Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
Given a functor between categories one may ask for each morphism that, if given a lift of its target
there be a universal lift of
There may also be other lifts of , but the universal one is essentially unique, as usual for anything having a universal property. Specifically, in is essentially uniquely determined by its target and its image in , and is called a cartesian morphism. A morphism which is cartesian relative to is called opcartesian or cocartesian.
If there are enough cartesian morphisms in , they may be used to define functors
between the fibers of over and .
This way a functor with enough Cartesian morphisms – called a Cartesian fibration or Grothendieck fibration – determines and is determined by a fiber-assigning functor .
This has its analog in higher categories.
(cartesian morphism)
Let be a functor. A morphism in the category is strongly cartesian with respect to (nowadays often just cartesian), or (strongly) -cartesian if for every , for every and every such that , there exists a unique such that and :
In imprecise words: for all commuting triangles in (involving as above) and all lifts through of its 2-horn to (involving as above), there is a unique refinement to a lift of the entire commuting triangle.
We can make this definition slightly more explicit by working with the fibres of : let denote , the set of objects living over ; and, for , and , let denote , the set of morphisms from to that maps to .
Then for , and , a morphism is cartesian iff for any , and , there is a unique with . This can be expressed in the following diagram, where the upper objects and morphisms live in over their corresponding data in :
If we pass to the nerve and of the categories, then in terms of diagrams in sSet this means that the morphism is -cartesian precisely if for all horn inclusions
such that the last edge of the 2-horn is the given edge , a unique lift
exists.
(weak/local Cartesian morphisms)
There is a weaker universal property, originally devised by Grothendieck and Gabriel, where one requires the above lifting property only for . Morphisms satisfying this universal property have in recent years been called locally Cartesian morphisms, although historically they have been called simply Cartesian, or sometimes weak Cartesian. For the case of Grothendieck fibered categories the notion of weak Cartesian morphisms already coincides with that of actual Cartesian morphisms.
Equivalently, a morphism is called locally -Cartesian (relative to a functor ) if it is Cartesian with respect to the projection functor where is the (homotopy) pullback of along the functor classifying the arrow in .
(Grothendieck fibration)
If for every morphism in and every lift of its target there is at least one lift which has as its target the chosen one and is a -cartesian morphism in the strong sense, one says that is a fibered category (also called Grothendieck fibration). Equivalently,
and
We discuss equivalent reformulations of the above definition of Cartesian morphism that lend themselves better to generalization to higher category theory.
For the following, we need this notation: let
by the overcategory of over the object ;
the corresponding overcategory of over ;
the category whose objects
are objects of equipped with morphisms to and such that the obvious triangle commutes, and whose morphisms are morphisms between these tip objects such that all diagrams in sight commute.
similarly for .
The condition that is a Cartesian morphism with respect to is equivalent to the condition that the functor
into the (strict) pullback of the obvious projection along the projection induced by the commutativity of
is a surjective equivalence, and this in turn is equivalent to it being an isomorphism of categories.
It is immediate to see that being an isomorphism of categories is equivalent to the condition that is a Cartesian morphism. We discuss that just the condition that is a surjective equivalence already implies that it is an isomorphism of categories.
So assume now that is a surjective equivalence.
Notice that objects in the pullback category are compatible pairs
We have that being surjective on object means that every such pair is in the image of some object
and hence that every filler exists . Assume two such fillers and . Then by the fact that an equivalence of categories is a surjection (even an isomorphism) on corresponding hom-sets, it follows that there exists (even uniquely) a morphism in connecting them
such that this maps under to the identity morphism in the pullback category. But in particular this maps to the morphism
in and evidently is the identity there if and only if is the identity. Hence this maps also to the identity in the pullback category if and only if is the identity. So must be the identity. So if two lifts of an object through the surjective equivalence exist, they must already be equal. Hence the surjective equivalence is even an isomorphism on objects and hence an isomorphism of categories.
The notion of cartesian morphism generalizes from category theory to (∞,1)-category theory. The definition given above can be rephrased as a pullback relation between homsets, which we can take as an abstract definition
(cartesian edge in an -category)
Let be a functor of (∞,1)-categories. Then, a morphism in is -cartesian if and only if the commutative square induced by the distributivity of over composition
is a pullback square for every .
Similarly, the reformulation in terms of slice/cone categories generalizes directly, and is indeed equivalent to the definition given above:
Let be a functor of (∞,1)-categories. Then, a morphism in is -cartesian iff the commutative square
is a pullback square.
The definition of being -cartesian can be described as a pullback square of hom-functors. By the contravariant (∞,1)-Grothendieck construction, classifies the right fibration, , and classifies the pullback of along ; that is by the comma category . There is a commutative diagram
The bottom-right square and wide rectangle are pullbacks by construction. By the pasting law, the bottom-left square is a pullback, and thus the top-left square is a pullback if and only if the tall rectangle is.
By the (∞,1)-Grothendieck construction, the top-left square is a pullback iff is -cartesian, so the proposition follows.
If , then is an equivalence of (∞,1)-categories, and composing its inverse with is the dependent sum .
To make this concrete, we can also discuss adaptations of the abstract idea to two different models of (∞,1)-category theory: quasi-categories and sSet categories.
We formulate a notion cartesian edge or cartesian morphism in a simplicial set relative to a morphism of simplicial sets. In the case that these simplicial sets are quasi-categories – i.e. simplicial set incarnations of (∞,1)-categories – this yields a notion of cartesian morphisms in -categories.
Let be a morphism of simplicial sets. Let be an edge in , i.e. a morphism .
Recall the notion of over quasi-category obtained from the notion of join of quasi-categories. Using this we obtain simplicial sets , , and in generalization of the categories considered in the above definition of cartesian morphisms in categories.
(cartesian edge in a simplicial set)
Let be an inner Kan fibration of simplicial sets.
Then a morphism in is -cartesian if the induced morphism
into the pullback in sSet is an acyclic Kan fibration.
This is HTT, def 2.4.1.1.
The morphism as above, for an inner fibration, is -cartesian precisely if for all and all right outer horn inclusions
(with the th horn of the -simplex) such that the last edge of the horn is the given edge , a lift
exists.
This is HTT remark 2.4.1.4.
This means that an inner fibration with a collection of -cartesian morphisms in specified satisfies the same kind of condition as a right fibration , the only difference being that not all right outer horns inclusion are required to have lifts, but only those where the last edge of the horn maps to a cartesian morphism.
In this sense a Cartesian fibration is a generalization of a right fibration.
If is an inner fibration of quasi-categories then a morphism in is -Cartesian precisely if for all objects in the diagram
of hom-objects in a quasi-category is a homotopy pullback square (in sSet equipped with its standard model structure).
This is HTT, prop. 2.4.4.3.
Let and be simplicially enriched categories and a sSet-enriched functor.
A morphism is -cartesian if it is so under the homotopy coherent nerve in the sense of quasi-categories above, i.e. if
is an acyclic Kan fibration.
If and are enriched in Kan complexes and if is hom-wise a Kan fibration, then
is an inner fibration;
a morphism in is an -cartesian morphism precisely if for all objects in the diagram
is a homotopy pullback square in sSet equipped with its standard model structure.
This is HTT, prop. 2.4.1.10.
For a functor, if in a diagram
in the two vertical morphisms are vertical with respect to (meaning that and ) and if the two horizontal morphisms are -Cartesian morphisms, then this square is a pullback square.
If
is another cone over , then its image under is
Since , another lift of the right horn of this is given by
which gives a unique filler by the fact that is Cartesian.
But this produces now two fillers – namely the original and the just obtained – of the horn
over
Since is Cartesian, these two fillers must be equal. This means that the morphism is a cone morphism and unique as such. Hence the original square is a pullback.
This appears as Elephant, lemma 1.3.3.
For a category, a morphism in is cartesian with respect to the terminal functor precisely if it is an isomorphism.
In particular all identity morphisms are cartesian.
This is trivial to see. The analog statement holds also for quasi-categories, where it is rather more nontrivial and quite useful:
For a quasi-category, a morphism in is cartesian with respect to the terminal morphism precisely if it is an equivalence.
More generally, for an inner fibration, a morphism in is an equivalence precisely if it is -cartesian and is an equivalence in .
The first statement is a proposition of Andre Joyal, slightly reformulated in the language of cartesian morphisms. It appears as HTT, prop 1.2.4.3. A proof appears below HTT, corollary 2.1.2.2.
The second statement is HTT, prop. 2.4.1.5.
Original reference:
Review:
For the 1-categorical case see for instance section B1.3 of
The -categorical version is in section 2.4 of
See also the references at Grothendieck fibration.
Last revised on December 28, 2023 at 11:40:03. See the history of this page for a list of all contributions to it.