A modular proof of the straightening theorem

Abstract: The development of ∞-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 realization 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 §51 of Joyal's 'Notes on quasi-categories': we factorize the striaghtening-unstraightening adjunction as the composite of three adjunctions (in fact, two adjunctions and an 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.


In person attendance is available in HN 1.33 for up to 52 people.

All attendees will be asked to check in using the CBR Covid-safe Check-In app or sign in on arrival.

Zoom attenence is also avalible. 

To join this seminar via Zoom please click here.

If you would like to join the seminar online and are not currently affiliated with ANU, please contact Martin Helmer at martin.helmer@anu.edu.au.