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\begin{document}

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\section*{Cartesian morphism}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{category_theory_2}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory_2}

[[!include quasi-category theory contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_categories}{In categories}\dotfill \pageref*{in_categories} \linebreak
\noindent\hyperlink{traditional_definition}{Traditional definition}\dotfill \pageref*{traditional_definition} \linebreak
\noindent\hyperlink{CartInOrdCatReformulation}{Reformulations}\dotfill \pageref*{CartInOrdCatReformulation} \linebreak
\noindent\hyperlink{in_categories_2}{In $(\infty,1)$-categories}\dotfill \pageref*{in_categories_2} \linebreak
\noindent\hyperlink{in_quasicategories}{In quasi-categories}\dotfill \pageref*{in_quasicategories} \linebreak
\noindent\hyperlink{in_categories_3}{In $sSet$-categories}\dotfill \pageref*{in_categories_3} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{pullback_along_cartesian_morphisms}{Pullback along Cartesian morphisms}\dotfill \pageref*{pullback_along_cartesian_morphisms} \linebreak
\noindent\hyperlink{cartesian_morphisms_and_equivalences}{Cartesian morphisms and equivalences}\dotfill \pageref*{cartesian_morphisms_and_equivalences} \linebreak
\noindent\hyperlink{variants}{Variants}\dotfill \pageref*{variants} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Given a [[functor]] $p\colon  X \to Y$ between categories one may ask for each [[morphism]] $f\colon  y_1 \to y_2$ if given a lift of its target

\begin{displaymath}
\itexarray{
    X &&& && \hat y_2
    \\
    \downarrow^p 
    \\
    Y &&& y_1 &\stackrel{f}{\to}& y_2
  }
\end{displaymath}
there is a \emph{universal} lift of $f$

\begin{displaymath}
\itexarray{
    X &&& \hat y_1 &\stackrel{\hat f}{\to}& \hat y_2
    \\
    \downarrow^p 
    \\
    Y &&& y_1 &\stackrel{f}{\to}& y_2
  }
  \,.
\end{displaymath}
There may also be other lifts of $f$, but the universal one is essentially unique, as usual for anything having a [[universal property]]. Specifically, $\hat f$ in $X$ is essentially uniquely determined by its target $\hat y_2$ and its image $f = p(\hat f)$ in $Y$, and is called a \textbf{cartesian morphism}. A morphism which is cartesian relative to $p^{op}\colon X^{op}\to Y^{op}$ is called \textbf{opcartesian} or \textbf{cocartesian}.

If there are enough cartesian morphisms in $Y$, they may be used to define [[functor]]s

\begin{displaymath}
f^* : X_{y_2} \to X_{y_1}
\end{displaymath}
between the [[fiber]]s of $p$ over $y_1$ and $y_2$.

This way a functor $p : X \to Y$ with enough Cartesian morphisms -- called a [[Cartesian fibration]] or [[Grothendieck fibration]] -- determines and is determined by a fiber-assigning functor $Y \to Cat^{op}$.

This has its analog in [[higher category theory|higher categories]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{in_categories}{}\subsubsection*{{In categories}}\label{in_categories}

\hypertarget{traditional_definition}{}\paragraph*{{Traditional definition}}\label{traditional_definition}

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(cartesian morphism)}

Let $p : X \to Y$ be a [[functor]]. A [[morphism]] $f : x_1 \to x_2$ in the [[category]] $X$ is \textbf{strongly cartesian} with respect to $p$ (nowdays often just \emph{cartesian}), or (strongly) \textbf{$p$-cartesian} if for every $x'\in X$, for every $h:x'\to x_2$ and every $u:p(x')\to p(x_1)$ such that $p(h) = p(f) u$, there exists a unique $v:x'\to x_1$ such that $h = f v$ and $u = p(v)$:

\begin{displaymath}
\itexarray{
    \forall x'
    \\
    \downarrow^{\mathrlap{\exists! v}} & \searrow^{\mathrlap{\forall h}}
    \\
    x_1 &\stackrel{f}{\to}& x_2
  }
  \;\;\;
  \;\;\;
  \stackrel{p}{\mapsto}
  \;\;\;
  \;\;\;
  \itexarray{
    p(x')
    \\
    \downarrow^{\mathrlap{\forall u}} & \searrow^{\mathrlap{p(h)}}
    \\
    p(x_1) &\stackrel{p(f)}{\to}& p(x_2)
  }
\end{displaymath}
In imprecise words: for all commuting triangles in $Y$ (involving $p(f)$ as above) and all lifts through $p$ of its 2-[[horn]] to $X$ (involving $f$ 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 $u = id_{p(x_1)}$, 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]] $N(X)$ and $N(Y)$ of the categories, then in terms of diagrams in [[sSet]] this means that the morphism $f : x \to y$ is $p$-cartesian precisely if for all [[horn]] inclusions

\begin{displaymath}
\itexarray{
    \Delta^{\{1,2\}}
    \\
    \downarrow & \searrow^f
    \\
    \Lambda_2[2] &\to& N(X)
    \\
    \downarrow && \downarrow^{\mathrlap{p}}
    \\
    \Delta[2] &\to& N(Y)
  }
\end{displaymath}
such that the last edge of the 2-[[horn]] is the given edge $f$, a unique lift $\sigma$

\begin{displaymath}
\itexarray{
    \Delta^{\{1,2\}}
    \\
    \downarrow & \searrow^f
    \\
    \Lambda_2[2] &\to& N(X)
    \\
    \downarrow &{}^\sigma\nearrow& \downarrow^{\mathrlap{p}}
    \\
    \Delta[2] &\to& N(Y)
  }
\end{displaymath}
exists.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
\textbf{(Grothendieck fibration)}

If for every morphism in $Y$ and every lift of its target there is at least one lift which has as its target the chosen one and is a $p$-cartesian morphism in the strong sense, one says that $p$ is a \textbf{fibered category} (also called [[Grothendieck fibration]]). Equivalently,

\begin{itemize}%
\item ($p$ is [[prefibered category]]) for every morphism in $Y$ and every lift of its target there is at least one lift through $p$ which has as its target the chosen one and is a $p$-cartesian morphism in the weak sense

\end{itemize}
\emph{and}

\begin{itemize}%
\item the composition of every two composable $p$-cartesian morphisms (in the weak sense) is a $p$-cartesian morphism (in the weak sense).

\end{itemize}
\end{defn}
\hypertarget{CartInOrdCatReformulation}{}\paragraph*{{Reformulations}}\label{CartInOrdCatReformulation}

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

\begin{itemize}%
\item $X/x_2$ by the [[overcategory]] of $X$ over the object $x_2$;


\item $Y/p(x_2)$ the corresponding [[overcategory]] of $Y$ over $p(x_2)$;


\item $X/f$ the category whose objects

\begin{displaymath}
Obj(X/f) = 
  \left\{

  \itexarray{
     && a
     \\
     &\swarrow && \searrow
     \\
     x_1 &&\stackrel{f}{\to}&& x_2
  }
  \right\}
\end{displaymath}
are objects $a$ of $X$ eqipped with morphisms to $x_1$ and $x_2$ such that the obvious triangle commutes, and whose morphisms are morphisms between these tip objects such that all diagrams in sight commute.


\item similarly for $Y/p(f)$.



\end{itemize}
\begin{prop}
\label{}\hypertarget{}{}
The condition that $f \in Mor X$ is a Cartesian morphism with respect to $p : X \to Y$ is equivalent to the condition that the functor

\begin{displaymath}
\phi : X/f \to X/{x_2} \times_{Y/{p(x_2)}} Y/p(f)
\end{displaymath}
into the (strict) [[pullback]] of the obvious projection $X/{x_2} \to Y/p(x_2)$ along the projection $Y/p(f) \to Y/p(x_2)$ induced by the commutativity of

\begin{displaymath}
\itexarray{
    X/f &\stackrel{\phi_2}{\to}& Y/p(f)
    \\
    {}^{\mathllap{\phi_1}}\downarrow && \downarrow
    \\
    X/{x_2} &\to& Y/{p(x_2)}
  }
\end{displaymath}
is a [[k-surjective functor|surjective equivalence]], and this in turn is equivalent to it being an [[isomorphism]] of categories.

\end{prop}
\begin{proof}
It is immediate to see that $\phi$ being an isomorphism of categories is equivalent to the condition that $f$ is a Cartesian morphism. We discuss that just the condition that $\phi$ is a surjective equivalence already implies that it is an isomorphism of categories.

So assume now that $\phi$ is a surjective equivalence.

Notice that objects in the pullback category are compatible pairs

\begin{displaymath}
\left(
  \left(
  \itexarray{
     && a 
     \\
     &&& \searrow
     \\
     x_1 &&\stackrel{f}{\to}&& x_2
  }
  \right) \in X/_{x_2}
  \;\;\;\;\;\;\,,\;\;\;\;\;\;
  \left(
  \itexarray{
     && b 
     \\
     & \swarrow && \searrow
     \\
     p(x_1) &&\stackrel{p(f)}{\to}&& p(x_2)
  }
  \right)
   \in Y/p(f)
  \right)
  \,.
\end{displaymath}
We have that $\phi$ being \emph{surjective} on object means that every such pair is in the image of some object

\begin{displaymath}
\left(
  \itexarray{
     && a 
     \\
     &{}^{\mathllap{g}}\swarrow&& \searrow
     \\
     x_1 &&\stackrel{f}{\to}&& x_2
  }
  \right)
  \in X_{f}
  \,,
\end{displaymath}
and hence that every filler \emph{exists} . Assume two such fillers $g$ and $g'$. 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 $X/f$ connecting them

\begin{displaymath}
\itexarray{
     && a
     \\
     &{}^{\mathllap{g}}\swarrow & \downarrow^{\mathrlap{h}} & \searrow
     \\
     && a
    \\
     \downarrow & {}^{\mathllap{g'}}\swarrow && \searrow & \downarrow
     \\
     x_1 &&\stackrel{f}{\to}&& x_2
  }
\end{displaymath}
such that this maps under $\phi$ to the identity morphism in the pullback category. But in particular this maps to the morphism

\begin{displaymath}
\itexarray{
     && a
     \\
     && \downarrow^{\mathrlap{h}} & \searrow
     \\
     && a
    \\
     & && \searrow & \downarrow
     \\
      &&&& x_2
  }
\end{displaymath}
in $X/{x_2}$ and evidently is the identity there if and only if $h$ is the identity. Hence this maps also to the identity in the pullback category if and only if $h$ is the identity. So $h$ must be the identity. So if two lifts of an object through the surjective equivalence $\phi$ exist, they must already be equal. Hence the surjective equivalence $\phi$ is even an isomorphism on objects and hence an isomorphism of categories.

\end{proof}
\hypertarget{in_categories_2}{}\subsubsection*{{In $(\infty,1)$-categories}}\label{in_categories_2}

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 [[simplicially enriched categories|sSet categories]].

\hypertarget{in_quasicategories}{}\paragraph*{{In quasi-categories}}\label{in_quasicategories}

We formulate a notion \textbf{cartesian edge} or \emph{cartesian morphism} in a [[simplicial set]] $X$ relative to a morphism $p : X \to Y$ of simplicial sets. In the case that these simplicial sets are [[quasi-category|quasi-categories]] -- i.e. simplicial set incarnations of [[(∞,1)-category|(∞,1)-categories]] -- this yields a notion of cartesian morphisms in $(\infty,1)$-categories.

Let $p : X \to Y$ be a morphism of [[simplicial set]]s. Let $f : x_1 \to x_2$ be an edge in $X$, i.e. a morphism $f : \Delta^1 \to X$.

Recall the notion of [[over quasi-category]] obtained from the notion of [[join of quasi-categories]]. Using this we obtain [[simplicial set]]s $X/f$, $X/{x_2}$, $S/p(f)$ and $S/p(x_2)$ in generalization of the categories considered in the above definition of cartesian morphisms in categories.

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(cartesian edge in a simplicial set)}

Let $p : X \to Y$ be an [[inner Kan fibration]] of simplicial sets.

Then a [[morphism]] $f : x \to y$ in $X$ is \textbf{$p$-cartesian} if the induced morphism

\begin{displaymath}
X_{/f} \to X_{/y} \times_{Y_{/p(y)}} Y_{/p(f)}
\end{displaymath}
into the [[pullback]] in [[sSet]] is an acyclic [[Kan fibration]].

\end{defn}
This is [[Higher Topos Theory|HTT, def 2.4.1.1]].

\begin{prop}
\label{}\hypertarget{}{}
The morphism $f : x \to y$ as above, for $p : X \to Y$ an [[inner fibration]], is $p$-cartesian precisely if for all $n \geq 2$ and all right outer [[horn]] inclusions

\begin{displaymath}
\itexarray{
    \Delta^{\{n-1,n\}}
    \\
    \downarrow & \searrow^f
    \\
    \Lambda[n]_n &\to& X
    \\
    \downarrow && \downarrow^{\mathrlap{p}}
    \\
    \Delta[n] &\to& Y
  }
\end{displaymath}
(with $\Lambda[n]_n$ the $n$th [[horn]] of the $n$-[[simplex]]) such that the last edge of the horn is the given edge $f$, a lift $\sigma$

\begin{displaymath}
\itexarray{
    \Delta^{\{n-1,n\}}
    \\
    \downarrow & \searrow^f
    \\
    \Lambda[n]_n &\to& X
    \\
    \downarrow &{}^\sigma\nearrow& \downarrow^{\mathrlap{p}}
    \\
    \Delta[n] &\to& Y
  }
\end{displaymath}
exists.

\end{prop}
This is [[Higher Topos Theory|HTT remark 2.4.1.4]].

\begin{remark}
\label{}\hypertarget{}{}
This means that an [[inner fibration]] $p : X \to Y$ with a collection of $p$-cartesian morphisms in $X$ specified satisfies the same kind of condition as a \emph{[[right fibration]]} , the only difference being that not \emph{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]].

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
If $p : X \to Y$ is an [[inner fibration]] of [[quasi-categories]] then a morphism $f : x \to y$ in $X$ is $p$-Cartesian precisely if for all objects $a$ in $X$ the diagram

\begin{displaymath}
\itexarray{
      Hom_X(a,x) &\stackrel{Hom_X(a,f)}{\to}& Hom_X(a,y)
      \\
      \downarrow && \downarrow
      \\
      Hom_Y(p(a), p(x)) &\stackrel{Hom_Y(p(a), p(f))}{\to}& 
      Hom_Y(p(a), p(y))
    }
\end{displaymath}
of [[hom-object in a quasi-category|hom-objects in a quasi-category]] is a [[homotopy pullback]] square (in [[sSet]] equipped with its [[model structure on simplicial sets|standard model structure]]).

\end{prop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop. 2.4.4.3]].

\end{proof}
\hypertarget{in_categories_3}{}\paragraph*{{In $sSet$-categories}}\label{in_categories_3}

Let $C$ and $D$ be [[simplicially enriched categories]] and $F : C \to D$ a [[sSet]]-[[enriched functor]].

\begin{defn}
\label{}\hypertarget{}{}
A morphism $f : x \to y \in C$ is \textbf{$F$-cartesian} if it is so under the [[homotopy coherent nerve]] $N : sSet Cat \to sSet$ in the sense of quasi-categories above, i.e. if

\begin{displaymath}
N(C)_{/f} \to N(C)_{/y} \times_{N(D)_{/F(y)}} N(D)_{/F(f)}
\end{displaymath}
is an acyclic [[Kan fibration]].

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
If $C$ and $D$ are enriched in [[Kan complex]]es and if $F$ is hom-wise a [[Kan fibration]], then

\begin{itemize}%
\item $N(F) : N(C) \to N(D)$ is an [[inner fibration]];


\item a morphism $f :x \to y$ in $N(C)$ is an $N(F)$-cartesian morphism precisely if for all objects $a$ in $C$ the diagram

\begin{displaymath}
\itexarray{
    C(a,x) &\stackrel{C(a,f)}{\to}& C(a,y)
    \\
    \downarrow && \downarrow
    \\
    D(F(a), F(x)) &\stackrel{C(F(a), F(f))}{\to}& 
    D(F(a), F(y))
  }
\end{displaymath}
is a [[homotopy pullback]] square in [[sSet]] equipped with its [[model structure on simplicial sets|standard model structure]].



\end{itemize}
\end{prop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop. 2.4.1.10]].

\end{proof}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{pullback_along_cartesian_morphisms}{}\subsubsection*{{Pullback along Cartesian morphisms}}\label{pullback_along_cartesian_morphisms}

\begin{lemma}
\label{}\hypertarget{}{}
For $p : \mathcal{E} \to \mathcal{C}$ a functor, if in a diagram

\begin{displaymath}
\itexarray{
    A &\stackrel{f}{\to}& B
    \\
    {}^{\mathllap{g}}\downarrow && \downarrow^{\mathrlap{h}}
    \\
    C &\stackrel{k}{\to}& D
  }
\end{displaymath}
in $\mathcal{E}$ the two vertical morphisms are vertical with respect to $p$ (meaning that $p(g) = Id_p(A)$ and $p(h) = Id(B)$) and if the two horizontal morphisms are $p$-Cartesian morphisms, then this square is a [[pullback]] square.

\end{lemma}
\begin{proof}
If

\begin{displaymath}
\itexarray{
    Q &\stackrel{}{\to}& B
    \\
    \downarrow && \downarrow^{\mathrlap{h}}
    \\
    C &\stackrel{k}{\to}& D
  }
\end{displaymath}
is another cone over $C \to D \leftarrow B$, then its image under $p$ is

\begin{displaymath}
\itexarray{
    p(Q)
    \\
    \downarrow & \searrow
    \\
    p(C) &\stackrel{p(k)}{\to}& D
  }
  \,.
\end{displaymath}
Since $p(f) = p(k)$, another lift of the right horn of this is given by

\begin{displaymath}
\itexarray{
    Q
    \\
    & \searrow
    \\
    A &\stackrel{f}{\to}& B
  }
\end{displaymath}
which gives a unique filler $Q \to A$ by the fact that $f$ is Cartesian.

But this produces now two fillers -- namely the original $Q \to C$ and the $Q \to A \to C$ just obtained -- of the horn

\begin{displaymath}
\itexarray{
    Q &\to& B
    \\
    && \downarrow
    \\
    C &\stackrel{k}{\to}& D
  }
\end{displaymath}
over

\begin{displaymath}
\itexarray{
    p(Q)
    \\
    \downarrow & \searrow
    \\
    p(C) &\stackrel{p(k)}{\to}& D
  }
  \,.
\end{displaymath}
Since $k$ is Cartesian, these two fillers must be equal. This means that the morphism $Q \to A$ is a cone morphism and unique as such. Hence the original square is a pullback.

\end{proof}
This appears as [[Elephant|Elephant, lemma 1.3.3]].

\hypertarget{cartesian_morphisms_and_equivalences}{}\subsubsection*{{Cartesian morphisms and equivalences}}\label{cartesian_morphisms_and_equivalences}

\begin{lemma}
\label{}\hypertarget{}{}
For $C$ a [[category]], a [[morphism]] in $C$ is cartesian with respect to the [[terminal object|terminal]] [[functor]] $C \to *$ precisely if it is an [[isomorphism]].

In particular all [[identity]] morphisms are cartesian.

\end{lemma}
This is trivial to see. The analog statement holds also for [[quasi-categories]], where it is rather more nontrivial and quite useful:

\begin{prop}
\label{}\hypertarget{}{}
For $C$ a [[quasi-category]], a morphism in $C$ is cartesian with respect to the [[terminal object|terminal]] morphism $C \to *$ precisely if it is an [[equivalence in a quasi-category|equivalence]].

More generally, for $p : X \to Y$ an [[inner fibration]], a morphism $f$ in $X$ is an [[equivalence in a quasi-category|equivalence]] precisely if it is $p$-cartesian and $p(f)$ is an equivalence in $Y$.

\end{prop}
\begin{proof}
The first statement is a proposition of [[Andre Joyal]], slightly reformulated in the language of cartesian morphisms. It appears as [[Higher Topos Theory|HTT, prop 1.2.4.3]]. A proof appears below [[Higher Topos Theory|HTT, corollary 2.1.2.2]].

The second statement is [[Higher Topos Theory|HTT, prop. 2.4.1.5]].

\end{proof}
\hypertarget{variants}{}\subsection*{{Variants}}\label{variants}

[[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\ldots{}

I imagine that there would then be an $(\infty,1)$-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 $|f^*|$ for the anafunctor $f^*$ to be the cartesian morphisms over $f$, with domain and codomain giving the functions $\sigma$ and $\tau$. 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]].

\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item A Cartesian morphism is the special case of an [[initial lift]] of a structured [[cosink]] when the cosink is a singleton.

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The traditional reference is SGA I.6 (written by [[P. Gabriel]] and A. Grothendieck)

\begin{itemize}%
\item chapter 6 in A. Grothendieck, M. Raynaud et al. \emph{Rev\^e{}tements \'e{}tales et groupe fondamental} ([[SGA1]]), Lecture Notes in Mathematics \textbf{224}, Springer 1971 (retyped as \href{http://arxiv.org/abs/math/0206203}{math.AG/0206203}; published version Documents Math\'e{}matiques \textbf{3}, Soci\'e{}t\'e{} Math\'e{}matique de France, Paris 2003)

\end{itemize}
There are excellent lectures of Vistoli:

\begin{itemize}%
\item [[Angelo Vistoli]], \emph{Grothendieck topologies, fibered categories and descent theory}, p. 1--104 in [[FGA explained]], \href{http://www.ams.org/mathscinet-getitem?mr=2223406}{MR2223406}; \href{http://arxiv.org/abs/math/0412512}{math.AG/0412512}.

\end{itemize}
For the 1-categorical case see for instance section B1.3 of

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]}

\end{itemize}
The $(\infty,1)$-categorical version is in section 2.4 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}
See also the references at [[Grothendieck fibration]].

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