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\begin{document}

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\section*{pullback}

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

\hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits}

[[!include infinity-limits - contents]]

\hypertarget{equality_and_equivalence}{}\paragraph*{{Equality and Equivalence}}\label{equality_and_equivalence}

[[!include equality and equivalence - 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_category_theory}{In category theory}\dotfill \pageref*{in_category_theory} \linebreak
\noindent\hyperlink{in_type_theory}{In type theory}\dotfill \pageref*{in_type_theory} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{saturation}{Saturation}\dotfill \pageref*{saturation} \linebreak
\noindent\hyperlink{PullbackFunctor}{Pullback functor}\dotfill \pageref*{PullbackFunctor} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In the [[category]] [[Set]] a `pullback' is a [[subset]] of the [[cartesian product]] of two [[sets]]. Given a [[diagram]] of [[sets]] and [[functions]] like this:

\begin{displaymath}
\itexarray{
     && A &&&& B
      \\
      & 
      && {}_{f}\searrow
       &
       & \swarrow_g
      && 
     \\
     
     &&&&
     C
     &&&&
     
  }
\end{displaymath}
the `pullback' of this diagram is the [[subset]] $X \subseteq A \times B$ consisting of pairs $(a,b)$ such that the [[equation]] $f(a) = g(b)$ holds.

A pullback is therefore the [[categorical semantics]] of an \emph{[[equation]]}.

This construction comes up, for example, when $A$ and $B$ are [[fiber bundles]] over $C$: then $X$ as defined above is the [[product]] of $A$ and $B$ in the category of fiber bundles over $C$. For this reason, a pullback is sometimes called a \textbf{fibered product} (or \emph{fiber product} or \emph{fibre product}).

In this case, the fiber of $A \times_C B$ over a [[generalized element|(generalized)]] element $x$ of $C$ is the ordinary [[product]] of the fibers of $A$ and $B$ over $x$. In other words, the fiber product is the product taken fiber-wise. Of course, the fiber of $A$ at the generalized element $x\colon I \to C$ is itself a fibre product $I \times_C A$; the terminology depends on your point of view.

Note that there are maps $p_A : X \to A$, $p_B : X \to B$ sending any $(a,b) \in X$ to $a$ and $b$, respectively. These maps make this [[commuting diagram|square commute]]:

\begin{displaymath}
\itexarray{
     &&&& X 
     \\& 
    &&
      {}^{p_A}\swarrow
      && \searrow^{p_B}
    \\
     && A &&&& B
      \\
      &
      && {}_f\searrow
       &
       & \swarrow_{g}
      &&
     \\
     &&&&
     C
     &&&&
  }
\end{displaymath}
In fact, the pullback is the [[universal property|universal]] solution to finding a commutative square like this. In other words, given \emph{any} [[commutative diagram|commutative square]]

\begin{displaymath}
\itexarray{
     &&&& Y
     \\& 
    &&
      {}^{q_A}\swarrow
      && \searrow^{q_B}
    \\
     && A &&&& B
      \\
      &
      && {}_f\searrow
       &
       & \swarrow_{g}
      &&
     \\
     &&&&
     C
     &&&&
  }
\end{displaymath}
there is a unique function $h: Y \to X$ such that

\begin{displaymath}
p_A h = q_A
\end{displaymath}
and

\begin{displaymath}
p_B h = q_B\,.
\end{displaymath}
Since this universal property expresses the concept of pullback purely arrow-theoretically, we can formulate it in any category. It is, in fact, a simple special case of a [[limit]].

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

\hypertarget{in_category_theory}{}\subsubsection*{{In category theory}}\label{in_category_theory}

A \textbf{pullback} is a [[limit]] of a [[diagram]] like this:

\begin{displaymath}
\itexarray{
     && a &&&& b
      \\
      & 
      && {}_{f}\searrow
       &
       & \swarrow_g
      && 
     \\
     
     &&&&
     c
     &&&&
     
  }
\end{displaymath}
Such a diagram is also called a \textbf{pullback diagram} or a [[cospan]]. If the [[limit]] exists, we obtain a commutative square

\begin{displaymath}
\itexarray{
     &&&& x 
     \\& 
    &&
      {}^{p_a}\swarrow
      && \searrow^{p_b}
    \\
     && a &&&& b
      \\
      &
      && {}_f\searrow
       &
       & \swarrow_{g}
      &&
     \\
     &&&&
     c
     &&&&
  }
\end{displaymath}
and [[generalized the|the]] object $x$ is also called the \textbf{pullback}. It is well defined up to unique [[isomorphism]]. It has the universal property already described above in the special case of the category $Set$.

The last commutative square above is called a \textbf{pullback square}.

The concept of pullback is dual to the concept of [[pushout]]: that is, a pullback in $C$ is the same as a pushout in the [[opposite category]] $C^{op}$.

\hypertarget{in_type_theory}{}\subsubsection*{{In type theory}}\label{in_type_theory}

In [[type theory]] a [[pullback]] $P$ in

\begin{displaymath}
\itexarray{
    P &\to& A
    \\
    \downarrow && \downarrow^{\mathrlap{f}}
    \\
    B &\stackrel{g}{\to}& C
  }
\end{displaymath}
is given by the [[dependent sum]] over the [[dependent type|dependent]] [[equality type]]

\begin{displaymath}
P = \sum_{a : A} \sum_{b : B} (f(a) = g(b))
  \,.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{prop}
\label{PullbacksAsEqualizers}\hypertarget{PullbacksAsEqualizers}{}
\textbf{(pullbacks as equalizers)}

If [[product]]s exist in $C$, then the pullback

\begin{displaymath}
\itexarray{
     a \times_c b &\to& a
     \\
     \downarrow && \downarrow^{\mathrlap{f}}
     \\
     b &\stackrel{g}{\to}& c
  }
\end{displaymath}
is equivalently the [[equalizer]]

\begin{displaymath}
a \times_c b \to a \times b \stackrel{\overset{f p_1}{\longrightarrow}}{\underset{g p_2}{\longrightarrow}}
  c
\end{displaymath}
of the two morphisms induced by $f$ and $g$ out of the [[product]] of $a$ with $b$.

\end{prop}
\begin{prop}
\label{PullbackPreservesMonomorphisms}\hypertarget{PullbackPreservesMonomorphisms}{}
\textbf{(pullbacks preserve monomorphisms and isomorphisms)}

Pullbacks preserve [[monomorphisms]] and [[isomorphisms]]:

If

\begin{displaymath}
\itexarray{
    a &\overset{}{\longrightarrow}& b
    \\
    {}^{\mathllap{f^\ast g}}\downarrow & & \downarrow^{g}
    \\
    c &\underset{f}{\longrightarrow}& d
  }
\end{displaymath}
is a pullback square in some category then:

\begin{enumerate}%
\item if $g$ is a [[monomorphism]] then $f^\ast g$ is a monomorphism;


\item if $g$ is an [[isomorphism]] then $f^\ast g$ is an isomorphism.



\end{enumerate}
On the other hand that $f^\ast g$ is a monomorphism does not imply that $g$ is a monomorphism.

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
\textbf{([[pasting law for pullbacks]])}

In any category consider a diagram of the form

\begin{displaymath}
\itexarray{
     a &\to& b &\to& c
     \\ 
     \downarrow && \downarrow && \downarrow
     \\
     d &\to& e &\to& f
  }
\end{displaymath}
There are three commuting squares: the two inner ones and the outer one.

Suppose the right-hand inner square is a pullback, then:

The square on the left is a pullback if and only if the outer square is.

\end{prop}
\begin{proof}
Pasting a morphism $x \to a$ with the outer square gives rise to a commuting square over the (composite) bottom and right edges of the diagram. The square over the cospan in the left-hand inner square arising from $x \to a$ includes a morphism into $b$, which if $b$ is a pullback induces the same commuting square over $d \to e \to f$ and $c \to d$. So one square is universal iff the other is.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
The converse implication does not hold: it may happen that the outer and the left square are pullbacks, but not the right square.

\end{prop}
\begin{proof}
For instance let $i : a \to b$ be a [[split monomorphism]] with [[retract]] $p : b \to a$ and consider

\begin{displaymath}
\itexarray{
     a & \stackrel{=}{\to} & a & \stackrel{=}{\to} & a
     \\ 
     \downarrow^{\mathrlap{=}} && \downarrow^{\mathrlap{i}} && \downarrow^{\mathrlap{=}}
     \\
     a &\stackrel{i}{\to}& b &\stackrel{p}{\to}& a
  }
\end{displaymath}
Then the left square and the outer rectangle are pullbacks but the right square cannot be a pullback unless $i$ was already an [[isomorphism]].

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
On the other hand, in the [[(∞,1)-category]] of [[∞-groupoids]], there is a sort of ``partial converse''; see [[homotopy pullback\#HomotopyFiberCharacterization]].

\end{remark}
\hypertarget{saturation}{}\subsubsection*{{Saturation}}\label{saturation}

The [[saturated class of limits|saturation]] of the class of pullbacks is the class of limits over categories $C$ whose groupoid reflection $\Pi_1(C)$ is trivial and such that $C$ is [[L-finite category|L-finite]].

\hypertarget{PullbackFunctor}{}\subsubsection*{{Pullback functor}}\label{PullbackFunctor}

If $f : X \to Y$ is a [[morphism]] in a [[category]] $C$ with pullbacks, there is an induced pullback [[functor]] $f^* : C/Y \to C/X$, sometimes also called [[base change]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[fiber product]], [[base change]]


\item [[wide pullback]]


\item [[lax pullback]], [[comma object]]


\item [[(∞,1)-pullback]], [[homotopy pullback]]



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

\begin{itemize}%
\item [[Robert Paré]], \emph{Simply connected limits}. Can. J. Math., Vol. XLH, No. 4, 1990, pp. 731-746, \href{http://math.ca/10.4153/CJM-1990-038-6}{CMS}

\end{itemize}
[[!redirects pullbacks]] [[!redirects pullback square]] [[!redirects pullback diagram]] [[!redirects pullback squares]] [[!redirects pullback diagrams]] [[!redirects fiber product]] [[!redirects fiber products]] [[!redirects fibre product]] [[!redirects fibre products]] [[!redirects fibered product]] [[!redirects fibered products]] [[!redirects fibred product]] [[!redirects fibred products]]



\end{document}