nLab
universal fibration of (infinity,1)-categories

Idea

The universal fibration of (∞,1)-categories is the generalized universal bundle of $(\mathrm{\infty }\mathrm{,1})$-categories in that it is Cartesian fibration

$$p:Z\to (\mathrm{\infty }\mathrm{,1}){\mathrm{Cat}}^{\mathrm{op}}$$p : Z \to (\infty,1)Cat^{op}

over the opposite category of the (∞,1)-category of (∞,1)-categories such that

• its fiber $p^{-1}(C)$ over $C\in (\mathrm{\infty }\mathrm{,1})\mathrm{Cat}$ is just the $(\mathrm{\infty }\mathrm{,1})$-category $C$ itself;

• every Cartesian fibration $p:C\to D$ arises as the pullback of the universal fibration along an (∞,1)-functor $S_p:D\to (\mathrm{\infty }\mathrm{,1}){\mathrm{Cat}}^{\mathrm{op}}$.

Recall from the discussion at generalized universal bundle and at stuff, structure, property that for n-categories at least for low $n$ the corresponding universal object was the $n$-category $n{\mathrm{Cat}}_{*}$ of pointed $n$-categories. $Z$ should at least morally be $(\mathrm{\infty }\mathrm{,1}){\mathrm{Cat}}_{*}$.

Definition

…see section 3.3.2 of HTT…

Restriction to $\mathrm{\infty }$-Groupoids

The universal fibration of $(\mathrm{\infty }\mathrm{,1})$-categories restricts to a Cartesian fibration $Z{\mid }_{\mathrm{\infty }\mathrm{Grpd}}\to \mathrm{\infty }{\mathrm{Grpd}}^{\mathrm{op}}$ over ∞-Grpd by pullback along the inclusion morphism $\mathrm{\infty }\mathrm{Grpd}\hookrightarrow (\mathrm{\infty }\mathrm{,1})\mathrm{Cat}$

$$\begin{array}{ccc}Z{\mid }_{\mathrm{\infty }\mathrm{Grpd}}& \to & Z\\ \downarrow & & \downarrow \\ \mathrm{\infty }{\mathrm{Grpd}}^{\mathrm{op}}& \hookrightarrow & (\mathrm{\infty }\mathrm{,1}){\mathrm{Cat}}^{\mathrm{op}}\end{array}.$$\array{ Z|_{\infty Grpd} &\to& Z \\ \downarrow && \downarrow \\ \infty Grpd^{op} &\hookrightarrow& (\infty,1)Cat^{op} } \,.

The ∞-functor $Z{\mid }_{\mathrm{\infty }\mathrm{Grpd}}\to \mathrm{\infty }{\mathrm{Grpd}}^{\mathrm{op}}$ is even a right fibration and it is the universal right fibration.

Proposition

The following are equivalent:

• An ∞-functor $p:C\to D$ is a right Kan fibration.

• Every functor $S_p:D\to (\mathrm{\infty }\mathrm{,1})\mathrm{Cat}$ that classifies $p$ as a Cartesian fibration factors through ∞-Grpd.

• There is a functor $G_p:D\to \mathrm{\infty }\mathrm{Grpd}$ that classifies $p$ as a right Kan fibration.

Proof

This is proposition 3.3.2.5 in HTT.

Models

For concretely constructing the relation between Cartesian fibrations $p:E\to C$ of (∞,1)-categories and (∞,1)-functors $F_p:C\to (\mathrm{\infty }\mathrm{,1})\mathrm{Cat}$ one may use a Quillen equivalence between suitable model categories of marked simplicial sets.

For $C$ an (∞,1)-category regarded as a quasi-category (i.e. as a simplicial set with certain properties), the two model categories in question are

• the projective global model structure on simplicial presheaves on $[C,\mathrm{SSet}]$ – this models the (∞,1)-category of (∞,1)-functors $(\mathrm{\infty }\mathrm{,1})\mathrm{Func}(C,(\mathrm{\infty }\mathrm{,1})\mathrm{Cat})$.

• the covariant model structure on the over category $\mathrm{SSet}/C$ – this models the $(\mathrm{\infty }\mathrm{,1})$-category of Cartesian fibrations over $C$.

The Quillen equivalence between these is established by the relative nerve? construction

$$N_{-}(C):[C,\mathrm{SSet}]\to \mathrm{SSet}/C.$$N_{-}(C) : [C,SSet] \to SSet/C \,.

By the adjoint functor theorem this functor has a left adjoint

$$F_{-}(C):\mathrm{SSet}/C\to [C,\mathrm{SSet}].$$F_{-}(C) : SSet/C \to [C,SSet] \,.

For $p:E\to C$ a left Kan fibration the functor $F_p(C):C\to \mathrm{SSet}$ sends $c\in \mathrm{Obj}(C)$ to the fiber $p^{-1}(c):=E{\times }_C\{c\}$

$$F_p(C):c\mapsto p^{-1}(c).$$F_p(C) : c \mapsto p^{-1}(c) \,.

(See remark 3.2.5.5 of HTT).

References

The universal fibration as such is discussed in section 3.3.2 of

• Jacob Lurie, Higher Topos Theory

The concrete description in terms of model theory on marked simplicial sets is in section 3.2. A simpler version of this is in section 2.2.1