Course Identification

simplicial sets, filling conditions: Kan complexes, Quasi-categories, (various flavors of) fibrations.  

The language of higher category theory: mapping spaces, functors, , the infinity categories of spaces and of categories. 

Limits and colimits, adjunctions, Kan extensions, Yoneda Lemma. 

Presentable infinity-categories of 

Examples: infinity categories and topology, derived infinity categories, infinity categories of sheaves.   

The participants will learn how to speak and apply the basic language of higher categories, and how to relate it to constructions and questions from other areas of mathematics. 

Higher Topos Theory - Jacob Lurie 

Introduction to Infinity Categories - Markus Land.