# nLab quasi-category

Contents

### Context

#### $(\infty,1)$-Category theory

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

# Contents

## Idea

The notion of quasi-category is a geometric model for (∞,1)-category.

In analogy to how a Kan complex is a model in terms of simplicial sets of an ∞-groupoid – also called an (∞,0)-category – a quasi-category is a model in terms of simplicial sets of an (∞,1)-category.

###### Warning

In older literature, such as The Joy of Cats, the term “quasicategory” was sometimes used for a “very large” category whose objects are large categories or otherwise built out of proper classes, but nowadays this usage is fairly archaic. See also metacategory.

## Definition

###### Definition

A quasi-category or weak Kan complex is a simplicial set $C$ satisfying the following equivalent conditions

• all inner horns in $C$ have fillers. This means that the lifting condition given at Kan complex is imposed only for horns $\Lambda^i[n]$ with $0 \lt i \lt n$.

• the morphism of simplicial sets

$sSet(\Delta[2],C) \to sSet(\Lambda^1[2],C)$

(induced from the inner horn inclusion $\Lambda^1[2] \to \Delta[2]$) is an acyclic Kan fibration.

The equivalence of these two definitions is due to Andre Joyal and recalled as HTT, corollary 2.3.2.2. Quasi-categories are the fibrant objects in the model structure for quasi-categories.

###### Remark

The second condition says manifestly that a quasi-category is a simplicial set in which composition of any two composable edges is defined up to a contractible space of choices. This is the coherence law on composition.

###### Definition

An algebraic quasi-category is a quasi-category equipped with a choice of inner horn fillers.

While quasi-categories provide a geometric definition of higher categories, algebraic quasi-categories provide an algebraic definition of higher categories. For more details on this see model structure on algebraic fibrant objects.

## Properties

### Relation to simplicially enriched categories

The homotopy coherent nerve relates quasi-categories with another model for $(\infty,1)$-categories: simplicially enriched categories.

### Higher associahedra in quasi-categories

While the geometric definition of (∞,1)-category in terms of quasi-categories elegantly captures all the higher categorical data automatically, it may be of interest in applications to explicitly extract the associators and higher associators encoded by this structure, that would show up in any algebraic definition of the same categorical structure, such as algebraic quasi-categories.

For a discussion of this see

• Emily Riehl, Associativity data in an $(\infty,1)$-category (pdf blog)

## Examples

The two basic examples for quasi-categories are

Since the nerve of a category is a Kan complex iff the category is a groupoid, quasi-categories are a minimal common generalization of Kan complexes and nerves of categories.

By the homotopy hypothesis-theorem every Kan complex arises, up to equivalence, as the fundamental ∞-groupoid of a topological space.

Analogously, every directed topological space $X$ has naturally a fundamental (∞,1)-category given by a quasi-category whose $k$-cells are maps $\Delta^k_{Top} \to X$ that map the 1-skeleton of the topological simplex in an order-preserving way to directed paths in $X$.

The directed homotopy theory that would state that this or a similar construction exhausts all quasicategories up to equivalence, does not quite exist yet.

## Constructions in quasi-categories

The point of quasi-categories is that they are supposed to provide a fully homotopy-theoretic refinement of the ordinary notion of category. In particular, all the familiar constructions of category theory have natural analogs in the context of quasi-categories. See for instance

## References

The notion of quasi-categories were originally defined, under the name weak Kan complexes in:

The main theorem of Vogt (1973) involved a category of homotopy coherent diagrams defined on a topologically enriched category and showed it was equivalent to a quotient category of the category of (commutative) diagrams on the same category.

• J.-M. Cordier, Sur la notion de diagramme homotopiquement cohérent, Cahiers de Top. Géom. Diff., 23, (1982), 93 –112,

defined the homotopy coherent nerve of any simplicially enriched category. This generalised the nerve of an ordinary category. In

• J.-M. Cordier and Tim Porter, Vogt’s theorem on categories of homotopy coherent diagrams, Math. Proc. Cambridge Philos. Soc., 100, (1986), 65–90,

it was shown that this homotopy coherent nerve was a quasi-category if the simplicial enrichment was by Kan complexes.

A systematic study of SSet-enriched categories in this context is in

The importance of quasi-categories as a basis for category theory has been particularly emphasized in work by André Joyal

For several years Joyal has been preparing a textbook on the subject which never became publically available, but an extensive writeup of lecture notes is:

Meanwhile Jacob Lurie, building on Joyal’s work, has considerably pushed the theory further. A comprehensive discussion of the theory of $(\infty,1)$-categories in terms of the models quasi-category and simplicially enriched category is in

An overview of the material there is contained in

Textbook accounts:

The relation between quasi-categories and simplicially enriched categories was discussed in detail in

Further survey:

An in-depth study of adjunctions between quasi-categories and the monadicity theorem is given in

Last revised on May 30, 2023 at 10:21:47. See the history of this page for a list of all contributions to it.