nLab universal fibration of (infinity,1)-categories




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 \colon 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_*.


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

One description of the universal cartesian fibration is given in section 3.3.2 of HTT as the contravariant (∞,1)-Grothendieck construction applied to the identity functor ((,1)Cat op) op(,1)Cat((\infty,1)Cat^{op})^{op} \to (\infty,1)Cat.

We can also give a more direct description:


Z opZ^{op} is equivalent to the full subcategory of (,1)Cat [1](\infty,1)Cat^{[1]} spanned by the morphisms of the form [C x/C][C_{x/} \to C] for small (∞,1)-categories CC and objects xCx \in C.

The universal fibration Z(,1)Cat opZ \to (\infty,1)Cat^{op} is opposite to the target evaluation.

Dually, the universal cocartesian fibration is Z(,1)CatZ' \to (\infty,1)Cat where ZZ' is the (∞,1)-category of arrows of the form [C /xC][C_{/x} \to C].


This is the proof idea of this mathoverflow post.

By proposition of Higher Topos Theory, there are presentable fibrations RFib(,1)CatRFib \to (\infty,1)Cat and (,1)Cat [1]tgt(,1)Cat(\infty,1)Cat^{[1]} \xrightarrow{tgt} (\infty,1)Cat classifying functors C𝒫(C)C \mapsto \mathcal{P}(C) and C(,1)Cat /CC \mapsto (\infty,1)Cat_{/C}.

By proposition of Higher Topos Theory, the yoneda embedding j:C𝒫(C)j : C \to \mathcal{P}(C) is a natural transformation, and the covariant Grothendieck construction provides a cocartesian functor ZRFibZ' \to RFib. Since it is fiberwise fully faithful and () !(-)_! preserves representable presheaves, we can identify ZZ' with the full subcategory of RFibRFib consisting of the representable presheaves.

The Grothendieck construction provides a fully faithful 𝒫(C)(,1)Cat /C\mathcal{P}(C) \to (\infty,1)Cat_{/C} whose essential image is the right fibrations. The contravariant Grothendieck construction a cartesian functor RFib(,1)Cat [1]RFib \to (\infty,1)Cat^{[1]}. Since it is fiberwise fully faithful and pullbacks preserve right fibrations, we can identify RFibRFib with the full subcategory of (,1)Cat [1]( \infty,1)Cat^{[1]} spanned by right fibrations.

By the relationship between the covariant and contravariant Grothendieck constructions, the universal cartesian fibration is classified by op:((,1)Cat op) op(,1)Catop : ((\infty,1)Cat^{op})^{op} \to (\infty,1)Cat.


A key point of this description is that for any small (∞,1)-category CC, the functor xC /xx \mapsto C_{/x} (where xyx \to y acts by composition) is a fully faithful functor C(,1)Cat /CC \to (\infty,1)Cat_{/C}. Dually, xC x/x \mapsto C_{x/} is a fully faithful functor C op(,1)Cat /CC^{op} \to (\infty,1)Cat_{/C}

The hom-spaces of the universal cocartesian fibration can be described as

Z([C /xC],[D /yD])Core(eval xy) Z'([C_{/x} \to C], [D_{/y} \to D]) \simeq Core(eval_x \downarrow y)

where eval x:D CDeval_x : D^C \to D. This should be compared with the lax slice 2-category construction. In fact, ZZ' can be constructed by taking the underlying (∞,1) category of the lax coslice (or colax, depending on convention) over the point of the (∞,2)-category of (∞,1)-categories.

For \infty-Groupoids


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} &\longrightarrow& Z \\ \big\downarrow && \big\downarrow \\ \infty Grpd^{op} &\hookrightarrow& (\infty,1)Cat^{op} } \,.

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.


The universal left fibration is the forgetful functor Grpd *Grpd\infty Grpd_* \to \infty Grpd. Its opposite is the universal right fibration.

(Lurie 2009, Prop,, Cisinski 2019, Sec. 5.2, for the further restriction to the universal Kan fibration see also Kapulkin & Lumsdaine 2021)


The following are equivalent:


This is proposition in HTT.


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 of HTT).


Textbook accounts:

The direct description of the universal fibration is discussed at

Discussion of the universal Kan fibration as a categorical semantics for a univalent type universe in homotopy type theory:

Discussion of fibrations via (∞,2)-category theory

On the categorical semantics of homotopy type theory in simplicial sets/ \infty -groupoids:

Last revised on August 26, 2022 at 14:22:08. See the history of this page for a list of all contributions to it.