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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

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

from the category of pairs ( X , F ) {\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

• Basic example

• Grothendieck construction

• See also

• Footnotes

• References

• Further reading

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

( f ∗ ∗ , π π ) : Hom ⁡ ⁡ ( z , x ) → → Hom ⁡ ⁡ ( z , y ) × × Hom ⁡ ⁡ ( π π ( z ) , π π ( y ) ) Hom ⁡ ⁡ ( π π ( z ) , π π ( x ) ) {\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.cite-ref-1[1]cite-ref-2[2] Explicitly, thus, f : x → → y {\displaystyle f:x\to y} is cartesian if given

• g : z → → y {\displaystyle g:z\to y} and

• u : π π ( z ) → → π π ( x ) {\displaystyle u:\pi (z)\to \pi (x)}

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

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

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

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

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

• an object is a pair ( X , F ) {\displaystyle (X,F)} of a scheme X {\displaystyle X} and a quasi-coherent sheaf F {\displaystyle F} on it,

• a morphism f ¯ ¯ : ( X , F ) → → ( Y , G ) {\displaystyle {\overline {f}}:(X,F)\to (Y,G)} consists of a morphism f : X → → Y {\displaystyle f:X\to Y} of schemes and a sheaf homomorphism φ φ f : f ∗ ∗ G → → ∼ ∼ F {\displaystyle \varphi _{f}:f^{*}G{\overset {\sim }{\to }}F} on X {\displaystyle X} ,

• the composition g ¯ ¯ ∘ ∘ f ¯ ¯ {\displaystyle {\overline {g}}\circ {\overline {f}}} of g ¯ ¯ : ( Y , G ) → → ( Z , H ) {\displaystyle {\overline {g}}:(Y,G)\to (Z,H)} and above f ¯ ¯ {\displaystyle {\overline {f}}} is the (unique) morphism h ¯ ¯ {\displaystyle {\overline {h}}} such that h = g ∘ ∘ f {\displaystyle h=g\circ f} and φ φ h {\displaystyle \varphi _{h}} is ( g ∘ ∘ f ) ∗ ∗ H ≃ ≃ f ∗ ∗ g ∗ ∗ H → → f ∗ ∗ φ φ g f ∗ ∗ G → → φ φ f F . {\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

π π : QCoh → → Sch {\displaystyle \pi :\operatorname {QCoh} \to \operatorname {Sch} }

is a cartesian fibration,cite-ref-7[7] let f : X → → π π ( ( Y , G ) ) {\displaystyle f:X\to \pi ((Y,G))} be in QCoh {\displaystyle \operatorname {QCoh} } . Take

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

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

( f ∘ ∘ h ) ∗ ∗ G ≃ ≃ h ∗ ∗ f ∗ ∗ G = h ∗ ∗ F → → φ φ h H . {\displaystyle (f\circ h)^{*}G\simeq h^{*}f^{*}G=h^{*}F{\overset {\varphi _{h}}{\to }}H.}

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

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

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

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

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

• fibered category

cite-note-11. ↑ Kerodon, Definition 5.0.0.1.

cite-note-22. ↑ Khan 2022, Definition 3.1.1.

cite-note-33. ↑ Khan 2022, Definition 3.1.2.

cite-note-44. ↑ Vistoli 2008, Definition 3.1. and § 3.1.2.

cite-note-55. ↑ Vistoli 2008, Definition 3.6.

cite-note-66. ↑ Khan 2022, Construction 3.1.4.

cite-note-77. ↑ Khan 2022, Example 3.1.3.

cite-note-88. ↑ Khan 2022, Theorem 3.1.5.

cite-note-99. ↑ An introduction in Louis Martini, Cocartesian fibrations and straightening internal to an ∞-topos [arXiv:2204.00295]

• citerefkhan2022Khan, Adeel A. (2022). "A modern introduction to algebraic stacks".

• citerefkerodon"Kerodon".

• citerefmazel-gee2015Mazel-Gee, Aaron (2015). "A user's guide to co/cartesian fibrations". arXiv:1510.02402 [math.CT].

• citerefvistoli2008Vistoli, Angelo (September 2, 2008). "Notes on Grothendieck topologies, fibered categories and descent theory" (PDF).

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

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

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