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\begin{document}

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\section*{pasting law for pullbacks}

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

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

[[!include infinity-limits - contents]]

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

[[!include category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{related_statements}{Related statements}\dotfill \pageref*{related_statements} \linebreak


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

In [[category theory]], the \emph{pasting law} is a statement about (de-)composition of [[pullback]]/[[pushout]] [[diagrams]].

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be a [[category]] or more generally an [[(∞,1)-category]] or [[derivator]]. Consider a [[commuting diagram]] in $\mathcal{C}$ of the following shape:

\begin{displaymath}
\itexarray{
    x & \longrightarrow & y & \longrightarrow & z
    \\
    \downarrow && \downarrow && \downarrow
    \\
    u & \longrightarrow & v & \longrightarrow & w
  }
\end{displaymath}
Then:

\begin{enumerate}%
\item if the right square is a pullback, then the total rectangle is a pullback precisely if the left square is a pullback.


\item if the left square is a [[pushout]], then the total rectangle is a pushout precisely if the right square is a pushout.



\end{enumerate}
\end{prop}
For \textbf{proof} see

\begin{itemize}%
\item for [[category theory]]: at \emph{\href{pullback#Pasting}{pullback -- pasting law}}


\item for [[(∞,1)-category theory]]: at \emph{\href{limit+in+a+quasi-category#PushoutPasting}{(∞,1)-limit -- pushout pasting law}}



\end{itemize}
\hypertarget{related_statements}{}\subsection*{{Related statements}}\label{related_statements}

In general, the implications in the above result do require the hypothesis (e.g. in the pullback case that the right square is a pullback). However, in some cases this can be omitted.

\begin{prop}
\label{}\hypertarget{}{}
Suppose we have a diagram of the above shape

\begin{displaymath}
\itexarray{
    x & \longrightarrow & y & \longrightarrow & z
    \\
    \downarrow && \downarrow && \downarrow
    \\
    u & \longrightarrow & v & \longrightarrow & w
  }
\end{displaymath}
in which the total rectangle (consisting of $x,z,u,w$) is a pullback, and moreover the induced map $y\to v\times z$ is a [[monomorphism]]. Then the left-hand square (consisting of $x,y,u,v$) is also a pullback.

\end{prop}
Another related statement involves a pair of rectangles and equalizers.

\begin{prop}
\label{}\hypertarget{}{}
Suppose $\mathcal{C}$ is any [[category]] with [[equalizers]] and that we have a diagram of the following shape:

\begin{displaymath}
\itexarray{
    x & \longrightarrow & y & \rightrightarrows & z
    \\
    \downarrow && \downarrow && \downarrow
    \\
    u & \longrightarrow & v & \rightrightarrows & w
  }
\end{displaymath}
such that the vertical arrows are all monic, the squares on the right are serially commutative, and the lower row is an equalizer. Then the upper row is an equalizer if and only if the left square is a pullback.

\end{prop}
[[!redirects pasting law for pullbacks]] [[!redirects pasting law for pushouts]] [[!redirects pasting law]] [[!redirects pullback lemma]]



\end{document}