nLab quasi-category

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.

See relation between quasi-categories and simplicial categories for more.

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

• Every Kan complex is, in particular, a quasi-category.

• The nerve of a category is a quasi-category.

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

• hom-object in a quasi-category

• equivalence in a quasi-category

• equivalence of quasi-categories

• join of quasi-categories

• over quasi-category

• terminal object in a quasi-category

• monomorphism in an (∞,1)-category

• limit in quasi-categories

• sub-quasi-category

• opposite quasi-category

• fibrations of quasi-categories

  • inner Kan fibration

  • Cartesian fibration

  • left Kan fibration/right Kan fibration

  • minimal Joyal fibration

Related concepts

One may try to further weaken the filler conditions in order to describe (∞,n)-categories for $n \gt 1$. One approach along these lines is the theory of weak complicial sets.

Or one may change the shape category to pass from simplicial sets to cellular sets. A quasi-category-like definition of (∞,n)-categories on these – n-quasicategories – is discussed at model structure on cellular sets.

References

Quasi-categories were originally defined in

• Michael Boardman, Rainer Vogt, Homotopy invariant algebraic structures on topological

  spaces_, Lecture Notes in Mathematics, Vol. 347. Springer-Verlag, 1973.

They occured as weak Kan complexes in

• Rainer Vogt, Homotopy limits and colimits, Math. Z., 134, (1973), 11–52.

Vogt’s main theorem 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.

Jean-Marc Cordier in

• 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

• J-M Cordier, Tim Porter Homotopy coherent category theory Trans. Amer. Math. Soc. 349 (1997), no. 1, 1-54. (pdf)

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

• André Joyal, Quasi-categories and Kan complexes, J. Pure Appl. Algebra, 175 (2002), 207-222.

For several years Joyal has been preparing a textbook on the subject. This still doesn’t quite exist, but an extensive writeup of lecture notes does:

• André Joyal, The Theory of Quasi-Categories and its Applications, lectures at Advanced Course on Simplicial Methods in Higher Categories, CRM 2008 (pdf, pdf)

and more recently, with more details

• André Joyal, Notes on quasi-categories, 2008 (pdf, pdf).

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

• Jacob Lurie, Higher Topos Theory .

An overview of the material there is contained in

• Denis-Charles Cisinski, Catégories supérieures et théorie des topos, Séminaire Bourbaki, 21.3.2015, pdf.

Comprehensive textbook account:

• Denis-Charles Cisinski, Higher Categories and Homotopical Algebra, Cambridge University Press (2019) [doi:10.1017/9781108588737, pdf]

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

• Dan Dugger, David Spivak, Rigidification of quasi-categories (arXiv:0910.0814)

• Dan Dugger, David Spivak, Mapping spaces in quasi-categories (arXiv:0911.0469)

Survey includes (see also the references at (infinity,1)-category):

Emily Riehl, Categorical Homotopy Theory, Cambridge University Press, 2014 (pdf, doi:10.1017/CBO9781107261457)

• Charles Rezk, Stuff about quasicategories, Lecture Notes for course at University of Illinois at Urbana-Champaign, 2016, version May 2017 (pdf, pdf)

• Charles Rezk, Introduction to quasicategories (2022) [pdf, pdf]

• Moritz Groth, A short course on ∞-categories (arXiv:1007.2925)

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

• Emily Riehl, Dominic Verity The 2-category theory of quasi-categories (arXiv), Homotopy coherent adjunctions and the formal theory of monads (arXiv)