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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 Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
. The study of such generalizations is known as
higher category 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 cate ...
. Quasi-categories were introduced by .
André Joyal André Joyal (; born 1943) is a professor of mathematics at the Université du Québec à Montréal who works on category theory. He was a member of the School of Mathematics at the Institute for Advanced Study in 2013, where he was invited to jo ...
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 set In mathematics, a simplicial set is an object composed 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 ...
s. 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 In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed model categories C and D is a special kind of adjunction between categories that induces an adjunction between the homotopy categories Ho(C) and Ho(D) via the ...
.


Definition

By definition, a quasi-category ''C'' is a
simplicial set In mathematics, a simplicial set is an object composed 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 ...
satisfying the inner Kan conditions (also called weak Kan condition): every inner horn in ''C'', namely a map of simplicial sets \Lambda^k to C where 0, has a filler, that is, an extension to a map \Delta to C. (See Kan fibration#Definitions for a definition of the simplicial sets \Delta /math> and \Lambda^k /math>.) The idea is that 2-simplices \Delta \to C are supposed to represent commutative triangles (at least up to homotopy). A map \Lambda^1 \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^ \to C^ 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 In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed be ...
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 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 ...
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 In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In ca ...
''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 In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a to ...
of ''X''. *More general than the previous example, every
Kan complex In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed ...
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 In mathematics, a Segal category is a model of an infinity category introduced by , based on work of Graeme Segal Graeme Bryce Segal FRS (born 21 December 1941) is an Australian mathematician, and professor at the University of Oxford. Biog ...
,
simplicially enriched category In mathematics, a simplicially enriched category, is a category enriched over the category of simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional genera ...
,
topological category In category theory, a discipline in mathematics, the notion of topological category has a number of different, inequivalent definitions. In one approach, a topological category is a category that is enriched over the category of compactly genera ...
, 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.


See also

*
Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called ' weak equivalences', ' fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstr ...
*
Stable infinity category A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...
*
∞-groupoid 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 (with the standard model structure). I ...
*
Higher category 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 cate ...
*
Globular set In category theory, a branch of mathematics, a globular set is a higher-dimensional generalization of a directed graph. Precisely, it is a sequence of sets X_0, X_1, X_2, \dots equipped with pairs of functions s_n, t_n: X_n \to X_ such that * s_n \c ...


References

* * * * * * * * Joyal's Catlab entry
The theory of quasi-categories
* * * * * * {{Category theory Homotopy theory Higher category theory