Straightening a la Joyal

Alexander Campbell – 29 July 2020

The development of oo-category theory in Lurie's book 'Higher Topos Theory' is founded on a series of rectification theorems, the first of which is the (unmarked) Straightening Theorem. This theorem states that, for each simplicial set A, the straightening--unstraightening adjunction is a Quillen equivalence between the category of simplicial presheaves over the homotopy coherent realisation of A (equipped with the projective Kan model structure) and the category of simplicial sets over A (equipped with the contravariant model structure, whose fibrant objects are the right fibrations over A). Lurie's proof of this theorem is notoriously difficult; alternative proofs -- substantially different from Lurie's proof and from each other -- have since been given by Stevenson and by Heuts and Moerdijk.

In this talk, I will present a new, simple proof of the Straightening Theorem. This proof is based on an idea which may be found in Section 51 of Joyal's 'Notes on quasi-categories': we factorise the straightening--unstraightening adjunction as the composite of three adjunctions (in fact, two adjunctions and one equivalence), each of which we show to be a Quillen equivalence. One of these adjunctions (the equivalence) is easily seen to be a Quillen equivalence (in fact, an equivalence of model categories). To prove that the remaining two adjunctions are Quillen equivalences, I will use my recent proof of Joyal's Cylinder Conjecture.