# nLab universal fibration of (infinity,1)-categories

### Context

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

(∞,1)-category theory

# Contents

## Idea

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

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

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

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

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

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 $\left(\infty ,1\right){\mathrm{Cat}}_{*}$.

## Definition

### For $\left(\infty ,1\right)$-categories

…see section 3.3.2 of HTT

### For $\infty$-Groupoids

###### Definition

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

$\begin{array}{ccc}Z{\mid }_{\infty \mathrm{Grpd}}& \to & Z\\ ↓& & ↓\\ \infty {\mathrm{Grpd}}^{\mathrm{op}}& ↪& \left(\infty ,1\right){\mathrm{Cat}}^{\mathrm{op}}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ Z|_{\infty Grpd} &\to& Z \\ \downarrow && \downarrow \\ \infty Grpd^{op} &\hookrightarrow& (\infty,1)Cat^{op} } \,.

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

It is closely related to the object classifier in ∞Grpd.

###### Proposition

The following are equivalent:

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

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

• There is a functor ${G}_{p}:D\to \infty \mathrm{Grpd}$ 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 ${F}_{p}:C\to \left(\infty ,1\right)\mathrm{Cat}$ 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 $\left[C,\mathrm{SSet}\right]$ – this models the (∞,1)-category of (∞,1)-functors $\left(\infty ,1\right)\mathrm{Func}\left(C,\left(\infty ,1\right)\mathrm{Cat}\right)$.

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

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

${N}_{-}\left(C\right):\left[C,\mathrm{SSet}\right]\to \mathrm{SSet}/C\phantom{\rule{thinmathspace}{0ex}}.$N_{-}(C) : [C,SSet] \to SSet/C \,.

${F}_{-}\left(C\right):\mathrm{SSet}/C\to \left[C,\mathrm{SSet}\right]\phantom{\rule{thinmathspace}{0ex}}.$F_{-}(C) : SSet/C \to [C,SSet] \,.

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

${F}_{p}\left(C\right):c↦{p}^{-1}\left(c\right)\phantom{\rule{thinmathspace}{0ex}}.$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

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

Revised on January 7, 2012 13:35:43 by Urs Schreiber (82.113.106.121)