Quasi-category

In mathematics, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a higher categorical generalization of a notion of a category introduced by Boardman & Vogt (1973). 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 Jacob Lurie (2009).

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.

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Definition

By the 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 \Lambda^k[n]\to C where 0<k<n, has a filler, that is, an extension to a map \Delta[n]\to C.

The idea is that 2-simplices \Delta[2] \to C are supposed to represent commutative triangles (at least up to homotopy). A map \Lambda^1[2] \to C 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 (non-obvious) consequence of the definition is that C^{\Delta[2]} \to C^{\Lambda^1[2]} 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. The homotopy category has objects as 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.

Examples

References

External links