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\begin{document}

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\section*{span rewriting}

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

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

[[!include category theory - contents]]

\hypertarget{span_rewriting}{}\section*{{Span rewriting}}\label{span_rewriting}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{double_pushout_rewriting}{Double pushout rewriting}\dotfill \pageref*{double_pushout_rewriting} \linebreak
\noindent\hyperlink{sesquipushout_rewriting}{Sesqui-pushout rewriting}\dotfill \pageref*{sesquipushout_rewriting} \linebreak
\noindent\hyperlink{single_pushout_rewriting}{Single pushout rewriting}\dotfill \pageref*{single_pushout_rewriting} \linebreak
\noindent\hyperlink{general_gluing_constructions}{General gluing constructions}\dotfill \pageref*{general_gluing_constructions} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Span rewriting is a collection of abstract [[category theory|category-theoretic]] methods of using [[spans]] to ``rewrite'' [[objects]] in a [[category]] by ``deleting'' part of them and replacing it with a substitute part that retains some of the same ``connections'' between the deleted part and the rest of the object.

The most common application is to the category of [[quivers]] (called [[directed graphs|(directed) graphs]] in the relevant literature) or related categories; thus span rewriting is often called ``graph rewriting''.

There is probably not much connection to algebraic [[rewriting]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

All the definitions below have the following context in common. A \textbf{production} or \textbf{rule} in a category (often the category [[Quiv]] of [[quivers]]) is a [[span]]

\begin{displaymath}
L \xleftarrow{l} K \xrightarrow{r} R.
\end{displaymath}
A \textbf{match} for this production is a morphism $f:L\to C$ for some object $C$. A \textbf{derivation} of a match along a production is supposed to be a new object obtained by ``deleting the image of $L$ and replacing it with $R$''.

A production is \textbf{left-linear} if $l$ is a [[monomorphism]], and \textbf{linear} if in addition $r$ is a monomorphism.

\hypertarget{double_pushout_rewriting}{}\subsubsection*{{Double pushout rewriting}}\label{double_pushout_rewriting}

For this definition, we work in an [[adhesive category]] (which includes all [[toposes]], hence in particular $Quiv$). Given a left-linear production $L \xleftarrow{l} K \xrightarrow{r} R$, a match $f:L\to C$ satisfies the \textbf{gluing condition} if the pair $(l,f)$ has a [[pushout complement]] consisting of $g:K\to E$ and $v:E\to C$. In this case the \textbf{(double-pushout) derivation} associated to the match is the pushout of $g$ along $r$, yielding a pair of pushout squares

\begin{displaymath}
\itexarray{ L & \xleftarrow{l} & K & \xrightarrow{r} & R \\
^f\downarrow && \downarrow^{\mathrlap{g}} && \downarrow \\
C & \xleftarrow{v}  & E & \to & D.}
\end{displaymath}
The restriction of $l$ to be monic is necessary to ensure that pushout complements are essentially unique when they exist. Often we further restrict $r$ to be monic to ensure overall good behavior, obtaining the notion of \textbf{linear production}.

\hyperlink{LS}{Lack and Sobocinski} describe the intuition in this way:

\begin{quote}%
in order to perform the rewrite, we need to match $L$ as a substructure of a redex $C$. The structure $K$, thought of as a substructure of $L$, is exactly the part of $L$ which is to remain invariant as we apply the rule to $C$. Finally, parts of $R$ which are not in $K$ should be added to produce the final result of the rewrite.

Thus, an application of a rewrite rule consists of three steps. First we must match $L$ as a substructure of the redex C; secondly, we delete all of parts of the redex matched by $L$ which are not included in $K$. Thirdly, we add all of $R$ which is not contained in $K$, thereby producing a new structure $D$. The deletion and addition of structure is handled, respectively, by finding a pushout complement and constructing a pushout.


\end{quote}
\hypertarget{sesquipushout_rewriting}{}\subsubsection*{{Sesqui-pushout rewriting}}\label{sesquipushout_rewriting}

In an adhesive category, the pushout of a monomorphism is also a pullback. Thus, the pushout complements involved in double-pushout rewriting are also ``pullback complements''. Pullback complements are not in general unique, even in an adhesive category, but those arising as pushout complements do satisfy a [[universal property]]: they are [[final pullback complements]].

This suggests the following generalization. In an arbitrary category, given an arbitrary production $L \xleftarrow{l} K \xrightarrow{r} R$, the \textbf{(sesqui-pushout) derivation} associated to a match $f:L\to C$ is a diagram

\begin{displaymath}
\itexarray{ L & \xleftarrow{l} & K & \xrightarrow{r} & R \\
^f\downarrow && \downarrow^{\mathrlap{g}} && \downarrow \\
C & \xleftarrow{v}  & E & \to & D.}
\end{displaymath}
in which the left square is a final pullback complement and the right square is a pushout. Such a derivation may or may not exist, but if it does then it is essentially unique. Moreover, if we are in an adhesive category and $l$ is monic, then any double-pushout derivation is also a sesqui-pushout derivation.

More generally, given the construction of final pullback complements using [[exponential objects]], if the exponential $\Pi_f l$ exists and the [[counit]] $f^* \Pi_f l \to l$ is an isomorphism, then such a derivation exists with $E = \Pi_f l$. In particular, this happens in any [[locally cartesian closed category]] if $f$ and $l$ are both monomorphisms.

\hypertarget{single_pushout_rewriting}{}\subsubsection*{{Single pushout rewriting}}\label{single_pushout_rewriting}

We can also regard a left-linear production as a [[partial morphism]], in which case the universal property of a final pullback complement suggests the following different viewpoint. Given a left-linear production $L \xleftarrow{l} K \xrightarrow{r} R$, the \textbf{(single-pushout) derivation} associated to a match $f:L\to C$ is the [[pushout]] of $(l,r)$ and $f$ in the category of partial morphisms (if it exists).

It can be shown (see \hyperlink{CHHK}{CHHK}) that in a certain abstract context, any sesqui-pushout derivation of a left-linear production is also a single-pushout derivation; and that the converse holds if and only if the match is ``conflict-free'', and if and only if the induced partial morphism from $R$ to the target object $D$ is total.

\hypertarget{general_gluing_constructions}{}\subsubsection*{{General gluing constructions}}\label{general_gluing_constructions}

In \hyperlink{Lowe10}{Lowe10} and \hyperlink{Lowe12}{Lowe12} a yet more general construction is considered for span rewriting, involving $3\times 3$ diagrams consisting of a pullback, two final pullback complements, and a pushout.

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

\begin{itemize}%
\item H. Ehrig, M. Pfender, and H.J. Schneider, \emph{Graph-grammars: an algebraic approach}, In IEEE Conf. on Automata and Switching Theory, pages 167--180, 1973.


\item [[Steve Lack]] and [[Pawel Sobocinski]], \emph{Adhesive categories}, \href{http://users.ecs.soton.ac.uk/ps/papers/adhesive.pdf}{PDF}



\end{itemize}
\begin{itemize}%
\item Andrea Corradini, Tobias Heindel, Frank Hermann, and Barbara K\"o{}nig, \emph{Sesqui-pushout rewriting}, 2006, \href{https://link.springer.com/chapter/10.1007/11841883_4}{springerlink}, \href{http://www.ti.inf.uni-due.de/publications/koenig/icgt06b.pdf}{pdf}.

\end{itemize}
\begin{itemize}%
\item Michael L\"o{}we, \emph{Graph rewriting in Span-categories}, 2010, \href{https://link.springer.com/chapter/10.1007/978-3-642-15928-2_15}{springerlink}

\end{itemize}
\begin{itemize}%
\item Michael L\"o{}we, \emph{Refined graph rewriting in Span-categories}, 2012, \href{https://link.springer.com/chapter/10.1007/978-3-642-33654-6_8}{springerlink}

\end{itemize}
\begin{itemize}%
\item Nicolas Behr, Pawel Sobocinski, \emph{Rule Algebras for Adhesive Categories}, \href{https://arxiv.org/abs/1807.00785}{arXiv:1807.00785}

\end{itemize}
[[!redirects graph rewriting]] [[!redirects double pushout graph rewriting]] [[!redirects double pushout rewriting]] [[!redirects single pushout graph rewriting]] [[!redirects single pushout rewriting]] [[!redirects sesqui-pushout graph rewriting]] [[!redirects sesqui-pushout rewriting]]



\end{document}