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$]$

Discussion of straightening and unstraightening entirely within the context of quasi-categories:

• Denis-Charles Cisinski, Hoang Kim Nguyen, The universal coCartesian fibration [arXiv:2210.08945]

which (along the lines of the discussion of the universal left fibration from Cisinski 2019) allows to understand the universal coCartesian fibration as categorical semantics for the univalent type universe in directed homotopy type theory:

• Denis-Charles Cisinski, Hoang Kim Nguyen, Tashi Walde: Univalent Directed Type Theory, lecture series in the CMU Homotopy Type Theory Seminar (13, 20, 27 Mar 2023) [web, video 1:YT, 2:YT, 3:YT; slides 0:pdf, 1:pdf, 2:pdf, 3:pdf]

(see video 3 at 1:16:58 and slide 3.33).