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

Given a functor 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 {\displaystyle \pi }-cartesian or simply cartesian if the natural map

( f ∗ , π ) : Hom ⁡ ( z , x ) → Hom ⁡ ( z , y ) × Hom ⁡ ( π ( z ) , π ( y ) ) Hom ⁡ ( π ( z ) , 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. 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 ( z ) → π ( x ) {\displaystyle u:\pi (z)\to \pi (x)}

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

Then π {\displaystyle \pi } is called a cartesian fibration if for each morphism of the form f : s → π ( z ) {\displaystyle f:s\to \pi (z)} in S, there exists a π {\displaystyle \pi }-cartesian morphism g : a → z {\displaystyle g:a\to z} in C such that π ( g ) = f {\displaystyle \pi (g)=f}. 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}. Also, somehow informally, g {\displaystyle g} is said to be a final object among all lifts of f {\displaystyle f}.

A morphism φ : π → ρ {\displaystyle \varphi :\pi \to \rho } between cartesian fibrations over the same base S is a map (functor) over the base; i.e., π = ρ ∘ φ {\displaystyle \pi =\rho \circ \varphi } that sends cartesian morphisms to cartesian morphisms. Given φ , ψ : π → ρ {\displaystyle \varphi ,\psi :\pi \to \rho }, a 2-morphism θ : φ → 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 {\displaystyle \pi }, θ E : φ ( E ) → ψ ( E ) {\displaystyle \theta _{E}:\varphi (E)\to \psi (E)} maps to the identity map of the object ρ ( φ ( E ) ) = ρ ( ψ ( E ) ) {\displaystyle \rho (\varphi (E))=\rho (\psi (E))} under 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)}.

Basic example

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 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 h {\displaystyle \varphi _{h}} is ( g ∘ f ) ∗ H ≃ f ∗ g ∗ H → f ∗ φ g f ∗ G → 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 : QCoh → Sch {\displaystyle \pi :\operatorname {QCoh} \to \operatorname {Sch} }

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

f ¯ = ( f , 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 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 h {\displaystyle \varphi _{h}} exists such that g ¯ = f ¯ ∘ h ¯ {\displaystyle {\overline {g}}={\overline {f}}\circ {\overline {h}}}, then we have φ g {\displaystyle \varphi _{g}} is

( f ∘ h ) ∗ G ≃ h ∗ f ∗ G = h ∗ F → 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.

Grothendieck construction

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).

Roughly, the construction goes as follows: given a cartesian fibration π {\displaystyle \pi }, we let F 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 − 1 ( x ) {\displaystyle \pi ^{-1}(x)}. So, F 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 {\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 : 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 {\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.

See also

Footnotes

  • Khan, Adeel A. (2022). .
  • .
  • Mazel-Gee, Aaron (2015). "A user's guide to co/cartesian fibrations". arXiv: [].
  • Vistoli, Angelo (September 2, 2008). (PDF).

Further reading