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

$\text{<mrow data-mjx-texclass="ORD">QCoh</mrow>}\to \text{<mrow data-mjx-texclass="ORD">Sch</mrow>}$

from the category of pairs $(X,F)$ of schemes and quasi-coherent sheaves on them is a cartesian fibration (see § 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

$(f_{*},\pi ):\mathrm{Hom}(z,x)\to \mathrm{Hom}(z,y){\times }_{\mathrm{Hom}(\pi (z),\pi (y))}\mathrm{Hom}(\pi (z),\pi (x))$

is bijective. 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 $g^{\prime }:z\to x$ in ${\pi }^{-1}(u)$ such that $f\circ g^{\prime }=g$.

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 $\pi (g)=f$. 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 $f^{*}z$. Also, somehow informally, $g$ is said to be a final object among all lifts of $f$.

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

This way, all the cartesian fibrations over the fixed base category S determine the (2, 1)-category denoted by $\mathrm{Cart}(S)$.

Basic example

Let $\mathrm{QCoh}$ 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 $\bar{f}:(X,F)\to (Y,G)$ consists of a morphism $f:X\to Y$ of schemes and a sheaf homomorphism ${\phi }_f:f^{*}G\stackrel{\sim }{\to }F$ on $X$,
• the composition $\bar{g}\circ \bar{f}$ of $\bar{g}:(Y,G)\to (Z,H)$ and above $\bar{f}$ is the (unique) morphism $\bar{h}$ such that $h=g\circ f$ and ${\phi }_h$ is

  $(g\circ f{)}^{*}H\simeq f^{*}{g}^{*}H\stackrel{f^{*}{\phi }_g}{\to }{f}^{*}G\stackrel{{\phi }_f}{\to }F.$

To see the forgetful map

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

is a cartesian fibration, let $f:X\to \pi ((Y,G))$ be in $\mathrm{QCoh}$. Take

$\bar{f}=(f,{\phi }_f):(X,F)\to (Y,G)$

with $F=f^{*}G$ and ${\phi }_f=\mathrm{id}$. We claim $\bar{f}$ is cartesian. Given $\bar{g}:(Z,H)\to (Y,G)$ and $h:Z\to X$ with $g=f\circ h$, if ${\phi }_h$ exists such that $\bar{g}=\bar{f}\circ \bar{h}$, then we have ${\phi }_g$ is

$(f\circ h{)}^{*}G\simeq h^{*}{f}^{*}G=h^{*}F\stackrel{{\phi }_h}{\to }H.$

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

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

Grothendieck construction

Main article: Grothendieck construction

Given a category $S$, the Grothendieck construction gives an equivalence of ∞-categories between $\mathrm{Cart}(S)$ and the ∞-category of prestacks on $S$ (prestacks = category-valued presheaves).

Roughly, the construction goes as follows: given a cartesian fibration $\pi$, we let $F_{\pi }:S^{op}\to \text{<mrow data-mjx-texclass="ORD">Cat</mrow>}$ be the map that sends each object x in S to the fiber ${\pi }^{-1}(x)$. So, $F_{\pi }$ is a $\text{<mrow data-mjx-texclass="ORD">Cat</mrow>}$-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 $\pi :\text{<mrow data-mjx-texclass="ORD">QCoh</mrow>}\to \text{<mrow data-mjx-texclass="ORD">Sch</mrow>}$, then we get the map $X\mapsto \text{<mrow data-mjx-texclass="ORD">QCoh</mrow>}(X)$ 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.

See also

• fibered category

References

• Khan, Adeel A. (2022). "A modern introduction to algebraic stacks". https://www.preschema.com/lecture-notes/2022-stacks/. 
• "Kerodon". https://kerodon.net/. 
• Mazel-Gee, Aaron (2015). "A user's guide to co/cartesian fibrations". arXiv:1510.02402 [math.CT].
• Vistoli, Angelo (September 2, 2008). "Notes on Grothendieck topologies, fibered categories and descent theory". http://homepage.sns.it/vistoli/descent.pdf. 

Further reading

• https://ncatlab.org/nlab/show/Cartesian+fibration
• https://ncatlab.org/nlab/show/Cartesian+morphism
• https://ncatlab.org/nlab/show/Grothendieck+fibration