Higher Category Theory - Quasicategories

Quasicategories

Weak Kan complexes, or quasi-categories, are simplicial sets satisfying a weak version of the Kan condition. Joyal showed that they are a good foundation for higher category theory. Recently the theory has been systematized further by Jacob Lurie who simply calls them infinity categories, though the latter term is also a generic term for all models of (infinity,k) categories for any k.