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.

Definition

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.^[1][2] 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]

Basic example

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.

Grothendieck construction

Main article: Grothendieck construction

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]

See also

• fibered category

Footnotes

1. Kerodon, Definition 5.0.0.1.
2. Khan 2022, Definition 3.1.1.
3. Khan 2022, Definition 3.1.2.
4. Vistoli 2008, Definition 3.1. and § 3.1.2.
5. Vistoli 2008, Definition 3.6.
6. Khan 2022, Construction 3.1.4.
7. Khan 2022, Example 3.1.3.
8. Khan 2022, Theorem 3.1.5.
9. An introduction in Louis Martini, Cocartesian fibrations and straightening internal to an ∞-topos [arXiv:2204.00295]

References

• Khan, Adeel A. (2022). "A modern introduction to algebraic stacks".
• "Kerodon".
• 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" (PDF).

Further reading

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