Infinity Categories

Abstract

This text attempts to review and understand the fundamentals of ∞-categories via weak Kan complexes. We develop to some extent the construction of ∞ homs. We find a way to deal with 1-morphisms and equivalences and form the ∞-category of all ∞-categories. Finally we present J-model structure on simplicial sets, which represents ∞-categories as fibrant objects. At the end we briefly touch upon further concepts which are possible to be constructed in the theory, in correspondence with ordinary category theory.