker(s) Alexander Campbell (MSRI - Mathematical Sciences Research Institute)
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Joyal's Cylinder Conjecture


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.



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Joyal's Cylinder Conjecture

H.264 Video 24958_28308_8277_Joyal's_Cylinder_Conjecture.mp4