Cartesian fibration
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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 β 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 Β§ 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.
Contents
β’ Definition
β’ Basic example
β’ See also
β’ Footnotes
β’ References
β’ Further reading
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Definition
Given a functor Ο : C β S {\displaystyle \pi :C\to S} , a morphism f : x β y {\displaystyle f:x\to y} in C {\displaystyle C} is called Ο {\displaystyle \pi } -cartesian or simply cartesian if the natural map
( f β , Ο ) : Hom β‘ ( z , x ) β Hom β‘ ( z , y ) Γ Hom β‘ ( Ο ( z ) , Ο ( y ) ) Hom β‘ ( Ο ( z ) , Ο ( 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))}
β’ g : z β y {\displaystyle g:z\to y} and
β’ u : Ο ( z ) β Ο ( x ) {\displaystyle u:\pi (z)\to \pi (x)}
with Ο ( g ) = Ο ( f ) β u {\displaystyle \pi (g)=\pi (f)\circ u} , there exists a unique g β² : z β x {\displaystyle g':z\to x} in Ο β 1 ( u ) {\displaystyle \pi ^{-1}(u)} such that f β g β² = g {\displaystyle f\circ g'=g} .
Then Ο {\displaystyle \pi } is called a cartesian fibration if for each morphism of the form f : s β Ο ( z ) {\displaystyle f:s\to \pi (z)} in S, there exists a Ο {\displaystyle \pi } -cartesian morphism g : a β z {\displaystyle g:a\to z} in C such that Ο ( g ) = f {\displaystyle \pi (g)=f} .cite-ref-3[3] Here, the object a {\displaystyle a} is unique up to unique isomorphisms (if b β z {\displaystyle b\to z} is another lift, there is a unique b β 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 β 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 Ο : Ο β Ο {\displaystyle \varphi :\pi \to \rho } between cartesian fibrations over the same base S is a map (functor) over the base; i.e., Ο = Ο β Ο {\displaystyle \pi =\rho \circ \varphi } that sends cartesian morphisms to cartesian morphisms.cite-ref-5[5] Given Ο , Ο : Ο β Ο {\displaystyle \varphi ,\psi :\pi \to \rho } , a 2-morphism ΞΈ : Ο β Ο {\displaystyle \theta :\varphi \rightarrow \psi } is an invertible map (map = natural transformation) such that for each object E {\displaystyle E} in the source of Ο {\displaystyle \pi } , ΞΈ E : Ο ( E ) β Ο ( E ) {\displaystyle \theta _{E}:\varphi (E)\to \psi (E)} maps to the identity map of the object Ο ( Ο ( E ) ) = Ο ( Ο ( E ) ) {\displaystyle \rho (\varphi (E))=\rho (\psi (E))} under Ο {\displaystyle \rho } .
This way, all the cartesian fibrations over the fixed base category S determine the (2, 1)-category denoted by Cart β‘ ( S ) {\displaystyle \operatorname {Cart} (S)} .cite-ref-6[6]
Basic example
Let QCoh {\displaystyle \operatorname {QCoh} } be the category where
β’ 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,
β’ a morphism f Β― : ( X , F ) β ( Y , G ) {\displaystyle {\overline {f}}:(X,F)\to (Y,G)} consists of a morphism f : X β Y {\displaystyle f:X\to Y} of schemes and a sheaf homomorphism Ο f : f β G β βΌ F {\displaystyle \varphi _{f}:f^{*}G{\overset {\sim }{\to }}F} on X {\displaystyle X} ,
β’ the composition g Β― β f Β― {\displaystyle {\overline {g}}\circ {\overline {f}}} of g Β― : ( Y , G ) β ( Z , H ) {\displaystyle {\overline {g}}:(Y,G)\to (Z,H)} and above f Β― {\displaystyle {\overline {f}}} is the (unique) morphism h Β― {\displaystyle {\overline {h}}} such that h = g β f {\displaystyle h=g\circ f} and Ο h {\displaystyle \varphi _{h}} is ( g β f ) β H β f β g β H β f β Ο g f β G β Ο 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
Ο : QCoh β Sch {\displaystyle \pi :\operatorname {QCoh} \to \operatorname {Sch} }
is a cartesian fibration,cite-ref-7[7] let f : X β Ο ( ( Y , G ) ) {\displaystyle f:X\to \pi ((Y,G))} be in QCoh {\displaystyle \operatorname {QCoh} } . Take
f Β― = ( f , Ο f ) : ( X , F ) β ( Y , G ) {\displaystyle {\overline {f}}=(f,\varphi _{f}):(X,F)\to (Y,G)}
with F = f β G {\displaystyle F=f^{*}G} and Ο f = id {\displaystyle \varphi _{f}=\operatorname {id} } . We claim f Β― {\displaystyle {\overline {f}}} is cartesian. Given g Β― : ( Z , H ) β ( Y , G ) {\displaystyle {\overline {g}}:(Z,H)\to (Y,G)} and h : Z β X {\displaystyle h:Z\to X} with g = f β h {\displaystyle g=f\circ h} , if Ο h {\displaystyle \varphi _{h}} exists such that g Β― = f Β― β h Β― {\displaystyle {\overline {g}}={\overline {f}}\circ {\overline {h}}} , then we have Ο g {\displaystyle \varphi _{g}} is
( f β h ) β G β h β f β G = h β F β Ο h H . {\displaystyle (f\circ h)^{*}G\simeq h^{*}f^{*}G=h^{*}F{\overset {\varphi _{h}}{\to }}H.}
So, the required h Β― {\displaystyle {\overline {h}}} trivially exists and is unqiue.
Note some authors consider QCoh β {\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
Roughly, the construction goes as follows: given a cartesian fibration Ο {\displaystyle \pi } , we let F Ο : S o p β Cat {\displaystyle F_{\pi }:S^{op}\to {\textbf {Cat}}} be the map that sends each object x in S to the fiber Ο β 1 ( x ) {\displaystyle \pi ^{-1}(x)} . So, F Ο {\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 β F ( x ) {\displaystyle a\in F(x)} and then let Ο {\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 Ο : QCoh β Sch {\displaystyle \pi :{\textrm {QCoh}}\to {\textrm {Sch}}} , then we get the map X β¦ 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, Ο {\displaystyle \pi } is determined by such a map.
Lurie's straightening theorem generalizes the above equivalence to the equivalence between the β-category of cartesian fibrations over some β-category C and the β-category of β-prestacks on C.cite-ref-9[9]
See also
Footnotes
cite-note-44. β Vistoli 2008, Definition 3.1. and Β§ 3.1.2.
cite-note-55. β Vistoli 2008, Definition 3.6.
cite-note-99. β An introduction in Louis Martini, Cocartesian fibrations and straightening internal to an β-topos [arXiv:2204.00295]
References
β’ citerefkhan2022Khan, Adeel A. (2022). "A modern introduction to algebraic stacks".
β’ citerefkerodon"Kerodon".
β’ citerefmazel-gee2015Mazel-Gee, Aaron (2015). "A user's guide to co/cartesian fibrations". arXiv:1510.02402 [math.CT].
β’ citerefvistoli2008Vistoli, Angelo (September 2, 2008). "Notes on Grothendieck topologies, fibered categories and descent theory" (PDF).
Further reading
β’ https://ncatlab.org/nlab/show/Cartesian+fibration
β’ https://ncatlab.org/nlab/show/Cartesian+morphism
β’ https://ncatlab.org/nlab/show/Grothendieck+fibration