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
from the category of pairs of schemes and quasi-coherent sheaves on them is a cartesian fibration. 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, a morphism in is called -cartesian or simply cartesian if the natural mapis bijective. Explicitly, thus, is cartesian if given
- and
Then is called a cartesian fibration if for each morphism of the form in S, there exists a -cartesian morphism in C such that. Here, the object is unique up to unique isomorphisms. Because of this, the object is often thought of as the pullback of and is sometimes even denoted as. Also, somehow informally, is said to be a final object among all lifts of.
A morphism between cartesian fibrations over the same base S is a map over the base; i.e., that sends cartesian morphisms to cartesian morphisms. Given, a 2-morphism is an invertible map such that for each object in the source of, maps to the identity map of the object under.
This way, all the cartesian fibrations over the fixed base category S determine the -category denoted by.
Basic example
Let be the category where- an object is a pair of a scheme and a quasi-coherent sheaf on it,
- a morphism consists of a morphism of schemes and a sheaf homomorphism on,
- the composition of and above is the morphism such that and is
- :
is a cartesian fibration, let be in. Take
with and. We claim is cartesian. Given and with, if exists such that, then we have is
So, the required trivially exists and is unqiue.
Note some authors consider, the core of instead. In that case, the forgetful map restricted to it is also a cartesian fibration.
Grothendieck construction
Given a category, the Grothendieck construction gives an equivalence of ∞-categories between and the ∞-category of prestacks on .Roughly, the construction goes as follows: given a cartesian fibration, we let be the map that sends each object x in S to the fiber. So, is a -valued presheaf or a prestack. Conversely, given a prestack, define the category where an object is a pair with and then let be the forgetful functor to. Then these two assignments give the claimed equivalence.
For example, if the construction is applied to the forgetful, then we get the map that sends a scheme to the category of quasi-coherent sheaves on. Conversely, 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.