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 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).

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 $\Lambda ^{k}[n]\to C$ where $0<k<n$ , has a filler, that is, an extension to a map $\Delta [n]\to C$ . (See Kan fibration#Definitions for a definition of the simplicial sets $\Delta [n]$ and $\Lambda ^{k}[n]$ .)

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 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 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 $\tau _{1}$ 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. $\tau _{1}(C)=hC$ .

Examples

The nerve of a category is a quasi-category with the extra property that the filling of any inner horn is unique. Conversely a quasi-category such that any inner horn has a unique filling is isomorphic to the nerve of some category. The homotopy category of the nerve of C is isomorphic to C.
Given a topological space X, one can define its singular set S(X), also known as the fundamental ∞-groupoid of X. S(X) is a quasi-category in which every morphism is invertible. The homotopy category of S(X) is the fundamental groupoid of X.
More general than the previous example, every Kan complex is an example of a quasi-category. In a Kan complex all maps from all horns—not just inner ones—can be filled, which again has the consequence that all morphisms in a Kan complex are invertible. Kan complexes are thus analogues to groupoids - the nerve of a category is a Kan complex iff the category is a groupoid.

Variants

An (∞, 1)-category is a not-necessarily-quasi-category ∞-category in which all n-morphisms for n > 1 are equivalences. There are several models of (∞, 1)-categories, including Segal category, simplicially enriched category, topological category, complete Segal space. A quasi-category is also an (∞, 1)-category.
Model structure There is a model structure on sSet-categories that presents the (∞,1)-category (∞,1)Cat.
Homotopy Kan extension The notion of homotopy Kan extension and hence in particular that of homotopy limit and homotopy colimit has a direct formulation in terms of Kan-complex-enriched categories. See homotopy Kan extension for more.
Presentation of (∞,1)-topos theory All of (∞,1)-topos theory can be modeled in terms of sSet-categories. (ToënVezzosi). There is a notion of sSet-site C that models the notion of (∞,1)-site and a model structure on sSet-enriched presheaves on sSet-sites that is a presentation for the ∞-stack (∞,1)-toposes on C.

References

Boardman, J. M.; Vogt, R. M. (1973), Homotopy Invariant Algebraic Structures on Topological Spaces, Lecture Notes in Mathematics, vol. 347, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0068547, ISBN 978-3-540-06479-4, MR 0420609
Groth, Moritz, A short course on infinity-categories (PDF)
Joyal, André (2002), "Quasi-categories and Kan complexes", Journal of Pure and Applied Algebra, 175 (1): 207–222, doi:10.1016/S0022-4049(02)00135-4, MR 1935979
Joyal, André; Tierney, Myles (2007), "Quasi-categories vs Segal spaces", Categories in algebra, geometry and mathematical physics, Contemp. Math., vol. 431, Providence, R.I.: Amer. Math. Soc., pp. 277–326, arXiv:math.AT/0607820, MR 2342834
Joyal, A. (2008), The theory of quasi-categories and its applications, lectures at CRM Barcelona (PDF), archived from the original (PDF) on July 6, 2011
Joyal, A., Notes on quasicategories (PDF)
Lurie, Jacob (2009), Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, arXiv:math.CT/0608040, ISBN 978-0-691-14049-0, MR 2522659
Joyal's Catlab entry: The theory of quasi-categories
quasi-category at the nLab
infinity-category at the nLab
fundamental+category at the nLab
Bergner, Julia E (2011). "Workshop on the homotopy theory of homotopy theories". arXiv:1108.2001 [math.AT].
(∞, 1)-category at the nLab
Hinich, Vladimir (2017-09-19). "Lectures on infinity categories". arXiv:1709.06271 [math.CT].
Toën, Bertrand; Vezzosi, Gabriele (2005), "Homotopical Algebraic Geometry I: Topos theory", Advances in Mathematics, 193 (2): 257–372, arXiv:math.AG/0207028, doi:10.1016/j.aim.2004.05.004