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 § 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.

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

Definition

Given a functor $\pi :C\to D$, 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. 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 $f:x\to \pi (z)$ in D, there exists a $\pi$-cartesian morphism $g:a\to z$ in C such that $\pi (g)=f$.

A morphism $\varphi :\pi \to \pi '$ between cartesian fibrations over the same base S is a map (functor) over the base; i.e., $\pi =\pi '\circ \varphi$. Given $\varphi ,\psi :\pi \to \pi '$, 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 $\pi '(\varphi (E))=\pi '(\psi (E))$ under $\pi '$.

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

Note some authors require a morphism between cartesian fibrations to preserve cartesian morphisms but this requirement seems to be related to requiring fibers to be groupoids (and that’s why this condition is not required above).

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, 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 is also a cartesian fibration.

Grothendieck construction

Main article: 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).

Roughly, the construction goes as follows: given a cartesian fibration $\pi$, we let $F_{\pi }:C^{op}\to {\textbf {Cat}}$ be the map that sends each object x in C 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 $C$. 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.

See also

• fibered category

References

• Adeel A. Khan, A modern introduction to algebraic stacks, https://www.preschema.com/lecture-notes/2022-stacks/
• Kerodon, https://kerodon.net/
• Aaron Mazel-Gee, A user’s guide to co/cartesian fibrations (arXiv:1510.02402)

Further reading

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