Strategy for proving Joyal's conjecture

In the second talk on Joyal's Conjecture, I'll explain the proof strategy. It is based on the construction, due to Karol Szumilo, of the quasicategory of fractions (or frames) that turns a fibration category into a quasicategory with finite limits. By further analyzing this construction we will be able to find conditions on a fibration category that guarantee that its quasicategory of fractions is locally cartesian closed. Furthermore, these conditions will be trivially satisfied be the category appearing in the statement of Joyal's Conjecture.