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 JacobLurie ( 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.

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 where , has a filler, that is, an extension to a map . (See Kan fibration#Definition for a definition of the simplicial sets and .)

The idea is that 2-simplices are supposed to represent commutative triangles (at least up to homotopy). A map 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 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 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. .

- 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.

In mathematics, especially in category theory and homotopy theory, a **groupoid** generalises the notion of group in several equivalent ways. A groupoid can be seen as a:

**Algebraic topology** is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

In mathematics, a **simplicial set** is an object made up of "simplices" in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and J. A. Zilber.

In mathematics, specifically in homotopy theory in the context of a model category *M*, a **fibrant object***A* of *M* is an object that has a fibration to the terminal object of the category.

This is a glossary of properties and concepts in category theory in mathematics.

In mathematics, particularly in homotopy theory, a **model category** is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category of topological spaces or of chain complexes, via the acyclic model theorem. The concept was introduced by Daniel G. Quillen (1967).

In mathematics, particularly in algebraic topology, the ** n-skeleton** of a topological space

In category theory, a discipline within mathematics, the **nerve***N*(*C*) of a small category *C* is a simplicial set constructed from the objects and morphisms of *C*. The geometric realization of this simplicial set is a topological space, called the **classifying space of the category***C*. These closely related objects can provide information about some familiar and useful categories using algebraic topology, most often homotopy theory.

In mathematics, **higher category theory** is the part of category theory at a *higher order*, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology, where one studies algebraic invariants of spaces, such as their fundamental weak ∞-groupoid.

In category theory, a **weak n-category** is a generalization of the notion of strict

In mathematics, **Kan complexes** and **Kan fibrations** are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category for simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category. The name is in honor of Daniel Kan.

In mathematics, **directed algebraic topology** is a refinement of algebraic topology for **directed spaces**, topological spaces and their combinatorial counterparts equipped with some notion of direction. Some common examples of directed spaces are spacetimes and simplicial sets. The basic goal is to find algebraic invariants that classify directed spaces up to directed analogues of homotopy equivalence. For example, homotopy groups and fundamental n-groupoids of spaces generalize to homotopy monoids and fundamental n-categories of directed spaces. Directed algebraic topology, like algebraic topology, is motivated by the need to describe qualitative properties of complex systems in terms of algebraic properties of state spaces, which are often directed by time. Thus directed algebraic topology finds applications in Concurrency, Network traffic control, General Relativity, Noncommutative Geometry, Rewriting Theory, and Biological systems.

In mathematical logic and computer science, **homotopy type theory** refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies.

In mathematics, a **weak equivalence** is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category.

In mathematics, more specifically in homotopy theory, a **simplicial presheaf** is a presheaf on a site taking values in simplicial sets. Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on a site. The notion was introduced by A. Joyal in the 1970s. Similarly, a **simplicial sheaf** on a site is a simplicial object in the category of sheaves on the site.

In category theory, a branch of mathematics, a (left) **Bousfield localization** of a model category replaces the model structure with another model structure with the same cofibrations but with more weak equivalences.

**Derived algebraic geometry** is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras, simplicial commutative rings or -ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements. Derived algebraic geometry can be thought of as an extension of this idea, and provides natural settings for intersection theory of singular algebraic varieties and cotangent complexes in deformation theory, among the other applications.

* Pursuing Stacks* is an influential 1983 mathematical manuscript by Alexander Grothendieck. The word "stack" refers to a possible generalization of scheme, a central object of study in algebraic geometry.

In category theory, a branch of mathematics, an **∞-groupoid** is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category of simplicial sets. It is an ∞-category generalization of a groupoid, a category in which every morphism is an isomorphism.

In mathematics, **homotopy theory** is a systematic study of situations in which maps come with homotopies between them. It originated as a topic in algebraic topology but nowadays it is studied as an independent discipline. Besides algebraic topology, the theory has also been in used in other areas of mathematics such as algebraic geometry and category theory.

- Boardman, J. M.; Vogt, R. M. (1973),
*Homotopy invariant algebraic structures on topological spaces*, Lecture Notes in Mathematics,**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.,**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,**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 in
*nLab* - infinity-category in
*nLab* - fundamental+category in
*nLab* - Bergner, Julia E (2011). "Workshop on the homotopy theory of homotopy theories". arXiv: 1108.2001 [math.AT].
- (∞, 1)-category in
*nLab* - Hinich, Vladimir (2017-09-19). "Lectures on infinity categories". arXiv: 1709.06271 [math.CT].

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