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 (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.
Given a functor , a morphism in is called -cartesian or simply cartesian if the natural map
is bijective. Explicitly, thus, is cartesian if given
with , there exists a unique in such that .
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 (if is another lift, there is a unique , which is shown to be an isomorphism). 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 (functor) over the base; i.e., that sends cartesian morphisms to cartesian morphisms. Given , a 2-morphism is an invertible map (map = natural transformation) 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 (2, 1)-category denoted by .
Let be the category where
To see the forgetful map
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.
Given a category , the Grothendieck construction gives an equivalence of âÂÂ-categories between and the âÂÂ-category of prestacks on (prestacks = category-valued presheaves).
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 âÂÂ-category of cartesian fibrations over some âÂÂ-category C and the âÂÂ-category of âÂÂ-prestacks on C.