nLab
quasi-category

Idea

The notion of quasi-category is a geometric definition of higher category that relaxes the Kan condition on a simplicial set.

Just as 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.

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 $0<i<n$.

• the morphism of simplicial sets

  $$\mathrm{sSet}(\Delta [2],C)\to \mathrm{sSet}(\Lambda [2{]}_1,C)$$sSet(\Delta[2],C) \to sSet(\Lambda[2]_1,C)

  (induced from the inner horn inclusion $\Lambda [2{]}_1\to \Delta [2]$) is an acyclic Kan fibration.

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.

The equivalence of these two definitions is due to Andre Joyal and recalled as HTT, corollary 2.3.2.2.

Remarks

• Compare with the definition of a Kan complex in which all horns are required to have fillers: a quasi-category is a structure slightly weaker than a Kan complex. Indeed, while we can think of a Kan complex as an ∞-groupoid (that is an $(\mathrm{\infty }\mathrm{,0})$-category), in which all morphisms are equivalences, a quasi-category is a model for an (∞,1)-category, in which only all k-morphisms for $k\ge 2$ are required to be equivalences.

• As the quasi-category condition is a weakening of the Kan complex condition, they have also been called weak Kan complexes and the corresponding condition, the weak Kan condition.

• The nerve of an ordinary category is always a quasi-category, while the nerve of a category is a Kan complex iff the category is a groupoid. In this sense quasi-categories are a “minimal common generalization” of Kan complexes and nerves of categories.

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

Higher associahedra in quasi-categories

While the geometric definition of (∞,1)-category in terms of quasi-categories eleganty 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 any any algebraic definition of the same categorical structure.

For a discussion of this see

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

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

• 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

• fibrations of quasi-categories

• opposite quasi-category

• equivalence of quasi-categories

Discussion

A previous version of this entry led to the following discussion.

Stephen Gaito: If we want to weaken this even further to provide a simplicial model of, for example, a (∞,2)-category, how would we do this?

Would we apply the lifting condition on all but three of the indicies… and if so which three? (The first, last and ????)

Mike Shulman: You may be looking for something along the lines of a weak complicial set.

References

Quasi-categories have originally been 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.

Cordier in

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

defined a homotopy coherent nerve of any simplicially enriched category, which 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 the 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 writup of lecture notes does:

• André Joyal, The theory of quasicategories and its applications lectures at Simplicial Methods in Higher Categories, (pdf)

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

• Jacob Lurie, Higher Topos Theory .

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)