nLab
limit in a quasi-category

Idea

One of the universal constructions in (∞,1)-category theory:

the notion of limit generalizes from category theory to quasi-categories straightforwardly using the generalization of over category and terminal object to over quasi-category and terminal object in a quasi-category.

Definition

For $K$ and $C$ two quasi-categories and $F:K\to C$ a morphism of quasi-categories, the limit over $F$ is, if it exists, the quasi-categorical terminal object in the over quasi-category $C_{/F}$:

$$\lim F:=\mathrm{TerminalObj}(C_{/F}).$$lim F := TerminalObj(C_{/F}) \,.

A quasi-categorical colimit is accordingly an limit in the opposite quasi-category.

Examples

Homotopy limits in Kan-enriched categories

Let $K$ and $C$ be categories enriched in Kan complexes and $F:K\to C$ a morphism of Kan-enriched categories (i.e. an SimpSet-functor). Then the homotopy limit of $F$ (computed for instance as described at weighted limit) coincides with the quasi-categorical limit of $F$ under the embedding of simplicial categories into quasi-categories.

This is theorem 4.2.4.1, p. 214 in Lurie’s book (see below).

Properties

Limits and colimits with values in $\mathrm{\infty }\mathrm{Grpd}$

Limits and colimits over a (∞,1)-functor with values in the (∞,1)-category ∞-Grpd of ∞-groupoids may be reformulation in terms of the universal fibration of (infinity,1)-categories.

Let ∞-Grpd be the (∞,1)-category of ∞-groupoids. Let the (∞,1)-functor $Z{\mid }_{\mathrm{Grpd}}\to \mathrm{\infty }{\mathrm{Grpd}}^{\mathrm{op}}$ be the universal ∞-groupoid fibration whose fiber over the object denoting some $\mathrm{\infty }$-groupoid is that very $\mathrm{\infty }$-groupoid.

Then let $X$ be any ∞-groupoid and

$$F:X\to \mathrm{\infty }\mathrm{Grpd}$$F : X \to \infty Grpd

an (∞,1)-functor. Recall that the coCartesian fibration $E_F\to X$ classified by $F$ is the pullback of the universal fibration of (∞,1)-categories $Z$ along F:

$$\begin{array}{ccc}{E}_F& \to & Z{\mid }_{\mathrm{Grpd}}\\ \downarrow & & \downarrow \\ X& \stackrel{F}{\to }& \mathrm{\infty }\mathrm{Grpd}\end{array}$$\array{ E_F &\to& Z|_{Grpd} \\ \downarrow && \downarrow \\ X &\stackrel{F}{\to}& \infty Grpd }

Proposition

Let the assumptions be as above. Then:

• The colimit of $F$ is equivalent to $E_F$:

  $$E_F\simeq \mathrm{colim}F$$E_F \simeq colim F

• The limit of $F$ is equivalent to the (∞,1)-groupoid of sections of $E_F\to X$

  $${\Gamma }_X(E_F)\simeq \lim F.$$\Gamma_X(E_F) \simeq lim F \,.

Proof

The statement for the colimit is corollary 3.3.4.6 in HTT. The statement for the limit is corollary 3.3.3.4.

If instead of ∞-Grpd the target is the (∞,1)-category of (∞,1)-categories then the latter statement is true with the $(\mathrm{\infty }\mathrm{,1})$-category of all sections replaced by (∞,1)-category of cartesian sections.

References

The definition of limit in quasi-categories is due to

• André Joyal, Quasi-categories and Kan complexes Journal of Pure and Applied Algebra 175 (2002), 207-222.

A brief survey is on page 159 of

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

A detailed account is in definition 1.2.13.4, p. 48 in

• Jacob Lurie, Higher Topos Theory