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

{\textrm {QCoh}}\to {\textrm {Sch}}

from the category of pairs $(X,F)$ of schemes and quasi-coherent sheaves on them is a cartesian fibration.^[1] 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.

A right fibration between simplicial sets is an example of a cartesian fibration.

Definition

Given a functor $\pi :C\to D$ , a morphism $f:x\to y$ in $C$ is called $\pi$ -cartesian if the natural map

(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.^[2]^[3] 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 $\pi ^{-1}(u)$ such that $f\circ g'=g$ .

Then $\pi$ is called a cartesian fibration if for each morphism of the form $f:x\to \pi (z)$ in D, there exists a $\pi$ -cartesian morphism $g:a\to z$ in C such that $\pi (g)=f$ . ^[4]

A morphism $\varphi :\pi \to \pi '$ between cartesian fibrations over the same base S is simply a map (functor) over the base; i.e., $\pi =\pi '\circ \varphi$ . Given $\varphi ,\psi :\pi \to \pi '$ , 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 $\pi '(\varphi (E))=\pi '(\psi (E))$ under $\pi '$ .

This way, all the cartesian fibrations over the fixed base category S determine the (2, 1)-category denoted by $\operatorname {Cart} (S)$ .

Grothendieck construction

Given a category $S$ , the Grothendieck construction gives an equivalence of ∞-categories between $\operatorname {Cart} (S)$ and the ∞-category of prestacks on $S$ (prestacks = category-valued presheaves).^[5]

Roughly, the construction goes as follows: given a cartesian fibration $\pi$ , we let $F_{\pi }:C^{op}\to {\textbf {Cat}}$ be the map that sends each object x in C to the fiber $\pi ^{-1}(x)$ . So, $F_{\pi }$ is a ${\textbf {Cat}}$ -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 $C$ . Then these two assignments give the claimed equivalence.

For example, if the construction is applied to the forgetful $\pi :{\textrm {QCoh}}\to {\textrm {Sch}}$ , then we get the map $X\mapsto {\textrm {QCoh}}(X)$ 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 the ∞-category of ∞-prestacks on C.^[6]

References

^ Khan, Example 3.1.3.
^ Kerodon, Definition 5.0.0.1.
^ Khan, Definition 3.1.1.
^ Khan, Definition 3.1.2.
^ Khan, Theorem 3.1.5.
^ An introduction in Louis Martini, Cocartesian fibrations and straightening internal to an ∞-topos [arXiv:2204.00295]

Adeel A. Khan, A modern introduction to algebraic stacks, https://www.preschema.com/lecture-notes/2022-stacks/
Kerodon, https://kerodon.net/
Aaron Mazel-Gee, A user’s guide to co/cartesian fibrations (arXiv:1510.02402)

Definition

Grothendieck construction

See also

References

Further reading