Cartesian Functor

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
:$\textrm \to \textrm$
from the category of pairs $(X, F)$ of schemes and quasi-coherent sheaves on them is a cartesian fibration (see ). 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 $\pi : C \to S$, a morphism $f : x \to y$ in $C$ is called $\pi$-cartesian or simply cartesian if the natural map
:$(f_*, \pi) : \operatorname(z, x) \to \operatorname(z, y) \times_ \operatorname(\pi(z), \pi(x))$
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 $\pi^(u)$ 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 $\operatorname(S)$.

Basic example

Let $\operatorname$ 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 $\overline : (X, F) \to (Y, G)$ consists of a morphism $f : X \to Y$ of schemes and a sheaf homomorphism $\varphi_f : f^* G \overset\to F$ on $X$,
*the composition $\overline \circ \overline$ of $\overline : (Y, G) \to (Z, H)$ and above $\overline$ is the (unique) morphism $\overline$ such that $h = g \circ f$ and $\varphi_h$ is
*:$(g \circ f)^*H \simeq f^* g^* H \overset\to f^*G \overset\to F.$
To see the forgetful map
:$\pi : \operatorname \to \operatorname$
is a cartesian fibration, let $f : X \to \pi((Y, G))$ be in $\operatorname$. Take
:$\overline = (f, \varphi_f) : (X, F) \to (Y, G)$
with $F = f^* G$ and $\varphi_f = \operatorname$. We claim $\overline$ is cartesian. Given $\overline : (Z, H) \to (Y, G)$ and $h : Z \to X$ with $g = f \circ h$, if $\varphi_h$ exists such that $\overline = \overline \circ \overline$, then we have $\varphi_g$ is
:$(f \circ h)^* G \simeq h^* f^* G = h^* F \overset\to H.$
So, the required $\overline$ trivially exists and is unqiue.

Note some authors consider $\operatorname^$, the core of $\operatorname$ instead. In that case, the forgetful map restricted to it is also a cartesian fibration.

Grothendieck construction

Given a category $S$, the Grothendieck construction gives an equivalence of ∞-categories between $\operatorname(S)$ and the ∞-category of prestacks on $S$ (prestacks = category-valued presheaves).

Roughly, the construction goes as follows: given a cartesian fibration $\pi$, we let $F_ : S^ \to \textbf$ be the map that sends each object ''x'' in ''S'' to the fiber $\pi^(x)$. So, $F_$ is a $\textbf$-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 $\pi : \textrm \to \textrm$, then we get the map $X \mapsto \textrm(X)$ 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''.An introduction in Louis Martini, Cocartesian fibrations and straightening internal to an ∞-topos rXiv:2204.00295/ref>

See also

* fibered category

Footnotes

References

*
*
*
* {{cite web , first=Angelo , last=Vistoli , url=http://homepage.sns.it/vistoli/descent.pdf , title=Notes on Grothendieck topologies, fibered categories and descent theory , date=September 2, 2008

Further reading

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

Homotopy theory