nLab universal fibration of (infinity,1)-categories

Idea

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

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

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

• its fiber $p^{-1}(C)$ 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

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 $((\infty,1)Cat^{op})^{op} \to (\infty,1)Cat$.

We can also give a more direct description:

Proposition

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

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

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

Proof

This is the proof idea of this mathoverflow post.

By proposition 5.5.3.3 of Higher Topos Theory, there are presentable fibrations $RFib \to (\infty,1)Cat$ and $(\infty,1)Cat^{[1]} \xrightarrow{tgt} (\infty,1)Cat$ classifying functors $C \mapsto \mathcal{P}(C)$ and $C \mapsto (\infty,1)Cat_{/C}$.

By proposition 5.3.6.2 of Higher Topos Theory, the yoneda embedding $j : C \to \mathcal{P}(C)$ is a natural transformation, and the covariant Grothendieck construction provides a cocartesian functor $Z' \to RFib$. Since it is fiberwise fully faithful and $(-)_!$ preserves representable presheaves, we can identify $Z'$ with the full subcategory of $RFib$ consisting of the representable presheaves.

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

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

Remark

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

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

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

where $eval_x : D^C \to D$. This should be compared with the lax slice 2-category construction. In fact, $Z'$ 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

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 $\infty Grpd \hookrightarrow (\infty,1)Cat$

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

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

(Lurie 2009, Prop, 3.3.2.7, Cisinski 2019, Sec. 5.2, for the further restriction to the universal Kan fibration see also Kapulkin & Lumsdaine 2021)

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

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

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

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

By the adjoint functor theorem this functor has a left adjoint

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

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).

Related concepts

• universal Kan fibration

• fibration of (∞,1)-categories?

References

Textbook accounts:

• Jacob Lurie, Section 3.3.2 of: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press 2009 (pup:8957, pdf)

  (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)

• Denis-Charles Cisinski, Section 5.2 of: Higher Categories and Homotopical Algebra, Cambridge University Press 2019 (doi:10.1017/9781108588737, pdf)

  (the universal left fibration)

The direct description of the universal fibration is discussed at

• MathOverflow: A construction of the universal cocartesian fibration

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

• Jacob Lurie, chapter 5 Fibrations of ∞-Categories (2021) in: Kerodon

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

• Chris Kapulkin, Peter LeFanu Lumsdaine, The Simplicial Model of Univalent Foundations (after Voevodsky), Journal of the European Mathematical Society 23 (2021) 2071–2126 $[$arXiv:1211.2851, doi:10.4171/jems/1050$]$