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

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

from the category of pairs ${\displaystyle (X,F)}$ 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.

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

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

is bijective.¹² Explicitly, thus, ${\displaystyle f:x\to y}$ is cartesian if given

• ${\displaystyle g:z\to y}$ and
• ${\displaystyle u:\pi (z)\to \pi (x)}$

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

Then ${\displaystyle \pi }$ is called a cartesian fibration if for each morphism of the form ${\displaystyle f:s\to \pi (z)}$ in S, there exists a ${\displaystyle \pi }$-cartesian morphism ${\displaystyle g:a\to z}$ in C such that ${\displaystyle \pi (g)=f}$.³ Here, the object ${\displaystyle a}$ is unique up to unique isomorphisms (if ${\displaystyle b\to z}$ is another lift, there is a unique ${\displaystyle b\to a}$, which is shown to be an isomorphism). Because of this, the object ${\displaystyle a}$ is often thought of as the pullback of ${\displaystyle z}$ and is sometimes even denoted as ${\displaystyle f^{*}z}$.⁴ Also, somehow informally, ${\displaystyle g}$ is said to be a final object among all lifts of ${\displaystyle f}$.

A morphism ${\displaystyle \varphi :\pi \to \rho }$ between cartesian fibrations over the same base S is a map (functor) over the base; i.e., ${\displaystyle \pi =\rho \circ \varphi }$ that sends cartesian morphisms to cartesian morphisms.⁵ Given ${\displaystyle \varphi ,\psi :\pi \to \rho }$, a 2-morphism ${\displaystyle \theta :\varphi \rightarrow \psi }$ is an invertible map (map = natural transformation) such that for each object ${\displaystyle E}$ in the source of ${\displaystyle \pi }$, ${\displaystyle \theta _{E}:\varphi (E)\to \psi (E)}$ maps to the identity map of the object ${\displaystyle \rho (\varphi (E))=\rho (\psi (E))}$ under ${\displaystyle \rho }$.

This way, all the cartesian fibrations over the fixed base category S determine the (2, 1)-category denoted by ${\displaystyle \operatorname {Cart} (S)}$.⁶

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

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

  ${\displaystyle (g\circ f)^{*}H\simeq f^{*}g^{*}H{\overset {f^{*}\varphi _{g}}{\to }}f^{*}G{\overset {\varphi _{f}}{\to }}F.}$

To see the forgetful map

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

is a cartesian fibration,⁷ let ${\displaystyle f:X\to \pi ((Y,G))}$ be in ${\displaystyle \operatorname {QCoh} }$. Take

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

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

${\displaystyle (f\circ h)^{*}G\simeq h^{*}f^{*}G=h^{*}F{\overset {\varphi _{h}}{\to }}H.}$

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

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

Given a category ${\displaystyle S}$, the Grothendieck construction gives an equivalence of ∞-categories between ${\displaystyle \operatorname {Cart} (S)}$ and the ∞-category of prestacks on ${\displaystyle S}$ (prestacks = category-valued presheaves).⁸

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

For example, if the construction is applied to the forgetful ${\displaystyle \pi :{\textrm {QCoh}}\to {\textrm {Sch}}}$, then we get the map ${\displaystyle X\mapsto {\textrm {QCoh}}(X)}$ that sends a scheme ${\displaystyle X}$ to the category of quasi-coherent sheaves on ${\displaystyle X}$. Conversely, ${\displaystyle \pi }$ 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.⁹