In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, âÂÂ-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 . 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 .
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 (see ).
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#Definitions 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, the left-adjoint of the nerve functor, and for a quasi-category C, we have .
An ordinary nerve of a category misses higher morphisms (e.g., a natural transformation between functors, which is a 2-morphism or a homotopy between paths). The homotopy coherent nerve of a simplicially-enriched category allows to capture such higher morphisms.
First we define as a "thickened" version of the category ( is a partially ordered set so can be viewed as a category). By definition, it has the same set of objects as does but the hom-simplicial-set from to is the nerve of where is the set of all subsets of containing and is partially ordered by inclusion. That is, in , a morphism looks like or none if . (Formally, is a cofibrant replacement of .)
Then is defined to be the simplicial set where each n-simplex is a simplicially-enriched functor from to . Moreover, if has the property that is a Kan complex for each pair of objects , then is an âÂÂ-category.
The functor from sSet to sSet-Cat is then defined as the left adjoint to . An important application is:
The theorem implies that a simplicial approach to the theory of âÂÂ-categories is equivalent (in the above weak sense) to a topological approach to that.
If X, Y are âÂÂ-categories, then the simplicial set , the internal Hom in sSet, is also an âÂÂ-category (more generally, it is an âÂÂ-category if X is only a simplicial set and Y is an âÂÂ-category.)
If are objects in an âÂÂ-category C, then is a Kan complex but is a priori not a functor. A functor that restricts to it can be constructed as follows.
Let S be a simplicial set and the sSet-enriched category generated by it. Since is a functor, gives a functor
where on the right is the 1-category of Kan complexes. Then, since is a left adjoint to , corresponds to
Taking to be an âÂÂ-category C, the above is the hom functor
which restricts to
See also: limits and colimits in an âÂÂ-category, core of an âÂÂ-category.
Given a functor between âÂÂ-categories, F is said to be an equivalence (in the sense of Joyal) if it is invertible in âÂÂ-Cat, the âÂÂ-category of (small) âÂÂ-categories.
Like in ordinary category theory, (with the presence of the axiom of choice), F is equivalence if and only if it is
Just like in ordinary category theory, one can consider a presheaf on an âÂÂ-category C. From the point of view of higher category theory, such a presheaf should not be set-valued but space-valued (for example, for a correct formulation of the Yoneda lemma). The homotopy hypothesis says that one can take an âÂÂ-groupoid, concretely a Kan complex, as a space. Given that, we take the category of "âÂÂ-presheaves" on C to be where is the âÂÂ-category of Kan complexes. A category-valued presheaf is commonly called a prestack. Thus, can be thought of consisting of âÂÂ-prestacks.
(With a choice of a functor structure on Hom), one then gets the âÂÂ-Yoneda embedding as in the ordinary category case:
There are at least two equivalent approaches to adjunctions. In Cisinski's book, an adjunction is defined just as in ordinary category theory. Namely, two functors are said to be an adjoint pair if there exists a 2-morphism such that the restriction to each pair of objects x in C, y in D,
is invertible in (recall the mapping spaces are Kan complexes).
In his book Higher Topos Theory, Lurie defines an adjunction to be a map that is both cartesian and cocartesian fibrations. Since is a cartesian fibration, by the Grothendieck construction of sort (straightening to be precise), one gets a functor
Similarly, as is also a cocartesian fibration, there is also Then they are an adjoint pair and conversely, an adjoint pair determines an adjunction.
Let be an object in an âÂÂ-category C. Then the following are equivalent:
Then is said to be final if any of the above equivalent condition holds. The final objects form a full subcategory, an âÂÂ-groupoid, that is either empty or contractible.
For example, a presheaf is representable if and only if the âÂÂ-category of elements for has a final object (as the representability amounts to saying the âÂÂ-category of elements is equivalent to a comma category over C).
More generally, a map between simplicial sets is called final if it belongs the smallest class of maps satisfying the following:
Then an object is final if and only if the map is a final map. Also, a map is called cofinal if is final.
Presheaf categories (discussed above) have some nice properties and their localizations also inherit such properties to some extent. An âÂÂ-category is called presentable if it is a localization of a presheaf category on an âÂÂ-category in the sense of Bousfield (the notion strongly depends on a choice of a universe, which is suppressed here. But one way to handle this issue is to manually keep track of cardinals. Another is to use the notion of an accessible âÂÂ-category as done by Lurie).
Cisinski notes that âÂÂAny [reasonable] algebraic structure defines a presentable âÂÂ-category," after taking a nerve. Thus, for example, "the category of groups, the category of abelian groups, the category of rings" are all (their nerves are) presentable âÂÂ-categories. Also, the nerve of a category of small sets is presentable.
The notion has an implication to theory of model categories. Roughly because of the above remark, all the typical model categories that are used in practice have nerves that are presentable; such a model category is called combinatorial. Precisely, we have: (Dugger) if C is a combinatorial model category, then the localization with respect to weak equivalences is a presentable âÂÂ-category and conversely, each presentable âÂÂ-category is of such form, up to equivalence.