Quasi-category

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

The idea is that 2-simplices ${\displaystyle \Delta [2]\to C}$ are supposed to represent commutative triangles (at least up to homotopy). A map ${\displaystyle \Lambda ^{1}[2]\to C}$ 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 ${\displaystyle C^{\Delta [2]}\to C^{\Lambda ^{1}[2]}}$ is a trivial Kan fibration. In other words, while the composition law is not uniquely defined, it is unique up to a contractible choice.

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

• 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.

• 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.

• Model category
• Stable infinity category
• ∞-groupoid
• Higher category theory
• Globular set

• 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].