Project 1: Basic aspects of quasi-categories
Instructor: Andre Joyal

The fundamental category of a simplicial set. Categorical equivalences. Adjoint
maps. Quasi-categories. Weak categorical equivalences. Inner fibrations, isofibrations. Join and slice. Left and right fibrations. The model structure for quasi-categories. Relation with the classical model structure on simplicial sets. Equivalence with complete Segal spaces, Segal categories and simplicial categories. The fibered model structures. Homotopy factorisation systems. The covariant and contravariant model structures over a base. Initial and terminal objects. Initial and final maps. Fully faithful maps, dominant maps. Minimal quasi-categories. Minimal fibrations. Morita equivalences. Localisation.

Reading list:
1) J.E. Bergner, "A survey of $(\infty,1)$-categories. In preparation.
2) A. Joyal, "Quasi-categories and Kan complexes" JPAA vol 175 (2002), 207-222.
3) A. Joyal, "The theory of quasi-categories I". In preparation.
4) A. Joyal and M. Tierney, "Quasi-categories vs Segal spaces". To appeara
5 ) A. Joyal, "Quasi-categories vs simplicial categories". In preparation.

Project 2: Extension of category theory to quasi-categories

Adjoint maps. Diagrams. Limits and colimits. Large quasi-categories. The quasi-category K of Kan complexes. Complete and cocomplete quasi-categories. Grothendieck fibrations. Proper and smooth maps. Kan extensions. Cylinders, distributors and spans. Duality. Yoneda Lemma. The universal left fibration over K. Trace and cotrace. Factorisation systems in quasi-categories. Quasi-algebra. Locally presentable quasi-categories. Quasi-varieties. Internal categories. Descent. Exact quasi-categories. Quasi-topos. Higher quasi-categories.

1) A. Joyal, "Quasi-categories and Kan complexes" JPAA vol 175 (2002), 207-222.
2) A. Joyal, "The theory of quasi-categories II". In preparation.
3) A. Joyal "The theory of quasi-categories in perspective". To appear
4) J. Lurie "Higher topos theory" In preparation.