cartesian fibration

In mathematics, especially homotopy theory, a cartesian fibration is, roughly, a map so that every lift exists that is a final object among all lifts. For example, the forgetful functor

: $\textrm{QCoh} \to \textrm{Sch}$

from the category of pairs $(X, F)$ of schemes and quasi-coherent sheaves on them is a cartesian fibration (see {{section link||Basic example}}). In fact, the Grothendieck construction says all cartesian fibrations are of this type; i.e., they simply forget extra data. See also: fibred category, prestack.

The dual of a cartesian fibration is called an op-fibration; in particular, not a cocartesian fibration.

A right fibration between simplicial sets is an example of a cartesian fibration.

Definition

Given a functor $\pi : C \to S$ , a morphism $f : x \to y$ in $C$ is called $\pi$ -cartesian or simply cartesian if the natural map

: $(f_*, \pi) : \operatorname{Hom}(z, x) \to \operatorname{Hom}(z, y) \times_{\operatorname{Hom}(\pi(z), \pi(y))} \operatorname{Hom}(\pi(z), \pi(x))$

is bijective.{{harvnb|Kerodon|loc=Definition 5.0.0.1.}}{{harvnb|Khan|2022|loc=Definition 3.1.1.}} Explicitly, thus, $f : x \to y$ is cartesian if given

$g: z \to y$ and
$u : \pi(z) \to \pi(x)$

with $\pi(g) = \pi(f) \circ u$ , there exists a unique $g' : z \to x$ in $\pi^{-1}(u)$ such that $f \circ g' = g$ .

Then $\pi$ is called a cartesian fibration if for each morphism of the form $f : s \to \pi(z)$ in S, there exists a $\pi$ -cartesian morphism $g : a \to z$ in C such that $\pi(g) = f$ .{{harvnb|Khan|2022|loc=Definition 3.1.2.}} Here, the object $a$ is unique up to unique isomorphisms (if $b \to z$ is another lift, there is a unique $b \to a$ , which is shown to be an isomorphism). Because of this, the object $a$ is often thought of as the pullback of $z$ and is sometimes even denoted as $f^* z$ .{{harvnb|Vistoli|2008|loc=Definition 3.1. and § 3.1.2.}} Also, somehow informally, $g$ is said to be a final object among all lifts of $f$ .

A morphism $\varphi : \pi \to \rho$ between cartesian fibrations over the same base S is a map (functor) over the base; i.e., $\pi = \rho \circ \varphi$ that sends cartesian morphisms to cartesian morphisms.{{harvnb|Vistoli|2008|loc=Definition 3.6.}} Given $\varphi, \psi : \pi \to \rho$ , a 2-morphism $\theta : \varphi \rightarrow \psi$ is an invertible map (map = natural transformation) such that for each object $E$ in the source of $\pi$ , $\theta_E : \varphi(E) \to \psi(E)$ maps to the identity map of the object $\rho(\varphi(E)) = \rho(\psi(E))$ under $\rho$ .

This way, all the cartesian fibrations over the fixed base category S determine the (2, 1)-category denoted by $\operatorname{Cart}(S)$ .{{harvnb|Khan|2022|loc=Construction 3.1.4.}}

Basic example

Let $\operatorname{QCoh}$ be the category where

an object is a pair $(X, F)$ of a scheme $X$ and a quasi-coherent sheaf $F$ on it,
a morphism $\overline{f} : (X, F) \to (Y, G)$ consists of a morphism $f : X \to Y$ of schemes and a sheaf homomorphism $\varphi_f : f^* G \overset{\sim}\to F$ on $X$ ,
the composition $\overline{g} \circ \overline{f}$ of $\overline{g} : (Y, G) \to (Z, H)$ and above $\overline{f}$ is the (unique) morphism $\overline{h}$ such that $h = g \circ f$ and $\varphi_h$ is
: $(g \circ f)^*H \simeq f^* g^* H \overset{f^*\varphi_g}\to f^*G \overset{\varphi_f}\to F.$

To see the forgetful map

: $\pi : \operatorname{QCoh} \to \operatorname{Sch}$

is a cartesian fibration,{{harvnb|Khan|2022|loc=Example 3.1.3.}} let $f : X \to \pi((Y, G))$ be in $\operatorname{QCoh}$ . Take

: $\overline{f} = (f, \varphi_f) : (X, F) \to (Y, G)$

with $F = f^* G$ and $\varphi_f = \operatorname{id}$ . We claim $\overline{f}$ is cartesian. Given $\overline{g} : (Z, H) \to (Y, G)$ and $h : Z \to X$ with $g = f \circ h$ , if $\varphi_h$ exists such that $\overline{g} = \overline{f} \circ \overline{h}$ , then we have $\varphi_g$ is

: $(f \circ h)^* G \simeq h^* f^* G = h^* F \overset{\varphi_h}\to H.$

So, the required $\overline{h}$ trivially exists and is unqiue.

Note some authors consider $\operatorname{QCoh}^{\simeq}$ , the core of $\operatorname{QCoh}$ instead. In that case, the forgetful map restricted to it is also a cartesian fibration.

Grothendieck construction

Given a category $S$ , the Grothendieck construction gives an equivalence of ∞-categories between $\operatorname{Cart}(S)$ and the ∞-category of prestacks on $S$ (prestacks = category-valued presheaves).{{harvnb|Khan|2022|loc=Theorem 3.1.5.}}

Roughly, the construction goes as follows: given a cartesian fibration $\pi$ , we let $F_{\pi} : S^{op} \to \textbf{Cat}$ be the map that sends each object x in S to the fiber $\pi^{-1}(x)$ . So, $F_{\pi}$ is a $\textbf{Cat}$ -valued presheaf or a prestack. Conversely, given a prestack $F$ , define the category $C_F$ where an object is a pair $(x, a)$ with $a \in F(x)$ and then let $\pi$ be the forgetful functor to $S$ . Then these two assignments give the claimed equivalence.

For example, if the construction is applied to the forgetful $\pi : \textrm{QCoh} \to \textrm{Sch}$ , then we get the map $X \mapsto \textrm{QCoh}(X)$ that sends a scheme $X$ to the category of quasi-coherent sheaves on $X$ . Conversely, $\pi$ is determined by such a map.

Lurie's straightening theorem generalizes the above equivalence to the equivalence between the ∞-category of cartesian fibrations over some ∞-category C and the ∞-category of ∞-prestacks on C.An introduction in Louis Martini, Cocartesian fibrations and straightening internal to an ∞-topos [arXiv:2204.00295]

Footnotes

References

{{cite web|first=Adeel A.|last=Khan|title=A modern introduction to algebraic stacks|url=https://www.preschema.com/lecture-notes/2022-stacks/|year=2022}}
{{cite web|title=Kerodon|url=https://kerodon.net/|ref=CITEREFKerodon}}
{{cite arxiv|first=Aaron|last=Mazel-Gee|title=A user’s guide to co/cartesian fibrations|eprint=1510.02402}}
{{cite web |first=Angelo |last=Vistoli |url=http://homepage.sns.it/vistoli/descent.pdf |title=Notes on Grothendieck topologies, fibered categories and descent theory |date=September 2, 2008}}