In mathematics, more specifically category theory, a weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex) is a generalization of the notion of a category. The study of such generalizations is known as higher category theory.
Generalization of a category were introduced by Boardman & Vogt (1973). André Joyal has much advanced the study of Generalization of a category showing that most of the usual basic category theory and some of the advanced notions and theorems have their analogues for Generalization of a category. An elaborate treatise of the theory of Generalization of a category has been expounded by Jacob Lurie (2009).
Generalization of a category are certain simplicial sets. Like ordinary categories, they contain objects (the 0-simplices of the simplicial set) and morphisms between these objects (1-simplices). But unlike categories, the composition of two morphisms need not be uniquely defined. All the morphisms that can serve as composition of two given morphisms are related to each other by higher order invertible morphisms (2-simplices thought of as "homotopies"). These higher order morphisms can also be composed, but again the composition is well-defined only up to still higher order invertible morphisms, etc.
The idea of higher category theory (at least, higher category theory when higher morphisms are invertible) is that, as opposed to the standard notion of a category, there should be a mapping space (rather than a mapping set) between two objects. This suggests that a higher category should simply be a topologically enriched category. The model of quasi-categories is, however, better suited to applications than that of topologically enriched categories, though it has been proved by Lurie that the two have natural model structures that are Quillen equivalent.