Given a functor between categories one may ask for each morphism if given a lift of its target
there is 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 source 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.
Let be a functor. A morphism in the category is (strongly) cartesian with respect to , or -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.
There is a weaker universal property, originally devised by Grothendieck and Gabriel, where one requires above lifting property only for , and traditionally also called simply cartesian, or rarely weak cartesian. In Grothendieck’s fibered categories (see below), cartesian in the strong sense and cartesian in the weak sense are equivalent properties of morphisms.
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 lift
If for every morphism in there is at least one lift through that is a -cartesian morphism in the strong sense, one says that is a Grothendieck fibration. Equivalently, for every morphism in there is at least one lift through that is a -cartesian morphism in the weak sense and the composition of every two composable -cartesian morphisms (in the weak sense) is a -cartesian morphism (in the weak sense).
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 notionation: let
by the overcategory of over the object ;
the corresponding overcategory of over ;
the category whose objects
are objects of eqipped 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. We describe it for two different incarnations of the notion of (∞,1)-category: 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 188.8.131.52.
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
This is HTT remark 184.108.40.206.
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).
Let and be simplicially enriched categories and an 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.
Pullback along Cartesian morphisms
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.
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
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.
Cartesian morphisms and equivalences
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 .
David Roberts: There would surely be an anafunctor version of this, that would require no choices whatsoever. It is unlikely that I would be able to find time to write this up, so my plea goes out to those in the know…
I imagine that there would then be an -version using whatever passes as anafunctors in that setting (dratted memory, failing at the first gate)
Mike Shulman: Yes, there would surely be such a version. (-: The simplest way would be to take the specifications for the anafunctor to be the cartesian morphisms over , with domain and codomain giving the functions and . Unique factorization would give you the values of morphisms.
David Roberts: just stumbled on this old comment - I’m reading Makkai more closely, and I’m convinced that basically anything defined by a universal property is given by a saturated anafunctor. So this is a heads up for posterity, that a map is a fibration iff the fairly obvious span of functors defines a saturated anafunctor?.
- A Cartesian morphism is the special case of an initial lift of a structured cosink when the cosink is a singleton.
The traditional reference is SGA I.6 (written by P. Gabriel and A. Grothendieck)
- chapter 6 in A. Grothendieck, M. Raynaud et al. Revêtements étales et groupe fondamental (SGA I), Lecture Notes in Mathematics 224, Springer 1971 (retyped as math.AG/0206203; published version Documents Mathématiques 3, Société Mathématique de France, Paris 2003)
There are excellent lectures of Vistoli:
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.