Joyal's cylinder conjecture

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Abstract

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.
Original languageEnglish
Article number107895
Pages (from-to)1-40
Number of pages40
JournalAdvances in Mathematics
Volume389
DOIs
Publication statusPublished - 8 Oct 2021
Externally publishedYes

Keywords

  • Quasi-category
  • Model category
  • Cylinder
  • Correspondence
  • Inner fibration
  • Covariant equivalence

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