equivalences in/of $(\infty,1)$-categories
Cartesian fibrations are one of the types of fibrations of quasi-categories.
A Cartesian fibration of quasi-categories – or more generally of simplicial sets – is a morphism that generalizes the notion of Grothendieck fibration from category theory to (∞,1)-category theory, specifically with (∞,1)-categories incarnated as quasi-categories:
It is an inner fibration that lifts also all right outer horn inclusions whose last edge is a cartesian morphism, and such that there is a sufficient supply of cartesian morphisms.
An inner fibration $p : C \to D$ may be thought of as a family of (infinity,1)-categories $(C_d)_{d \in D}$ which is functorial in $d$ only in the sense of correspondences. Then the condition of $p$ being a cartesian fibration ensures that the family is actually functorial. More precisely, if an (∞,1)-functor $p : C \to D$ is a Cartesian fibration, then it is possible to interpret its value over any morphism $f : d_1 \to d_2$ in $D$ as an (∞,1)-functor $p^{-1}(f) : p^{-1}(d_2) \to p^{-1}(d_1)$ between the fibers $p^{-1}(d_2)$ and $p^{-1}(d_1)$ over its source and target objects.
This way a Cartesian fibration $p : C \to D$ determines and is determined by an (∞,1)-functor $D^{op} \to (\infty,1)Cat$ into the (∞,1)-category of (∞,1)-categories. This is the content of the (∞,1)-Grothendieck construction.
Cartesian fibrations over $S$ are the fibrant objects in the model structure on marked simplicial over-sets over $S$.
A morphism $p : X \to Y$ in sSet is a Cartesian fibration (or Grothendieck fibration) if
it is an inner fibration
for every edge $f : x \to y$ of $Y$ and for every lift $\hat {y}$ of $y$ (that is: $p(\hat{y})=y$) there is a Cartesian edge $\hat{f} : \hat{x} \to \hat{y}$ in $X$ lifting $f$ (that is: such that $p(\hat f) = f$)
The morphism is a cocartesian fibration (or Cartesian opfibration, Grothendieck opfibration) if the opposite $p^{op} : X^{op} \to Y^{op}$ is a Cartesian fibration.
This is HTT, def. 2.4.2.1.
We have:
Every isomorphism of simplicial sets is a Cartesian fibration.
The composite of two Cartesian fibrations is again a Cartesian fibration.
This is HTT, prop. 2.4.2.3.
Every Cartesian fibration is a fibration in the Joyal-model structure for quasi-categories.
This is HTT, prop. 3.3.1.7.
Since a Cartesian fibration is in particular an inner fibration and since inner fibrations are stable under pullback in sSet, it follows that for $p : E \to C$ a Cartesian fibration, the fiber $E_x$ over every point $x \in C_0$ is a quasi-category
The difference between inner fibrations and Cartesian fibrations is that only for Cartesian fibrations is it generally guaranteed that these fibers over the points are functorially related over the morphisms in $C$. This is the content of the (∞,1)-Grothendieck construction.
But moreover:
The pullback of a Cartesian fibration in sSet is again a Cartesian fibration.
This is HTT, prop. 2.4.2.3.
We know from the discussion at inner fibration that the pullback is an inner fibration. It remains to check if it has enough Cartesian morphisms. By HTT, prop 2.4.1.3 we have that in a pullback diagram
a morphism $f \in E'$ is $p'$-Cartesian if $q(f)$ is $p$-Cartesian. Since the morphisms of $E'$ are pairs of morphisms $(\gamma, \hat f) \in C'_1 \times E_1$ and since by assumption $p$ is a Cartesian fibration, there is for $\gamma \in C'_1$ and $y \in E'_0$ such that $p'(y)$ is the target of the morphism $\gamma$ a Cartesian lift $\hat f \in E$ of $k(\gamma)$ such that $q(y)$ is the target of $\hat f$. Hence a Cartesian lift $(\gamma, \hat f)$ of $\gamma$ in $E'$ having $y$ as target.
We can test locally if a morphism is a Cartesian fibration:
An inner fibration $p : X \to Y$ is Cartesian precisely if for every $n$-simplex $\sigma : \Delta[n] \to Y$ the sSet-pullback $\sigma^* p : X \times_Y \Delta[n]$ in
is a Cartesian fibration.
This is HTT, cor. 2.4.2.10.
The pullback in sSet of a weak equivalence in the Joyal-model structure for quasi-categories along a Cartesian fibration is again a Joyal-weak equivalence
This is HTT, prop 3.3.1.3
Equivalences in $sSet_{Joyal}$ are stable under pullback along Cartesian fibrations:
if
is a pullback square in sSet with $T \to S$ a weak equivalence in $sSet_{Joyal}$ and $X \to S$ a Cartesian fibration, then $X \times_S T \to X$ is also a weak equivalence.
This is HTT, prop. 3.3.1.3.
The following proposition asserts that the ordinary pullback (in sSet) of Cartesian fibrations already models the correct homotopy pullback.
Let
be an pullback diagram in sSet of quasi-categories, where $X' \to S'$ is a Cartesian fibration. Then this is already a homotopy pullback diagram with respect to the model structure for quasi-categories.
This is HTT, prop 3.3.1.4. We factor the bottom morphism as $S \stackrel{\simeq}{\to} T \to S'$ into a weak equivalence and a fibration in $sSet_{Joyal}$. Then the right square in
is the ordinary pullback over a fibrant replacement of the original diagram hence is a homotopy pullback. The claim follows thus if $X \to T \times_{S'} X'$ is a weak equivalence, which it is by one of the above lemmas.
The notion of right fibration is a special case of that of Cartesian fibration:
The following are equivalent:
$p : X \to Y$ is a Cartesian fibration and every edge in $X$ is $p$-Cartesian
$p$ is a right fibration;
$p$ is a Cartesian fibration and every fiber is a Kan complex.
This is HTT, prop. 2.4.2.4.
There are also the “categorical fibrations”, the fibrations in the Joyal model structure for quasi-categories on $sSet$. These turn out not to have much of an intrinsic category theoretic meaning. By the following proposition one can understand the notion of Cartesian fibration as a suitable refinement of the notion of categorical fibration
Every Cartesian $p : X \to Y$ in sSet is a fibration with respect to the Joyal model structure for quasi-categories on sSet.
This is HTT, prop. 3.3.1.7.
However, when the base is an $\infty$-groupoid, then the difference between Cartesian fibrations and categorical fibrations disappears:
Let $p : X \to Y$ be a morphism of simplicial sets where $Y$ is a Kan complex. Then the following are equivalent:
$p$ is a Cartesian fibration
$p$ is a coCartesian fibration
$p$ is a categorical fibration
This is HTT, prop. 3.3.1.8.
There is however another model category structure, which does model Cartesian fibrations.
Let $sSet^+/S$ be the overcategory of the category of marked simplicial sets over $S$, equipped with the model structure on marked simplicial over-sets.
An object $X \to S$ is fibrant in that model category precisely if
the underlying morphism of simplicial sets $X \to S$ is a Cartesian fibration;
the marked edges in $X$ are precisely the Cartesian morphisms.
This is HTT, prop. 3.1.4.1.
By the (infinity,1)-Grothendieck construction a Cartesian fibration $K \to \Delta[1]$ corresponds to a morphism $\Delta[1]^{op} \to (\infty,1)Cat$, hence to an (infinity,1)-functor $f : C \to D$.
Obtaining this through the straightening functor above is tedious, but there is a more immediate way to characterize $f$:
For $p : K \to \Delta[1]$ a Cartesian fibration with specified equivalences
and
an (infinity,1)-functor $f : D \to C$ is associated to $p$ if there exists a commuting diagram
such that
$s|_{D \times \{1\}} = h_1$;
$s|_{D \times \{0\}} = h_0 \circ f$
for every object $x$ of $D$ we have that $s|_{\{x\} \times \Delta[1]}$ is a p-Cartesian morphism of $K$.
This is HTT, def. 5.2.1.1.
If $p : K \to \Delta[1]$ is both a Cartesian fibration as well as a coCartesian fibration, then it determines $(\infty,1)$-functors in both directions
Such a pair is a pair of adjoint (infinity,1)-functors.
… for the moment see HTT, section 3.2.2 …
for the moment see
Cartesian fibration, Cartesian fibration of dendroidal sets
Section 2.4.2 in