TY - JOUR

T1 - Joyal's cylinder conjecture

AU - Campbell, Alexander

N1 - Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/10/8

Y1 - 2021/10/8

N2 - 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.

AB - 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.

KW - Correspondence

KW - Covariant equivalence

KW - Cylinder

KW - Inner fibration

KW - Model category

KW - Quasi-category

UR - http://www.scopus.com/inward/record.url?scp=85110616042&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2021.107895

DO - 10.1016/j.aim.2021.107895

M3 - Article

SN - 0001-8708

VL - 389

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 107895

ER -