Quasi-category

In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. The study of such generalizations is known as higher category theory.
Quasi-categories were introduced by . 
André Joyal has much advanced the study of quasi-categories showing that most of the usual basic category theory and some of the advanced notions and theorems have their analogues for quasi-categories. An elaborate treatise of the theory of quasi-categories has been expounded by .
Quasi-categories 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.
Definition.
By definition, a quasi-category "C" is a simplicial set satisfying the inner Kan conditions (also called weak Kan condition): every inner horn in "C", namely a map of simplicial sets formula_1 where formula_2, has a filler, that is, an extension to a map formula_3. (See Kan fibration#Definition for a definition of the simplicial sets formula_4 and formula_5.)
The idea is that 2-simplices formula_6 are supposed to represent commutative triangles (at least up to homotopy). A map formula_7 represents a composable pair. Thus, in a quasi-category, one cannot define a composition law on morphisms, since one can choose many ways to compose maps.
One consequence of the definition is that formula_8 is a trivial Kan fibration. In other words, while the composition law is not uniquely defined, it is unique up to a contractible choice.
The homotopy category.
Given a quasi-category "C," one can associate to it an ordinary category "hC," called the homotopy category of "C". The homotopy category has as objects the vertices of "C." The morphisms are given by homotopy classes of edges between vertices. Composition is given using the horn filler condition for "n" = 2.
For a general simplicial set there is a functor formula_9 from sSet to Cat, known as the "fundamental category functor", and for a quasi-category "C" the fundamental category is the same as the homotopy category, i.e. formula_10.