For ordinary categories there is the notion of
Grothendieck fibration between two categories.
and the special case of a category fibered in groupoids.
The analog of this for quasi-categories are
inner fibrations – these correspond to bundles of quasi-categories : an inner fibration over the interval is the quasi-categorical analog of a cograph of a profunctor: it characterizes the fibers over the endpoints as quasi-categories. Notably having an inner fibration over the point says precisely that is a quasi-category.
categorical fibrations – these appear as the fibrations in the sense of model category theory in the Joyal model structure for quasi-categories . But they have no particular intrinsic meaning in higher category theory. In fact, there is also the model structure on marked simplicial sets which is Quillen equivalent to and in which the model-theoretic fibrations coincide precisely with the Cartesian fibrations that do have an intrinsic category theoretic meaning.
We list the different definitions in the order of their generality. The examples of each definition are also examples of the following definitions.
All morphisms in the following are morphisms of simplicial sets.
A morphism with left lifting property against all Kan fibrations is called anodyne.
A morphism of simplicial sets is a left fibration or left Kan fibration if it has the right lifting property with respect to all horn inclusions except the right outer ones. It is a right fibration or right Kan fibration if its extends against all horns except the left outer ones.
This is a Kan complex-enriched category and as such an incarnation of the (∞,1)-category of right fibrations. It is modeled by the model structure for right fibrations. For details on this see the discussion at (∞,1)-Grothendieck construction.
for every edge of
and every lift of (that is, ),
there is a Cartesian edge in lifting .
(HTT, def 220.127.116.11)
(HTT, p. 81).
The morphisms with the left lifting property against all inner fibrations are called inner anodyne.
By the small object argument we have that every morphism of simplicial sets may be factored as
with a left/right/inner anodyne cofibraiton and accordingly a left/right/inner Kan fibration.
Under the operation of forming the opposite quasi-category, left fibrations turn into right fibrations, and vice versa: if is a left fibration then is a right fibration.
Therefore it is sufficient to list properties of only one type of these fib rations, that for the other follows.
In classical homotopy theory, a continuous map of topological spaces is said to have the homotopy lifting property if it has the right lifting property with respect to all morphisms for the standard interval and every commuting diagram
there exists a lift making the two triangles
commute. For the point this can be rephrased as saying that the universal morphism induced by the commuting square commuting square
A morphism of simplicial sets is a left fibration precisely if the canonical morphism
is a trivial Kan fibration.
The notion of right fibration of quasi-categories generalizes the notion of category fibered in groupoids. This follows from the following properties.
It follows that the fiber of every right fibration over every point , i.e. the pullback
is a Kan complex.
For and quasi-categories that are ordinary categories (i.e. simplicial sets that are nerves of ordinary categories), a morphism is a right fibration precisely if the correspunding ordinary functor exhibits as a category fibered in groupoids over .
This is HTT, prop. 18.104.22.168.
A canonical class of examples of a fibered category is the codomain fibration. This is actually a bifibration. For an ordinary category, a bifiber of this is just a set. For an -category it is an -groupoid. Hence fixing only one fiber of the bifibration should yield a fibration in -groupoids. This is asserted by the following statement.
The collection of left anodyne morphisms (those with left lifting property against left fibrations) is equivalently for the following choices of :
the collection of all left horn inclusions
with and there is a unique morphism making the diagram
This is HTT, prop. 22.214.171.124
This is immediate, but let’s spell it out:
In any commutative diagram
by the above the bottom morphism is already unqiely specified by the remaining diagram.
By the above there exists a unique lift into
and by uniqueness of lifts into this must also make the lower square commute
chapter 2 of