LECTURE . FIBRATIONS
In our last lecture, we started to discuss the homotopy theory of quasicategories and of marked simplicial sets. We
ended with the following statement.
.. eorem. ere is a simplicial model structure on the category sSet+ , in which:
(..) e simplicial structure is given by
Map♯ (X, Y)m = Mor(X × (Δm )♯ , Y).
(..) e cofibrations are precisely those marked maps that are monomorphisms on their underlying simplicial sets.
(..) e fibrant objects are precisely the marked simplicial sets of the form X♮ for some quasicategory X.
We’ll state a generalization of this result, and we’ll give some of the elements of the proof.
e first kind of fibration we’ll consider is that of an inner fibration; this can be thought of as a mild relative version
of the notion of a quasicategory.
.. Definition. A map X . S of simplicial sets is said to be an inner fibration if it satisfies the right liing property
with respect to the inner horns Λnk . Δn for n ≥ 2 and 1 ≤ k ≤ n − 1.
.. Lemma. A map X .
a quasicategory.
S of simplicial sets is an inner fibration if and only if, for any simplex σ ∈ Sn , the fiber Xσ is
Proof. In one direction, inner fibrations are closed under pullback. In the other direction, suppose that, for any simplex σ ∈ Sn , the fiber Xσ is a quasicategory. Consider a square
Λ.nk
η
.
n
Δ.
X.
σ
S.
Since Xσ is a quasicategory, and since the hor-filler in Δn is unique, the map Xσ .
η.
Δn admits a section that extends
□
We may think of an inner fibration X . S as a family of quasicategories Xs , one for each vertex s ∈ S0 ; however,
the manner in which the family varies as one passes from vertex to vertex is a little subtle. An edge s . t is S doesn’t
necessarily give rise to a functor between Xs and Xt ; rather, one obtains a correspondence from Xs to Xt . We will not
pursue this point further. Instead, we’ll study a key special case.
.. Definition. Suppose p : X .
n ≥ 2 and every square
S an inner fibration. We say that an edge φ : Δ1 .
Λ.nn
η
X.
.
n
Δ.
S.
such that η|Δ{n−1,n} = φ, there exists an extension η : Δn .

X such that η = η|Λnn .
X is p-cartesian if for every
Dually, we say that φ is p-cocartesian if for every n ≥ 2 and every square
Λ.n0
η
X.
.
Δ.n
S.
such that η|Δ{0,1} = φ, there exists an extension η : Δn .
X such that η = η|Λn0 .
.. Lemma. Suppose p : X .
S an inner fibration. If p is an isomorphism, every edge is p-cartesian and p-cocartesian.
.. Lemma. Suppose p : X .
S an inner fibration. If
Y.
X.
.
q
T.
p
S.
is a pullback square, then an edge φ of Y is q-(co)cartesian if its image g(φ) is p-(co)cartesian.
.. Lemma. Suppose p : X . S and q : S . T two inner fibrations, and suppose φ an edge of X such that p(φ) is
q-(co)cartesian. en φ is p-(co)cartesian just in case it is q ◦ p-(co)cartesian.
.. Exercise. Suppose S a quasicategory, and suppose p : X . S an inner fibration. Show that an edge φ : Δ1 .
is an equivalence of X if and only if φ is p-(co)cartesian and p(φ) is an equivalence of S.
.. Proposition. Suppose S a quasicategory, suppose p : X . S an inner fibration, and suppose σ : Δ2 .
simplex.
Suppose σ|Δ{1,2} is p-cartesian. en σ|Δ{0,1} is p-cartesian if and only if σ|Δ{0,2} is so.
Dually, suppose σ|Δ{0,1} is p-cocartesian. en σ|Δ{1,2} is p-cocartesian if and only if σ|Δ{0,2} is so.
X
X a 2-
Proof. Write [x, y, z] for the vertices of σ. For clarity, let us prove the relevant liing conditions only in the case n = 2.
Suppose that x . y is p-cartesian, suppose u . z an edge, and suppose η ∈ S2 with vertices [p(u), p(x), p(z)].
Since S is a quasicategory, we may extend p(σ) and η to a 3-simplex τ ∈ S3 with vertices [p(u), p(x), p(y), p(z)]
contained therein. Since y . z is p-cartesian, we may li the face τ|Δ{0,2,3} to X, and since x . y is p-cartesian, we
may li the face τ|Δ{0,1,2} to X. Now since p is an inner fibration, we may li of the full 3-simplex τ and in particular
the 2-simplex η.
Conversely, suppose that x . z is p-cartesian, suppose u . y an edge, and suppose η ∈ S2 with vertices
[p(u), p(x), p(y)]. Since S is a quasicategory, we may extend p(σ) and η to a 3-simplex τ ∈ S3 given by the vertices [p(u), p(x), p(y), p(z)]. Since p is an inner fibration, we may li the face τ|Δ{0,2,3} to X, and since x . z is
p-cartesian, we may also li the face τ|Δ{0,1,3} to X. Now since y . z is p-cartesian, we may li the full 3-simplex τ
and in particular the 2-simplex η.
□
.. Definition. An inner fibration p : X . S is said to be a cartesian fibration if for any edge φ : x .
any vertex y ∈ Xy , there exits a p-cartesian edge φ : x . y of Xφ .
Dually, a cocartesian fibration is the opposite of a cartesian fibration.
.. Proposition. e following are equivalent for an inner fibration p : X .
y of S and
S.
(..) e map p is a cartesian fibration, and every edge in X is p-cartesian.
(..) e map p satisfies the right liing property with respect to all right horns Λnk . Δn , with n ≥ 1 and 1 ≤
k ≤ n.
(..) e map p is a cartesian fibration, and the fiber Xs over any vertex s ∈ S0 is a Kan complex.

Proof. First, that every edge is p-cartesian is the right liing property with respect to the inclusions Λnn . Δn for n ≥
2, and the condition that p-cartesian maps exist is the right liing property with respect to the inclusion Λ11 . Δ1 .
So the first two conditions are equivalent.
at the second condition implies the third follows from the fact that if an object satisfies the right liing property
with respect to all right horns Λnk . Δn , with n ≥ 1 and 1 ≤ k ≤ n, it is a Kan complex. is follows from our
result last time: X is a Kan complex just in case hX is a groupoid, which is in turn equivalent to admitting all right
horn-fillers.
Finally, we must show that if p is cartesian fibration whose fibers are Kan complexes, then every edge φ : x . y
of X is p-cartesian. By assumption there is a p-cartesian edge φ′ : x′ . y covering p(φ), and so there is a 2-simplex
[x, x′ , y] covering the degenerate 2-cell [p(x), p(x), p(y)]. In particular, the map x . x′ lies in Xp(x) , whence it is an
equivalence. Now it’s easy to see that φ is p-cartesian as well.
□
Here is a key construction, to which we will appeal many times.
.. eorem. Suppose p : X . S a cartesian fibration, and suppose q : Y .
maps Tp Y . S and Tp Z . S via the uniersal properties
MorS (K, Tp Y) ∼
= MorS (X ×S K, Y).
S a cocartesian fibration. Define new
en Tp Y . S is a cocartesian fibration, and an edge of Tp Y is cocartesian just in case the induced map Δ1 ×S X .
carries p-cartesian edges to q-cocartesian edges.
Y
.. Notation. When p : X . S is a cartesian fibration; write X♮ for the marked simplicial set in which the marked
edges are the p-cartesian maps.
.. eorem. Suppose S a simplicial set. ere is a simplicial model structure on the category sSet+
/S , in which:
(..) e simplicial structure is given by
Map♯S (X, Y)m = MorS (X × (Δm )♯ , Y).
(..) e cofibrations are precisely those marked maps over S that are monomorphisms on their underlying simplicial
sets.
(..) e fibrant objects are precisely the marked simplicial sets of the form X♮ for some cartesian fibration X . S.
Sketch of proof. Consider the set of cofibrantly generated model structures on sSet+
/S in which the cofibrations are
precisely the monomorphisms; order this set by inclusion of the classes of weak equivalences. e resulting poset
contains a least element; this is the minimal model structure of Cisinski. Now Bousfield localize this model category
so that the fibrant objects are precisely of the form form X♮ for some cartesian fibration X . S. To do this, let S be
the union of the following four sets:
{(Λnk )♭ .
(Δn )♭ | n ≥ 2, 1 ≤ k ≤ n − 1}
(Δn , S) | n ≥ 1, S = s0 (Δn0 ) ∪ Δ{n−1,n} }
{(Λnn , S ∩ Λnn ) .
{(Δ2 , (Λ21 )1 ) .
{K♭ .
(Δ2 )♯ }
K♯ | K is a Kan complex}
One may confirm that this model structure is simplicial by inspecting the relevant pushout products and showing
that they are transfinite compositions of pushouts of elements of these four sets.
□
Obviously, this model structure is not self-dual. e dual model structure has as its fibrant objects the cocartesian
fibrations.
.. Example. For any relative category (C, wC), form the marked simplicial set NC, and mark the edges corresponding to weak equivalences. Form a fibrant replacement in sSet+ with the model structure described above. e
result is a quasicategory N(C, wC) that contains the homotopical information of (C, wC). Doing this with the relative category QCat yields Cat∞ , the quasicategory of ∞-categories.

.. Example. Here’s a universal example. Consider the forgetful functor NQCat . NsSet, and form the BousfieldKan model of its homotopy colimit. is is some (large) simplicial set U′ with a map U′ . NQCat. Now form a
fibrant replacement of the composite map U′ . Cat∞ in sSet+
Cat∞ with the cocartesian model structure. e result
is a cocartesian fibration U . Cat∞ whose fiber over a vertex C is (equivalent to) the quasicategory C itself.
Now for any diagram of ∞-categories X : S . Cat∞ , one may form a corresponding cocartesian fibration
S ×Cat∞ U . S, whose fiber over any vertex s ∈ S0 is the quasicategory X(s). It turns out that this assignment
establishes an equivalence of quasicategories
Fun(S, Cat∞ ) ≃ N(sSet+
/S )f ,
where the latter is given this cocartesian model structure.
