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

### Context

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

(∞,1)-category theory

# Contents

## Idea

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

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

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

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

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

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 $n Cat_*$ of pointed $n$-categories. $Z$ should at least morally be $(\infty,1)Cat_*$.

## Definition

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

…see section 3.3.2 of HTT

### For $\infty$-Groupoids

###### Definition

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

$\array{ Z|_{\infty Grpd} &\to& Z \\ \downarrow && \downarrow \\ \infty Grpd^{op} &\hookrightarrow& (\infty,1)Cat^{op} } \,.$
###### Remark

The ∞-functor $Z|_{\infty Grpd} \to \infty Grpd^{op}$ is even a right fibration and it is the universal right fibration. In fact it is (when restricted to small objects) the object classifier in the (∞,1)-topos ∞Grpd, see at object classifier – In ∞Grpd.

###### Proposition

The following are equivalent:

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

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

• There is a functor $G_p : D \to \infty Grpd$ 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 $F_p : C \to (\infty,1)Cat$ 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 Quillen equivalence between these is established by the relative nerve? construction

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

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

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

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

(See remark 3.2.5.5 of HTT).

## References

The universal fibration as such is discussed in section 3.3.2 of

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 November 3, 2014 13:57:53 by Urs Schreiber (185.26.182.33)