nLab
universal fibration of (infinity,1)-categories

Contents

Idea

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

p:Z(,1)Cat 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)p^{-1}(C) over C(,1)CatC \in (\infty,1)Cat is just the (,1)(\infty,1)-category CC itself;

  • every Cartesian fibration p:CDp : C \to D arises as the pullback of the universal fibration along an (∞,1)-functor S p:D(,1)Cat opS_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 nn the corresponding universal object was the nn-category nCat *n Cat_* of pointed nn-categories. ZZ should at least morally be (,1)Cat *(\infty,1)Cat_*.

Definition

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

…see section 3.3.2 of HTT

For \infty-Groupoids

Definition

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

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

The ∞-functor Z| GrpdGrpd opZ|_{\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:

Proof

This is proposition 3.3.2.5 in HTT.

Models

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

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

By the adjoint functor theorem this functor has a left adjoint

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

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

F p(C):cp 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

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)