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 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 $f\circ g'=g$. See also an argument in § Basic example, which may be less mysterious.

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 simply 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 $\operatorname {Cart} (S)$.

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 an invertible 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 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 ${\overline {f}}:(X,F)\to (Y,G)$ be in $\operatorname {QCoh}$. 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 must have:

$\varphi _{g}=\varphi _{h}\circ h^{*}\varphi _{f}.$

Since $h^{*}\varphi _{f}$ is invertible, the above equation uniquely determines $\varphi _{h}$.

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