Encyclopedia

Cartesian fibration

Wikipedia contributorsAdapted for Fathomly · View authors

2 min read
Source updated August 5, 2026

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.

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.

01Definition

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. 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 $\pi (g)=f$ . 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 $\varphi :\pi \to \rho$ 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. 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 $\operatorname {Cart} (S)$ .

02Basic 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 unique.

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.

03Grothendieck 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 $F_{\pi }:S^{op}\to {\textbf {Cat}}$ 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.

Watch videos about Cartesian fibrationExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Cartesian fibration”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Cartesian fibration

01Definition

02Basic example

03Grothendieck construction

Sources and credits

Related topics

Fibred category