texlive[74843] Master: qworld (3apr25)
commits+karl at tug.org
commits+karl at tug.org
Thu Apr 3 21:52:38 CEST 2025
Revision: 74843
https://tug.org/svn/texlive?view=revision&revision=74843
Author: karl
Date: 2025-04-03 21:52:38 +0200 (Thu, 03 Apr 2025)
Log Message:
-----------
qworld (3apr25)
Modified Paths:
--------------
trunk/Master/texmf-dist/doc/latex/qworld/README.md
trunk/Master/texmf-dist/tex/latex/qworld/qworld.sty
trunk/Master/tlpkg/libexec/ctan2tds
Added Paths:
-----------
trunk/Master/texmf-dist/doc/latex/qworld/qworld.pdf
trunk/Master/texmf-dist/doc/latex/qworld/qworld.tex
trunk/Master/texmf-dist/doc/latex/qworld/qworld_eng.pdf
trunk/Master/texmf-dist/doc/latex/qworld/qworld_eng.tex
Removed Paths:
-------------
trunk/Master/texmf-dist/doc/latex/qworld/README.TEXLIVE
Deleted: trunk/Master/texmf-dist/doc/latex/qworld/README.TEXLIVE
===================================================================
--- trunk/Master/texmf-dist/doc/latex/qworld/README.TEXLIVE 2025-04-03 19:52:19 UTC (rev 74842)
+++ trunk/Master/texmf-dist/doc/latex/qworld/README.TEXLIVE 2025-04-03 19:52:38 UTC (rev 74843)
@@ -1,9 +0,0 @@
-The files below have been removed in the TeX Live installation of
-the current package, possibly because of missing source (e.g., a
-PDF with no source), possibly because their only purpose is to support
-nonfree fonts or other software, possibly for other reasons.
-More info: https://tug.org/texlive/pkgcontrib.html#requirements
-
-If this is a concern, please contact the package maintainer.
-
- qworld.pdf
Modified: trunk/Master/texmf-dist/doc/latex/qworld/README.md
===================================================================
--- trunk/Master/texmf-dist/doc/latex/qworld/README.md 2025-04-03 19:52:19 UTC (rev 74842)
+++ trunk/Master/texmf-dist/doc/latex/qworld/README.md 2025-04-03 19:52:38 UTC (rev 74843)
@@ -1,54 +1,86 @@
-# QWorld
+# The QWorld Package
+
## Overview
-QWorld is a LaTeX package designed to simplify the drawing of complex diagrammatic calculations in monoidal category theory. This package provides a concise and intuitive set of commands, allowing users to efficiently render string diagrams with minimal effort. This package is particularly useful for researchers working in category theory, quantum theory, and related fields where diagrammatic reasoning plays a crucial role.
+
+QWorld is a LaTeX package designed to simplify the drawing of complex graphical calculus in monoidal category theory. This package provides a concise and intuitive set of commands, allowing users to efficiently render string diagrams with minimal effort. This package is particularly useful for researchers working in category theory, quantum theory, and related fields where diagrammatic reasoning plays a crucial role.
+
## Installation
-Once the package is registered on CTAN, it can be installed using:
+
+Once the package is available in a TeX distribution via CTAN, you can use it in your document by adding:
+
```
\usepackage{qworld}
```
-Alternatively, to install manually:
-1. Download the `.sty` file of the package.
-2. Place it in your LaTeX working directory or a suitable location within your TeX distribution.
-3. Add `\usepackage{qworld}` to the preamble of your document.
+
+Alternatively, to install it manually:
+
+1. Download the package's `.sty` file.
+2. Place it in your LaTeX working directory or an appropriate directory within your TeX distribution.
+3. Add \usepackage{qworld} to the preamble of your document.
4. Run `texhash` if necessary to update the file database.
## Usage
+
Below is a basic example of how to use qworld
+
```
\documentclass{article}
\usepackage{qworld}
+
\begin{document}
- \begin{equation}
- \q{
- \qcap\qwire\n\qwire\qcup
- }=\q{
- \qwireup[vertical length=2]
- }
- \end{equation}
+ \[
+ \q{
+ \qcap\qwire\n
+ \qwire\qcup
+ }=\q{
+ \qwireup[vlen=2]
+ }
+ \]
\end{document}
```
+
For more detailed usage instructions, refer to the package documentation.
+
## License
+
This package is distributed under the [LaTeX Project Public License (LPPL)](https://www.latex-project.org/lppl.txt), version 1.3c or later.
+
## Copyright
+
© 2025 Niina Ryota
This Package consists of the following files:
+
- `qworld.sty` (the main package file)
- `qworld.pdf` (documentation)
The current maintainer of this package is **Niina Ryota**.
+
## Modification Policy
+
Under the terms of the LPPL:
+
- Any modifications must **not** be distributed under the original package name.
- Any modified versions must clearly state that they are derived from this package.
- The original author (Niina Ryota) must always be credited in derived works.
## Contact
+
Author: Niina Ryota
Email: qworld-package at googlegroups.com
-## Version History
-* v1.0.0 (2025-03-31): Initial release.
+## Change Log
+### 1.1.0 (2025-04-03)
+- The following commands have been updated. The ID, which was previously a required argument for these commands, has been made optional. Users can now specify the ID via an optional key, allowing for more flexible usage of the commands.
+ - `\symbolI`, `\symbols`, `\symbolS`, `\symbolO`, `\symboln`, `\symbolN`.
+ - Specific changes:
+ - Old format: `\symbolI[i]{ID}{text}[above right]`
+ - New format: `\symbolI[i]{text}[above right, id=ID]`
+- English documentation has been added to the package.
+
+### 1.0.0 (2025-04-01)
+- Initial release of the package.
+
## Contributions
+
Bug reports, feature requests, and contributions are welcome. Please contact qworld-package at googlegroups.com for inquires or to submit issues.
Added: trunk/Master/texmf-dist/doc/latex/qworld/qworld.pdf
===================================================================
(Binary files differ)
Index: trunk/Master/texmf-dist/doc/latex/qworld/qworld.pdf
===================================================================
--- trunk/Master/texmf-dist/doc/latex/qworld/qworld.pdf 2025-04-03 19:52:19 UTC (rev 74842)
+++ trunk/Master/texmf-dist/doc/latex/qworld/qworld.pdf 2025-04-03 19:52:38 UTC (rev 74843)
Property changes on: trunk/Master/texmf-dist/doc/latex/qworld/qworld.pdf
___________________________________________________________________
Added: svn:mime-type
## -0,0 +1 ##
+application/pdf
\ No newline at end of property
Added: trunk/Master/texmf-dist/doc/latex/qworld/qworld.tex
===================================================================
--- trunk/Master/texmf-dist/doc/latex/qworld/qworld.tex (rev 0)
+++ trunk/Master/texmf-dist/doc/latex/qworld/qworld.tex 2025-04-03 19:52:38 UTC (rev 74843)
@@ -0,0 +1,2372 @@
+\documentclass{ltjsarticle}
+
+\usepackage{qworld}
+
+\usepackage{bookmark}
+\usepackage{fontspec}
+\usepackage{pdfpages}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{amsfonts}
+\usepackage{mathtools}
+\usepackage{ascmac}
+\usepackage{fancybox}
+\usepackage{float}
+\usepackage{xparse}
+\usepackage{bbm}
+\usepackage{mathrsfs}
+%\numberwithin{equation}{subsection}
+
+\usepackage{framed}
+\newenvironment{longshadowbox}{%
+ \def\FrameCommand{\fboxsep=\FrameSep \shadowbox}%
+ \MakeFramed {\FrameRestore}}%
+{\endMakeFramed}
+
+\usepackage{url}
+\usepackage{hyperref}
+\hypersetup{colorlinks=true, linkcolor=blue}
+
+\usepackage{iitem}
+
+\usepackage{multirow}
+
+\usepackage{here}
+
+\usepackage{fontawesome}
+
+\usepackage{xspace}
+\newcommand{\qworldname}{\texttt{QWorld}\xspace}
+\newcommand{\tikzname}{Ti\emph{k}Z\xspace}
+\newcommand{\texlivename}{\TeX{} \textsf{Live}\xspace}
+
+\setcounter{tocdepth}{3}
+
+\usepackage{listings}
+\lstset{basicstyle=\ttfamily}
+\usepackage[most]{tcolorbox}
+\newtcblisting{mybox}[2][]{%
+ colback=red!5!white,
+ colframe=red!75!black,
+ fonttitle=\bfseries,
+ title=#2,
+ #1
+}
+
+\tcbset
+{
+ codeinfo/.style=
+ {
+ colframe=cyan!90,
+ coltitle=cyan!60,
+ colbacktitle=black,
+ fonttitle=\small\sffamily\bfseries,
+ fontupper=\small,
+ fontlower=\small
+ },
+ exampletitleinfo/.style 2 args=
+ {
+ codeinfo,
+ title={例 \thetcbcounter #1},
+ label={#2}
+ },
+ failurecodeinfo/.style=
+ {
+ colframe=red!90,
+ coltitle=red!60,
+ colbacktitle=black,
+ fonttitle=\small\sffamily\bfseries,
+ fontupper=\small,
+ fontlower=\small
+ },
+ failureexampletitleinfo/.style 2 args=
+ {
+ failurecodeinfo,
+ title={失敗例 \thetcbcounter #1},
+ label={#2}
+ }
+}
+\newtcblisting{texexp}[1]
+{
+ codeinfo,
+ #1
+}
+\newtcblisting[auto counter,number within=section]{codebox}[3][center lower,listing side text,righthand width=3.5cm,bicolor,colbacklower=white]
+{
+ exampletitleinfo={#2}{#3},
+ breakable,
+ #1
+}
+
+\newtcblisting[use counter from=codebox,number within=section]{CODEBOX}[3][center lower,bicolor,colbacklower=white]
+{
+ exampletitleinfo={#2}{#3},
+ breakable,
+ #1
+}
+\newtcolorbox[use counter from=codebox]{codeboxspec}[3][]
+{
+ exampletitleinfo={#2}{#3},
+ breakable,
+ #1
+}
+\newtcblisting{texfexp}[1]
+{
+ failurecodeinfo,
+ #1
+}
+\newtcblisting[auto counter,number within=section]{fcodebox}[3][center lower,listing side text,righthand width=3.5cm,bicolor,colbacklower=white]
+{
+ failureexampletitleinfo={#2}{#3},
+ breakable,
+ #1
+}
+
+\newtcblisting[use counter from=fcodebox,number within=section]{FCODEBOX}[3][center lower,bicolor,colbacklower=white]
+{
+ failureexampletitleinfo={#2}{#3},
+ breakable,
+ #1
+}
+\newtcolorbox[use counter from=fcodebox]{fcodeboxspec}[3][]
+{
+ failureexampletitleinfo={#2}{#3},
+ breakable,
+ #1
+}
+
+\begin{document}
+\title{
+ \Huge \textbf{\qworldname}\\[4pt]
+ \scriptsize\texttt{Version 1.1.0} \\[4pt]
+}
+\author{
+ \textbf{新名 亮太} \\[4pt]
+}
+\date{\today}
+\maketitle
+
+\begin{abstract}
+ \qworldname は, モノイダル圏における図式計算を直感的かつ効率的に描画するために設計された\LaTeX パッケージである.
+ 特に, 圏論的手法が重要な役割を果たす量子論, トポロジカル量子計算, 及び関連分野における図式的推論の可視化を支援する.
+ 本パッケージは\tikzname\cite{tikz-manual} を基盤としており, シンプルなコマンドによって複雑な図式を容易に描画できる.
+\end{abstract}
+
+\tableofcontents
+\section{導入}
+本稿では, \qworldname パッケージのインストール方法, 基本的な使用法, 応用例を体系的に解説する.
+\qworldname は, モノイダル圏に基づく図式言語の組版に特化した\LaTeX パッケージであり,
+モノイダル圏, フロベニウス構造, 組紐, ホップ代数, シンメトリー, 双対性, ピボタル構造, ダガーなど, 多岐にわたる圏論的概念を図式表現するための直感的なコマンドセットを提供する.
+\section{基本的な使い方}
+\subsection{基本要素}
+\qworldname パッケージのの最も基本的なコマンドを以下に示す.
+これらは図式を描画するための基礎となる.
+\begin{itemize}
+ \item \texttt{\textbackslash q}: 図の素子を配置するためのキャンバス.
+ \item \texttt{\textbackslash qbox}: ボックス\q[scale=0.5]{\qbox}の描画.
+ \item \texttt{\textbackslash qwire}: ワイヤー\q[scale=0.5]{\qwire}の描画.
+ \item \texttt{\textbackslash qcirc}: 合成.
+\end{itemize}
+ これらのコマンドは, キャンバスである\verb|\q{...}|内に記述する.
+ 例えば\verb|\q{\qbox}|と記述すると, ボックス\q[scale=0.5]{\qbox}が描画される.\par
+ \(\q[scale=0.5]{\qbox}\)と\(\q[scale=0.5]{\qbox}\)を合成するためには, \verb|\qcirc|を使用する.
+ これは数学における合成\(g\circ f\) (\verb|\(g\circ f\)|) と似ている.
+ \begin{codebox}{}{Ex:20250317}
+ \q{ \qbox \qcirc \qbox }
+ \end{codebox}
+ また, \verb|\qbox| を並置すると, 描画結果でもボックス\(\q[scale=0.5]{\qbox}\)が横に並ぶ:
+ \begin{codebox}{}{boxs2}
+ \q{ \qbox \qbox \qbox }
+ \end{codebox}
+ \verb|\qcirc| を用いて複数のボックスを合成することも可能である:
+ \begin{codebox}{}{boxsbr}
+ \q
+ {
+ \qbox \qbox \qbox
+ \qcirc
+ \qbox \qbox \qbox
+ }
+ \end{codebox}
+ なお, \verb|\n|は\verb|\qcirc|と同義である:
+ \begin{codebox}{}{boxsn}
+ \q
+ {
+ \qbox \qbox \qbox \n
+ \qbox \qbox \qbox
+ }
+\end{codebox}
+\subsubsection{識別番号}
+\verb|\q{...}|内に配置された全てのコマンドには, 実行順に基づいて自動的に固有の識別番号が割り当てられる.
+例えば, 先の例 \ref{boxsn} では各\verb|\qbox|に番号が振られる:
+\[\q{\qloop[2]{\qloop[3]{\qbox[show id={true}]}\n}}\]
+詳細は, \pageref{subsub:show id}ページの\ref{subsub:show id}項を参照せよ.
+\subsubsection{バウンディングボックス}
+\[\q{
+ \qcaprev \qcap \n
+ \qwire \qbraid \qwire \n
+ \qwire \qbraid \qwire \n
+ \qcup \qcuprev
+ \qbb[red][1,2]
+ \bbsymbol{Measure}
+ \bbsymbol[W]{Output}[left]
+ \bbsymbol[NW]{Computation}[above left]
+ \bbsymbol[NE]{Physics}[above right]
+ \qbb[yellow][3,5,6,8]
+ \bbsymbol{Braid}
+ \bbsymbol[W]{Compute}[left]
+ \qbb[cyan][9,10]
+ \bbsymbol{Create}
+ \bbsymbol[W]{Initialize}[left]
+}\]
+次の例 \ref{ex:hopflink} をみよ:
+\begin{codebox}{}{ex:hopflink}
+ \q{
+ \qcaprev \qcap \n
+ \qwire \qbraid \qwire \n
+ \qwire \qbraid \qwire \n
+ \qcup \qcuprev
+ }
+\end{codebox}
+\verb|\qcaprev|は\q[scale=0.6]{\qcaprev}を,
+\verb|\qcap|は\q[scale=0.6]{\qcap}を,
+\verb|\qwire|は\q[scale=0.5]{\qwire}を,
+\verb|\qbraid|は\q[scale=0.5]{\qbraid}を,
+\verb|\qcup|は\q[scale=0.6]{\qcup}を,
+\verb|\qcuprev|は\q[scale=0.6]{\qcuprev}を, 各々描画する.
+全体として, コマンドの配置の通りに描画されていることが分かる.\par
+ 各素子には, バウンディングボックスが付随する.
+ 以下に例を示す:
+ \begin{itemize}
+ \item \q{\qcaprev}のバウンディングボックスを可視化すると\q{\qcaprev\qbball}.
+ \item \q{\qbraid}のバウンディングボックスを可視化すると\q{\qbraid\qbball}.
+ \item \q{\qwire}のバウンディングボックスを可視化すると\q{\qwire\qbball}.
+ \item \q{\qcuprev}のバウンディングボックスを可視化すると\q{\qcuprev\qbball}.
+ \end{itemize}
+ バウンディングボックスを可視化するコマンドと,
+ 可視化したバウンディングボックス近辺にテキストを置くコマンドは次の通りである:
+ \begin{itemize}
+ \item 可視化
+ \iitem \texttt{\textbackslash qbb}: 特定の素子のバウンディングボックスを表示.
+ \iitem \texttt{\textbackslash qbball}: 全ての素子のバウンディングボックスを一度に表示.
+ \iitem \texttt{\textbackslash qBBall}: キャンバス全体を囲むバウンディングボックスをを表示.
+ \item テキストを置く
+ \iitem \texttt{\textbackslash bbsymbol}: バウンディングボックスにテキストを配置.
+ \end{itemize}
+
+ \begin{verbatim}
+ \qbb[<色>][<ID列>]
+ \end{verbatim}
+ \begin{itemize}
+ \item 第一引数 (オプション): 色を指定. 未指定の場合は無色.
+ \item 第二引数: 対象となる素子の識別番号のリストを指定し, それらを含むバウンディングボックスを表示.
+ \end{itemize}
+ \begin{codebox}{}{Ex:20250320}
+ \q{\qbox\qbb[1]}
+ \end{codebox}
+ \begin{codebox}{}{ex:202503270454}
+ \q{
+ \qcaprev \qcap \n
+ \qwire \qbraid \qwire \n
+ \qwire \qbraid \qwire \n
+ \qcup \qcuprev
+ \qbb[4,7]
+ }
+ \end{codebox}
+ \begin{CODEBOX}{}{Ex:20250321}
+ \q{\qbox\qbox\qbox\qbox\qbox\qbox\qbb[yellow][2,...,5]}
+ \end{CODEBOX}
+ \begin{codebox}{}{Ex:20250222}
+ \q
+ {
+ \qbox\qbox\qbox\n
+ \qbox\qbox\qbox
+ \qbb[green][2,5]
+ }
+ \end{codebox}
+ \begin{codebox}{}{Ex:20250318}
+ \q{
+ \qcaprev \qcap \n
+ \qwire \qbraid \qwire \n
+ \qwire \qbraid \qwire \n
+ \qcup \qcuprev
+ \qbb[blue][4,7]
+ }
+ \end{codebox}
+ \begin{verbatim}
+ \qbball[<色>]
+ \end{verbatim}
+ 複数 (一個でも良いが) のバウンディングボックスを一度に可視化する.
+ \begin{codebox}[center lower,listing side text,righthand width=5cm,bicolor,colbacklower=white]{}{Ex:202503270511}
+ \q{
+ \qcaprev \qcap \n
+ \qwire \qbraid \qwire \n
+ \qwire \qbraid \qwire \n
+ \qcup \qcuprev
+ \qbball
+ }
+ \end{codebox}
+ 各素子のバウンディングボックスが重なることなく組み合っていることが判る.
+ \qworldname パッケージは, 図式描画に伴う\textbf{細かく面倒な座標設定の多くを省略できる}.
+
+ \begin{verbatim}
+ \qBBall[<色>]
+ \end{verbatim}
+ キャンバス内の全素子を囲むバウンディングボックスを表示する.
+ \begin{codebox}[center lower,listing side text,righthand width=5cm,bicolor,colbacklower=white]{}{Ex:20250316}
+ \q{
+ \qcaprev \qcap \n
+ \qwire \qbraid \qwire \n
+ \qwire \qbraid \qwire \n
+ \qcup \qcuprev
+ \qBBall
+ }
+ \end{codebox}
+ \begin{verbatim}
+ \bbsymbol[<方角>]{<text>}[<オプション>]
+ \end{verbatim}
+ 直前のバウンディングボックスにテキストを配置する.
+ \begin{itemize}
+ \item 第一引数 (オプション): \texttt{N} (北), \texttt{S} (南), \texttt{E} (東), \texttt{S} (西), \texttt{NW} (北西), \texttt{NE} (北東), \texttt{SE} (南東), \texttt{SW} (南西) のいずれかを指定. デフォルトは\texttt{E}.
+ \item 第二引数: 配置するテキスト.
+ \item 第三引数 (オプション): 位置調整 (例: \texttt{above}, \texttt{below}, \texttt{left}, \texttt{right}). デフォルトは\texttt{right}.
+ \end{itemize}
+ \begin{CODEBOX}{}{Ex:bbsymbol}
+ \q[scale=5]{
+ \qbox
+ \qbb[1]
+ \bbsymbol{東 (デフォルト)}
+ \bbsymbol[W]{西}[left]
+ \bbsymbol[N]{北}[above]
+ \bbsymbol[S]{南}[below]
+ \bbsymbol[NE]{北東}[above right]
+ \bbsymbol[NW]{北西}[above left]
+ \bbsymbol[SE]{南東}[below right]
+ \bbsymbol[SW]{南西}[below left]
+ \bbsymbol[C]{重心}[]
+ }
+ \end{CODEBOX}
+ \begin{CODEBOX}{}{Ex:20250315}
+ \q
+ {
+ \qcaprev \qcap \n
+ \qwire \qbraid \qwire \n
+ \qwire \qbraid \qwire \n
+ \qcup \qcuprev
+ \qbb[red][1,2]
+ \bbsymbol{Measure}
+ \bbsymbol[W]{Output}[left]
+ \bbsymbol[NW]{Computation}[above left]
+ \bbsymbol[NE]{Physics}[above right]
+ \qbb[yellow][3,5,6,8]
+ \bbsymbol{Braid}
+ \bbsymbol[W]{Compute}[left]
+ \qbb[cyan][9,10]
+ \bbsymbol{Create}
+ \bbsymbol[W]{Initialize}[left]
+ }
+ \end{CODEBOX}
+
+ \subsubsection{数式モード}
+ 図式言語を含んだ数式を記述する場合は, 数式モードを使用する.
+ \begin{CODEBOX}{}{IntroEx}
+ \[
+ \operatorname{Tr}\left(\q{\qbraid\qcirc\qbraid}\right)
+ =\q{
+ \qcap \qcap \n
+ \qwire \qbraidinv \qwire \n
+ \qwire \qbraidinv \qwire \n
+ \qcuprev \qcuprev
+ }
+ \]
+ \end{CODEBOX}
+
+ \subsection{インストール}
+ \qworldname パッケージのスタイルファイル\verb|qworld.sty|はCTANからダウンロード可能である.
+ \TeX のパッケージディレクトリに配置し,
+ プリアンブルに以下を記述することで使用可能となる:
+ \begin{verbatim}
+ \usepackage{qworld}
+ \end{verbatim}
+
+ 依存パッケージは以下の通りである:
+ \begin{itemize}
+ \item \tikzname
+ \iitem \texttt{cd}
+ \iitem \texttt{positioning}
+ \iitem \texttt{arrows}
+ \iitem \texttt{arrows.meta}
+ \iitem \texttt{calc}
+ \iitem \texttt{intersections}
+ \iitem \texttt{shapes.symbols}
+ \iitem \texttt{shapes.geometric}
+ \iitem \texttt{shapes.misc}
+ \iitem \texttt{decorations.pathreplacing}
+ \iitem \texttt{decorations.markings}
+ \iitem \texttt{decorations.pathmorphing}
+ \item \texttt{pgffor}
+ \item \texttt{ifthen}
+ \item \texttt{xparse}
+ \item \texttt{xfp}
+ \item \texttt{xstring}
+ \end{itemize}
+ これらのパッケージは\texlivename ディストリビューションに標準で含まれており, 特別な手間をかける必要はない.
+\subsection{基本的なカスタマイズ}
+本節では, \qworldname のコマンドをカスタマイズする方法を解説する.
+ \subsubsection{ボックスのカスタマイズ}
+ \label{subsubsec:boxcustomization}
+キーを使い, ボックスの経線の数や配置を調整できる.
+例えば\q[scale=0.5]{\qbox[p=20,hlen=3]}や\q[scale=0.5]{\qbox[p=10,N={1,2,3,8,9,10},S={4,5,6,7},hlen=3]}などを描画可能である.
+これらのキーはワイヤー以外の多くの素子にも適用可能である.
+主要なキーは以下の通り:
+\begin{itemize}
+ \item \texttt{n}
+ \iitem ボックス\(\q[scale=0.5]{\qbox[p=2, P={}]}\)の北縁において経線\q[scale=0.31]{\qwire}を生やすことができる点 (このような点を\texttt{n}端子と呼ぶこととする) の数.
+ \iitem 例えば \texttt{n=3}とすれば, ボックスの北縁に \(3\) 点等間隔に打つことができる.
+ \iitem デフォルト値は \texttt{n=1}.
+ \iitem \texttt{n}端子を最北端まで垂直移動させた点を\texttt{N}端子と呼ぶ.
+ \iitem \texttt{n}端子及び\texttt{N}端子は西 (左) から順に\(1,2,...\)と添え字付けられる.
+ \iitem \(i\)番目の\texttt{n}端子に経線を生やすとは, \(i\)番目の\texttt{n}端子と\(i\)番目の\texttt{N}端子を線分で結ぶことをいう.
+ \item \texttt{N}
+ \iitem 経線を実際に生やす\texttt{n}端子の添え字の集合.
+ \iitem 例えば \texttt{n=3}とした上で\texttt{N=\{2\}}とすれば, \(3\)点のうち\(2\)番目の\texttt{n}端子に経線を生やすことができる.
+ \iitem \texttt{n}が未指定で, \texttt{N}が指定されているときは, 自動で \(\texttt{n}\coloneq\max\texttt{N}\) となる.
+ \iitem \texttt{N}が未指定で, \texttt{n}が指定されているときは, 自動で \(\texttt{N}\coloneq\{1,...,\texttt{n}\}\) となる.
+ \iitem \texttt{n}端子から実際に生えた経線の最北端を出力点, あるいは\texttt{O}端子と呼ぶこととする.
+ \item \texttt{s}
+ \iitem ボックスの南縁において経線を生やすことができる点 (このような点を\texttt{s}端子と呼ぶこととする) の数.
+ \iitem 例えば \texttt{s=4}とすれば, ボックスの南縁に\(4\)点等間隔に打つことができる.
+ \iitem デフォルト値は\texttt{s=1}.
+ \iitem \texttt{s}端子を最南端まで垂直移動させた点を\texttt{S}端子と呼ぶ.
+ \iitem \texttt{s}端子及び\texttt{S}端子は西 (左) から順に\(1,2,...\)と添え字付けられる.
+ \iitem \(i\)番目の\texttt{s}端子に経線を生やすとは, \(i\)番目の\texttt{s}端子と\(i\)番目の\texttt{S}端子を線分で結ぶことをいう.
+ \item \texttt{S}
+ \iitem 経線を実際に生やす\texttt{s}端子の添え字の集合.
+ \iitem 例えば \texttt{s=4}とした上で \texttt{S=\{1,4\}} とすれば, \(4\)点のうち\(1\)番目と\(4\)番目の点に経線を生やすことができる.
+ \iitem \texttt{s}が未指定で, \texttt{S}が指定されているときは, 自動で \(\texttt{s}\coloneq\max \texttt{S}\) となる.
+ \iitem \texttt{S}が未指定で, \texttt{s}が指定されているときは, 自動で \(\texttt{S}\coloneq\{1,...,\texttt{s}\}\) となる.
+ \iitem \texttt{s}端子から実際に生えた経線の最南端を入力点, あるいは\texttt{I}端子と呼ぶこととする.
+ \item \texttt{hlen}
+ \iitem 水平方向のバウンディングボックスの長さ.
+ \iitem 例えば\texttt{hlen=2}とすれば, 水平方向のバウンディングボックスの長さが\(2\)となる.
+ \iitem 唯一つの例外を除いて, 最西端の経線とバウンディングボックスの境界線との距離, 最東端の経線とバウンディングボックスの境界線との距離はともに\(\fpeval{0.5*\qwireinterval}\)である.
+ \iitem \texttt{hlen}が未指定であるとき, \texttt{n}端子及び\texttt{s}端子は, 間隔を\(1\)に保って並べられる.
+ \iitem \texttt{hlen}が指定されているとき, \texttt{n}端子及び\texttt{s}端子は, 長さ\(\texttt{hlen}-\fpeval{\qwireinterval}\)の区間を等間隔に並べられる. このときの端子の間隔は\texttt{hlen}及び端子の数に依存する.
+ \iiitem 端子の数が\(1\)つであるときは, 区間の中心に置かれる. このとき, \(\texttt{hlen}>1\) であるならば, 最西端の経線とバウンディングボックスの境界線との距離, 及び最東端の経線とバウンディングボックスの境界線との距離はともに\(\frac{\texttt{hlen}}{2}>\fpeval{0.5*\qwireinterval}\)である.
+ \item \texttt{vlen}
+ \iitem 垂直方向のバウンディングボックスの長さ (高さ).
+ \iitem デフォルト値は \texttt{vlen=1}.
+ \item \texttt{id}
+ \iitem ボックスの識別子.
+ \iitem 例えば\texttt{id=ID}とすれば, ここまで登場した様々な点に次のような名前が自動的に付けられる:
+ \iiitem (西から数えて) \(i\)番目の\texttt{n}端子: (n-\(i\)-ID)
+ \iiitem \(i\)番目の\texttt{s}端子: (s-\(i\)-ID)
+ \iiitem \(i\)番目の\texttt{n}端子をボックスの北縁へ垂直移動させた点: (N-\(i\)-ID)
+ \iiitem \(i\)番目の\texttt{s}端子をボックスの南縁へ垂直移動させた点: (S-\(i\)-ID)
+ \iiitem バウンディングボックスの最北西端: (NW-ID)
+ \iiitem バウンディングボックスの最北東端: (NE-ID)
+ \iiitem バウンディングボックスの最南西端: (SW-ID)
+ \iiitem バウンディングボックスの最南東端: (SE-ID)
+ \iiitem バウンディングボックスの最北端: (N-ID)
+ \iiitem バウンディングボックスの最南端: (S-ID)
+ \iiitem バウンディングボックスの最西端: (W-ID)
+ \iiitem バウンディングボックスの最東端: (E-ID)
+ \iiitem バウンディングボックスの重心: (C-ID)
+ \iiitem ボックスの最北西端: (nw-ID)
+ \iiitem ボックスの最北東端: (ne-ID)
+ \iiitem ボックスの最南西端: (sw-ID)
+ \iiitem ボックスの最南東端: (se-ID)
+ \iiitem ボックスの最北端: (n-ID)
+ \iiitem ボックスの最南端: (s-ID)
+ \iiitem ボックスの最西端: (w-ID)
+ \iiitem ボックスの最東端: (e-ID)
+ \iiitem \(i\) 番目の出力点: (O-\(i\)-ID)
+ \iiitem \(i\) 番目の入力点: (I-\(i\)-ID)
+ \iitem 指定の有無にかかわらず, \texttt{\textbackslash q\{...\}}内に置かれた各コマンドには識別番号が振られている. 各々\texttt{ID}部分をその番号に置き換えても同じ点を表す. 詳細は \ref{subsub:show id} 項を確認されたし.
+ \item \texttt{p}, \texttt{P}
+ \iitem 正整数\texttt{k}に対して, \texttt{p=k}は\texttt{n=k}, \texttt{s=k}を意味する.
+ \iitem 正整数の有限集合\texttt{K}に対して, \texttt{P=K}は\texttt{N=K}, \texttt{S=K}を意味する.
+ \item \texttt{name}
+ \iitem \texttt{name=<text>} により, ボックスの重心に\texttt{<text>}が置かれる.
+\end{itemize}
+\begin{CODEBOX}{}{ex:boxpoints}
+ \def\pradius{0.4pt}
+ \q[scale=5]
+ {
+ \qbox
+ [
+ n=3,N={2},
+ s=4,S={1,4},
+ hlen=2,
+ vlen=2,
+ id={ID},
+ name={重心 (C-ID)}
+ ]\n
+ \qbb[ID]
+ \node [above] at (NE-ID) {NE-ID};
+ \node [above] at (NW-ID) {NW-ID};
+ \node [below] at (SE-ID) {SE-ID};
+ \node [below] at (SW-ID) {SW-ID};
+ \fill (NE-ID) circle (\pradius);
+ \fill (NW-ID) circle (\pradius);
+ \fill (SE-ID) circle (\pradius);
+ \fill (SW-ID) circle (\pradius);
+ \node [above] at (ne-ID) {ne-ID};
+ \node [above] at (nw-ID) {nw-ID};
+ \node [below] at (se-ID) {se-ID};
+ \node [below] at (sw-ID) {sw-ID};
+ \fill (ne-ID) circle (\pradius);
+ \fill (nw-ID) circle (\pradius);
+ \fill (se-ID) circle (\pradius);
+ \fill (sw-ID) circle (\pradius);
+ \foreach\i in {1,...,3}
+ {
+ \symboln[\i]{n-\i-ID}[below]
+ \fill (n-\i-ID) circle (\pradius);
+ \symbolN[\i]{N-\i-ID}
+ \fill (N-\i-ID) circle (\pradius);
+ }
+ \foreach\i in {1,...,4}
+ {
+ \symbols[\i]{s-\i-ID}[above]
+ \fill (s-\i-ID) circle (\pradius);
+ \symbolS[\i]{S-\i-ID}[below]
+ \fill (S-\i-ID) circle (\pradius);
+ }
+ \draw[line width=0.2pt] (NW-ID)
+ to [bend left=60] node [fill=white, midway] {hlen} (NE-ID);
+ \draw[line width=0.2pt] (NW-ID)
+ to [bend right=60] node [fill=white, midway] {vlen} (SW-ID);
+ \draw [red, <->] (C-ID -| W-ID)
+ -- node [midway, auto, font=\footnotesize] {0.5} (C-ID -| S-1-ID);
+ \draw [red, <->] (C-ID -| S-4-ID)
+ -- node [midway, auto, font=\footnotesize] {0.5} (C-ID -| E-ID);
+ \qwire[arrowtype={loosely dotted}, dom={s-1-ID}, cod={n-1-ID}]
+ \qwire[arrowtype={loosely dotted}, dom={s-4-ID}, cod={n-3-ID}]
+ \node (O1prime) at ([shift={(0.25,0.25)}]O-1-ID){O-1-ID};
+ \draw[->,red] (O1prime) to ([shift={(0.05,0.05)}]O-1-ID);
+ \node (I1prime) at ([shift={(0.5,-0.5)}]I-1-ID){I-1-ID};
+ \node (I2prime) at ([shift={(0.5,-0.5)}]I-2-ID) {I-2-ID};
+ \draw[->,red](I1prime) to ([shift={(0.05,-0.05)}]I-1-ID);
+ \draw[->,red](I2prime) to ([shift={(0.05,-0.05)}]I-2-ID);
+ }
+\end{CODEBOX}
+\verb|\qwire|の\verb|arrowtype|キーと\verb|dom|, \verb|cod|キーは
+\pageref{subsubsec:wirecustomization}ページの\ref{subsubsec:wirecustomization}項にて説明する.
+\paragraph{ラベル (\texttt{\textbackslash symbol})}
+\begin{verbatim}
+ \symbolI[<番号>]{<label>}[<オプション>]
+\end{verbatim}
+例えば, \verb|\symbolI[2]{text}[below right, id=<ID>]| により,
+\verb|\q{\qbox[p=3, id=ID]}| (\(\q[scale=0.5]{\qbox[p=3, id=ID]}\)) の\(2\)番目の入力点 (I-2-ID) の右下に``text''が置かれる.
+番号とオプションの指定のないときは, デフォルト値\(1\), below, \verb|id=<直前のもの>| が各々適用される.
+他にもラベル表示コマンドとして,
+\begin{verbatim}
+ \symbolS[<番号>]{<label>}[<オプション>]
+\end{verbatim}
+\begin{verbatim}
+ \symbols[<番号>]{<label>}[<オプション>]
+\end{verbatim}
+\begin{verbatim}
+ \symbolO[<番号>]{<label>}[<オプション>]
+\end{verbatim}
+\begin{verbatim}
+ \symbolN[<番号>]{<label>}[<オプション>]
+\end{verbatim}
+\begin{verbatim}
+ \symboln[<番号>]{<label>}[<オプション>]
+\end{verbatim}
+がある. \verb|\symbolS|, \verb|\symbols|はそれぞれ点 (S-\(i\)-ID), (s-\(i\)-ID) に\textless label\textgreater を置く.
+番号\(i\)のデフォルト値は \(1\), オプションのデフォルト値は below である.
+\verb|\symbolO|, \verb|\symbolN|, \verb|\symboln| はそれぞれ点 (O-\(i\)-ID), (N-\(i\)-ID), (n-\(i\)-ID) に\textless label\textgreater を置く.
+番号\(i\)のデフォルト値は \(1\), オプションのデフォルト値は above である.
+\begin{codebox}[center lower,listing side text,righthand width=5cm,bicolor,colbacklower=white]{}{Ex:20250311}
+ \q{
+ \qbox[p=5, S={1,4,5}, id=ID]
+ \symbolO{A}
+ \symbolN[3]{B}
+ \symboln[5]{C}[below left]
+ \symbolS[2]{D}
+ \symbols[3]{E}
+ \symbolI[2]{F}
+ \symbolI[3]{G}[below right]
+ }
+\end{codebox}
+\begin{CODEBOX}{}{Ex:20250310}
+ \q{
+ \qbox[p=10,N={1,2,5,6,9,10}, id=ID]
+ \foreach\i in {1,...,10}
+ {
+ \symbolI[\i]{\(x_{\i}\)}
+ }
+ \foreach\j in {1,...,6}
+ {
+ \symbolO[\j]{\(y_{\j}\)}
+ }
+ }
+\end{CODEBOX}
+
+\subsubsection{繰り返し}
+\label{subsub:forloop}
+\begin{verbatim}
+ \qloop[<回数>]{<コマンド>}
+\end{verbatim}
+\verb|\qloop|コマンドを用いれば, 指定した回数だけコマンドを繰り返せる:
+\begin{CODEBOX}{}{Ex:20250305}
+ \q{ \qloop[10]{\qbox} }
+\end{CODEBOX}
+\verb|\qloop|コマンドは\verb|\q|外でも使用可能である:
+\begin{codebox}[center lower,listing side text,righthand width=5.5cm,bicolor,colbacklower=white]{}{Ex:202503220930}
+ \qloop[30]{僕ボブ}
+\end{codebox}
+\subsubsection{識別番号の表示}
+\label{subsub:show id}
+\verb|\q{...}| 内のコマンドには自動的に識別番号が付与される.
+\texttt{show id=true}を指定すると, それらが表示される.
+\begin{CODEBOX}{}{Ex:20250303}
+ \q{
+ \qbox[show id={true}] \qbox[show id={true}] \qbox[show id={true}]\n
+ \qbox[show id={true}] \qbox[show id={true}] \qbox[show id={true}]
+ }
+\end{CODEBOX}
+\begin{CODEBOX}{}{Ex:20250302}
+ \q
+ {
+ \qloop[10]
+ {
+ \qloop[10]
+ {
+ \qbox[show id={true}]
+ }\n
+ }\n
+ }
+\end{CODEBOX}
+
+\subsubsection{色}
+
+ボックスに色を指定できる:
+\begin{itemize}
+ \item \texttt{color}
+ \iitem ボックス内部の色, 例えば\texttt{color=green}により\q[scale=0.5]{\qbox[color={green}]}
+ \item \texttt{frame color}
+ \iitem ボックスの枠の色, 例えば\texttt{frame color=cyan}により\q[scale=0.5]{\qbox[frame color={cyan}]}
+ \item \texttt{morphism color}
+ \iitem ボックス内の射の色, 例えば\texttt{morphism colon=red}により\q[scale=0.5]{\qbox[morphism color={red}, name={\(f\)}]}
+\end{itemize}
+\begin{CODEBOX}{}{Ex:20250259}
+ \q
+ {
+ \qbox
+ [
+ name={\(f\)},
+ color=green,
+ frame color=cyan,
+ morphism color=red
+ ]
+ }
+\end{CODEBOX}
+
+\paragraph{\texttt{N}, \texttt{S}, 等差数列}
+\qworldname は\texttt{pgffor}パッケージをロードしているため, リストに対して\verb|...|を用いた省略記法が使用できる:
+\begin{itemize}
+ \item \(1,...,10\): 公差\(1\)の等差数列
+ \item \(1,3,...,11\): 最初の二項に基づいた公差の等差数列
+\end{itemize}
+\begin{CODEBOX}{}{Ex:20250258}
+ \q{
+ \qbox[N={1,...,10}, S={1,3,...,11}]
+ }
+\end{CODEBOX}
+\begin{CODEBOX}{}{Ex:20250257}
+ \q{
+ \qbox[N={1,...,4,6,8,...,20}, S={1,...,20}, hlen=10]
+ }
+\end{CODEBOX}
+\begin{CODEBOX}{}{Ex:20250256}
+ \q{
+ \qbox[N={1,...,4,7,10,...,19,20,21,...,25}, S={1,...,25}, hlen=10]
+ }
+\end{CODEBOX}
+\subsubsection{相対座標}
+素子の位置座標を相対的に指定するコマンドである.
+なお, \(x\), \(y\) は整数である必要はなく, \verb|2^1.5|や\verb|cos(pi)|, \verb|sqrt(2)|などの\verb|\fpeval|の評価可能式であればよい.
+\begin{itemize}
+ \item \texttt{\textbackslash qmv[\(x\),\(y\)]}
+ \iitem (\(x\), \(y\)) だけ並行移動する.
+ \iitem デフォルトで\(x=y=0\).
+ \item \texttt{\textbackslash n[\(y\)]}
+ \iitem \(y\)だけ垂直降下し (\(Y\mapsto Y-y\)), X座標を\(0\)にする.
+ \iitem デフォルトで \(y=1\) であり, 改行のような挙動をみせる.
+ \item \texttt{\textbackslash qspace[\(x\)]}
+ \iitem \(x\) だけ水平移動する.
+ \iitem デフォルトで \(x=1\) であり, スペースキーのような挙動をみせる.
+ \iitem \texttt{\textbackslash qmv[\(x\),0]} と同義である.
+ \item \texttt{\textbackslash qreset[\(x\),\(y\)]}
+ \iitem 点 (\(x\),\(y\)) に移動する.
+\end{itemize}
+\q[scale=2.35]{
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (I) at (qworldat);
+\qbox
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (II) at (qworldat);
+\n
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (III) at (qworldat);
+\qspace
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (IV) at (qworldat);
+\qspace[0.5]
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (V) at (qworldat);
+\n[2]
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (VI) at (qworldat);
+\qmv[-2,-1]
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (VII) at (qworldat);
+\n
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (VIII) at (qworldat);
+\qspace[-3]
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (IX) at (qworldat);
+\qreset
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (X) at (qworldat);
+\qreset[-2,1]
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (XI) at (qworldat);
+\qbball
+\qwireup[dom={I}, cod={II},
+ label={Auto-update}, label side={below},
+ label at={midway}, color=cyan]
+\qwireup[dom={II},cod={III},
+ label={\texttt{\textbackslash n}},
+ label side={below}, color=cyan]
+\qwireup[dom={III},cod={IV},
+ label={\texttt{\textbackslash qspace}},
+ label side={below}, color=cyan]
+\qwireup[dom={IV},cod={V},
+ label={\texttt{\textbackslash qspace[0.5]}},
+ label at={at end}, label side={above right}, color=cyan]
+\qwireup[dom={V},cod={VI},
+ label={\texttt{\textbackslash n[2]}},color=cyan]
+\qwireup[dom={VI},cod={VII},
+ label={\texttt{\textbackslash qmv[-2,-1]}},color=cyan]
+\qwireup[dom={VII},cod={VIII},
+ label={\texttt{\textbackslash n}},color=cyan]
+\qwireup[dom={VIII},cod={IX},
+ label={\texttt{\textbackslash qspace[-3]}},
+ label side={below},color=cyan]
+\qwireup[dom={IX},cod={X},
+ label={\texttt{\textbackslash qreset}},color=cyan]
+\qwireup[dom={X},cod={XI},
+ label={\texttt{\textbackslash qreset[-2,1]}},color=cyan]
+\draw[help lines] (-4,-6) grid (2,2);
+}
+
+\subsubsection{絶対座標}
+\texttt{at}キーを用いれば, 絶対座標で位置を指定できる.
+\verb|\q{\qbox[at={2,3}]}|は\verb|\q{\qreset[2,3]\qbox}|と同義である.
+\begin{CODEBOX}{絶対座標}{Ex:20250254}
+ \q{
+ \coordinate (a) at (0,{\fpeval{2*sqrt(3)}});
+ \qbox[at=a]
+ \qbox[at={-2,0}]
+ \qbox[at={2,0}]
+ \fill[red] (a) circle (2pt);
+ \fill[red] (-2,0) circle (2pt);
+ \fill[red] (2,0) circle (2pt);
+ \node[below left] at (a) {a};
+ \node[below left] at (-2,0) {(-2,0)};
+ \node[below left] at (2,0) {(2,0)};
+ }
+\end{CODEBOX}
+\TeX 言語はチューリング完全である\footnote{例えば, チューリングマシンをシミュレートできることがLiterateProgramsの例で示されてる\cite{literateprograms}. また, Overleafの記事では, \TeX を用いたフィボナッチ数列の計算方法が紹介されており, そのチューリング完全性の実用性が理解できる\cite{overleaf}.}.
+すなわち, この世に存在するあらゆるアルゴリズムは \TeX で記述可能である.
+物理シミュレーション (天気予報や化学反応予測),
+経済シミュレーション (投資戦略),
+さらには人工知能の学習アルゴリズム (画像認識や自然言語処理) ------.\par
+どれも\TeX で実装できる.\par
+しかし, それを実際にやろうと思う人間は果たしてどれほどいるだろうか?
+筆者はソートアルゴリズムですら\TeX で書きたくない.
+それは表計算ソフトでスーパーマリオを作るようなものだからである.\par
+------ところが, である.\par
+筆者は``天啓''を受けた
+\footnote{元ネタ: 2011年1月20日, 匿名掲示板4chanにおいて``Sleep Sort''というソートアルゴリズムが公開された\cite{geniussorting,gfxhatena}. このアルゴリズムは\texttt{Bash}スクリプトを用いて各入力値をスレッドとして非同期に実行し, それぞれのスレッドの終了タイミング (指定された``スリープ''時間) を用いてリストをソートする手法である.}.
+そして\texttt{at}キーを用いたソートアルゴリズムを作り上げた.
+\begin{codebox}{}{Ex:genius}
+ \newcommand{\AtSort}[1][1]{
+ \q
+ {
+ \foreach \i in {#1}
+ {
+ \qbox[name={\i}, at={0, -\i}]\n
+ }
+ }
+ }
+ \AtSort[4,1,3,-1,6,2]
+\end{codebox}
+動く. 確かに動く.\par
+だが, 一つ問題がある.\par
+\textbf{リストの要素に絶対値の大きな値が含まれていると, 出力が紙面からはみ出してしまうのである}.
+
+\subsubsection{ワイヤーのカスタマイズ}
+\label{subsubsec:wirecustomization}
+\ref{subsubsec:boxcustomization} 項で登場したボックス\q[scale=0.5]{\qbox}のキーの中で, \texttt{id}, \texttt{vlen} はワイヤーでも有効である.
+ワイヤー特有のキーとして, 基本的なものは次の通りである:
+\begin{itemize}
+ \item \texttt{dom}: 入力点.
+ \item \texttt{cod}: 出力点.
+ \iitem デフォルト値は\texttt{dom}を \(1\) だけ北へ垂直移動した位置.
+ \item \texttt{label}: ワイヤーにラベルを付ける.
+ \item \texttt{label at}: ラベルのワイヤー上の位置.
+ \iitem デフォルトで midway.
+ \item \texttt{label side}: ラベルとワイヤーの位置関係.
+ \iitem デフォルトで right.
+\end{itemize}
+\begin{codebox}{}{ex:202503260951}
+ \q{
+ \qbox[n=3,hlen=1.5]
+ \qbox[at={1,2}, s=2, hlen=1.5]
+ \qbox[at={-1,4}, s=2, hlen=1.5]
+ \qwire[dom={N-1-1}, cod={S-1-3}]
+ \qwire[dom={N-2-1}, cod={S-1-2}]
+ \qwire[dom={N-3-1}, cod={S-2-2}]
+ \qwire[dom={N-1-2}, cod={S-2-3}]
+ }
+\end{codebox}
+ワイヤー\q[scale=0.6]{\qwire}のバウンディングボックスは, \q[scale=0.5]{\qbox}と同じである:
+\begin{codebox}{}{ex:202503220926}
+ \q
+ {
+ \qwire\qbox\qwire\n
+ \qbox\qwire\qbox
+ \qbball
+ }
+\end{codebox}
+ワイヤーのバウンディングボックスは, \texttt{dom}を入力点, \texttt{cod}を出力点とするようなボックスのそれと一致する:
+\begin{itemize}
+ \item \(\q[scale=0.5]{\qwire}\) と \(\q[scale=0.5]{\qbox}\)
+ \item \(\q[scale=0.5]{\qwire[dom={0,0},cod={1,1}]}\) と \(\q[scale=0.5]{\qbox[p=2,S={1},N={2}]}\)
+\end{itemize}
+
+\begin{CODEBOX}{}{Ex:20250309}
+ \q[scale=3]{
+ \coordinate (A) at (0,0);
+ \coordinate (B) at (2,3);
+ \qwire[dom=A, cod={1,1}, label={wire1}, label side={left}]
+ \qwire[dom={1,1}, cod=B, label={wire2}, label at={near end}]
+ \node[fill=red, circle] at (A) {A};
+ \node[fill=green, circle] at (B) {B};
+ \node[fill=yellow, circle] at (1,1) {(1,1)};
+ \qbball
+ }
+\end{CODEBOX}
+
+\begin{CODEBOX}{}{Ex:20250301}
+ \q{
+ \qloop[5]{
+ \qloop[10]{
+ \qbox[show id={true}]
+ }\n
+ }\n
+ \qspace[3]\qbox[s=2,show id={true},id={ego1}]\n\n
+ \qspace[6]\qbox[n=2,show id={true},id={ego2}]\n\n
+ \qloop[5]{
+ \qloop[10]{
+ \qbox[show id={true}]
+ }\n
+ }
+ \qwire[dom={O-1-55},cod={I-1-51}]
+ \qwire[dom={O-1-60},cod={I-1-52}]
+ \qwire[dom={O-1-51},cod={I-1-45}]
+ \qwire[dom={O-1-52},cod={I-2-51}]
+ \qwire[dom={O-2-52},cod={I-1-48}]
+ }
+\end{CODEBOX}
+
+\paragraph{arrowtype}
+ワイヤーのスタイルは, \texttt{arrowtype}キーを用いて指定する.
+\begin{codebox}[center lower,listing side text,righthand width=5cm,bicolor,colbacklower=white]{}{wirearrowtype}
+ \q{
+ \qwire[arrowtype=dotted]
+ \qwire[arrowtype=dashed]
+ \qwire[arrowtype={dash dot}]
+ \qwire[arrowtype={dash dot dot}]
+ \qwire[arrowtype={densely dotted}]
+ \qwire[arrowtype={loosely dotted}]
+ }
+\end{codebox}
+
+\begin{codebox}[center lower,listing side text,righthand width=5cm,bicolor,colbacklower=white]{}{boxarrowtype}
+ \q{
+ \qbox[dom arrowtype={dotted}]
+ \qbox[cod arrowtype={dotted}]
+ \qbox[arrowtype={dotted}]
+ }
+\end{codebox}
+
+\begin{codebox}[center lower,listing side text,righthand width=5cm,bicolor,colbacklower=white]{}{spiderarrowtype}
+ \q{
+ \qspider[dom arrowtype={dotted}]
+ \qspider[cod arrowtype={dotted}]
+ \qspider[arrowtype={dotted}]
+ }
+\end{codebox}
+
+\subsubsection{キャンバス}
+\begin{verbatim}
+ \q[オプション]
+ {
+ コマンド
+ コマンド
+ ...
+ }
+\end{verbatim}
+キャンバスのオプションで図式全体を変換できる.
+\begin{codebox}{}{Ex:20250253}
+ \q[scale=2,rotate=30]{\qbox}
+\end{codebox}
+キャンバスを傾けることもできる.
+\begin{CODEBOX}{}{Ex:20250252}
+ \def\anglerot{-55}
+ \tikzset
+ {
+ my slant style/.style={
+ yslant=-cot(\anglerot),
+ xscale=sin(-\anglerot)
+ }
+ }
+ \q[my slant style]
+ {
+ \qcaprev\qcap\n
+ \qwire\qbraidinv\qwiredown\n
+ \qwiredown\qbraidinv\qwire\n
+ \qcup\qcuprev
+ \qBBall
+ }
+\end{CODEBOX}
+\begin{CODEBOX}{}{Ex:20250251}
+ \def\anglerot{-60}
+ \tikzset
+ {
+ my slant style/.style={
+ yslant=-cot(\anglerot),
+ xscale=sin(-\anglerot)
+ }
+ }
+ \[\q[my slant style]
+ {
+ \qbraid\qwire\n
+ \qwire\qbraid\n
+ \qbraid\qwire
+ }
+ =\q[my slant style]
+ {
+ \qbraid\qwire\n
+ \qwire\qwire\qwire\n
+ \qbraid[num R=2]
+ \qbb[red][1]
+ \qbb[cyan][6]
+ }
+ =\q[my slant style]
+ {
+ \qbraid[ num R=2]\n
+ \qwire\qwire\qwire\n
+ \qwire\qbraid
+ \qbb[red][6]
+ \qbb[cyan][1]
+ }
+ =\q[my slant style]
+ {
+ \qwire\qbraid\n
+ \qbraid\qwire\n
+ \qwire\qbraid
+ }\]
+\end{CODEBOX}
+
+\section{モノイダル圏と\qworldname}
+本章では, \qworldname パッケージを用いてモノイダル圏の図式計算をどのように表現できるかを探る.
+\qworldname は, それ単体で機能し, 図式を扱うことができるが,
+数式をより柔軟に表現するために, その基盤となるパッケージ (\texttt{amsmath}, \texttt{amssymb}, \texttt{mathtools}) を使用する.
+再現の際は, 以下のようにパッケージを適宜ロードせよ:
+\begin{verbatim}
+ \usepackage{amsmath, amssymb, mathtools}
+ \usepackage{qworld}
+\end{verbatim}
+\begin{codebox}{ドメイン}{ex:dom2}
+ \[
+ \operatorname{dom}\left(
+ \q{
+ \qbox[name={\(f\)}]
+ \symbolI{\(x\)}
+ \symbolO{\(y\)}
+ }
+ \right)=\q{\qwire[label={\(x\)}]}
+ \]
+\end{codebox}
+\begin{codebox}{コドメイン}{ex:cod2}
+ \[
+ \operatorname{cod}\left(
+ \q{
+ \qbox[name={\(f\)}]
+ \symbolI{\(x\)}
+ \symbolO{\(y\)}
+ }
+ \right)=\q{\qwire[label={\(y\)}]}
+ \]
+\end{codebox}
+
+\begin{codebox}{合成 (垂直合成)}{def:composite}
+ \[
+ \q{
+ \qbox[name={\(f\)}]\qcirc\qbox[name={\(g\)}]
+ }=\q{\qbox[name={\(f\circ g\)}]}
+ \]
+\end{codebox}
+\begin{CODEBOX}{結合律}{Ex:20250250}
+ \[
+ \forall\q{\qbox[name={\(h\)}] \symbolI{\(z\)} \symbolO{\(w\)}},
+ \q{\qbox[name={\(g\)}] \symbolI{\(y\)} \symbolO{\(z\)}},
+ \q{\qbox[name={\(f\)}] \symbolI{\(x\)} \symbolO{\(y\)}}:
+ \q{
+ \qbox[name={\(h\)}]\qcirc\qbox[name={\(g\)}]\qcirc\qbox[name={\(f\)}]
+ \qbb[2,3]
+ }
+ =
+ \q{
+ \qbox[name={\(h\)}]\qcirc\qbox[name={\(g\)}]\qcirc\qbox[name={\(f\)}]
+ \qbb[1,2]
+ }
+ \]
+\end{CODEBOX}
+\begin{CODEBOX}{恒等律}{IdentityLaw}
+ \begin{equation}
+ \label{law:identity}
+ \forall\q
+ {
+ \qbox[name={\(f\)}]
+ \symbolS{\(x\)}
+ \symbolN{\(y\)}
+ }\colon
+ \q{
+ \qwire[label={\(y\)}]\n
+ \qbox[name={\(f\)}]
+ }=\q{
+ \qbox[name={\(f\)}]
+ }=\q{
+ \qbox[name={\(f\)}]\n
+ \qwire[label={\(x\)}]
+ }
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{テンソル積 (並列合成)}{monoidalproduct}
+ \[
+ \q
+ {
+ \qbox[name={\(f\)}] \qbox[name={\(g\)}]
+ \symbolO{\(Y_1\)}[id=1] \symbolO{\(Y_2\)}
+ \symbolI{\(X_1\)}[id=1] \symbolI{\(X_2\)}
+ }
+ =\q
+ {
+ \qbox[name={\(f\otimes g\)}, p=2]
+ \symbolO{\(Y_1\)} \symbolO[2]{\(Y_2\)}
+ \symbolI{\(X_1\)} \symbolI[2]{\(X_2\)}
+ }
+ =\q
+ {
+ \qbox[name={\(f\otimes g\)}]
+ \symbolO{\(Y_1\otimes Y_2\)}
+ \symbolI{\(X_1\otimes X_2\)}
+ }
+ \]
+\end{CODEBOX}
+
+\begin{CODEBOX}{交換則 (\(\bigotimes\)の共変関手性)}{InterchangeLaw}
+ \begin{equation}
+ \label{law:interchange}
+ \q
+ {
+ \qbox[name={\(f_2\)}]\qbox[name={\(g_2\)}]\n
+ \qbox[name={\(f_1\)}]\qbox[name={\(g_1\)}]
+ \qbb[1,2]
+ \qbb[3,4]
+ }
+ =\q
+ {
+ \qbox[name={\(f_2\)}]\qbox[name={\(g_2\)}]\n
+ \qbox[name={\(f_1\)}]\qbox[name={\(g_1\)}]
+ \qbb[2,4]
+ \qbb[1,3]
+ }
+ \end{equation}
+\end{CODEBOX}
+
+\begin{CODEBOX}{単位対象}{Ex:20250244}
+ \begin{gather}
+ \q{ \qwire[arrowtype=dotted]}
+ =\q{ \qunitob \qbball }\\
+ \q{ \qunitob \qbox[name={\(f\)}] \qbball }
+ =\q{ \qbox[name={\(f\)}] }
+ =\q{ \qbox[name={\(f\)}] \qunitob \qbball }\label{cond:unitob}
+ \end{gather}
+\end{CODEBOX}
+\begin{CODEBOX}{}{Ex:20250243}
+ \[
+ \q{\qunitob\qbball}
+ =\q{\qloop[2]{\qunitob}\qbball}
+ =\q{\qloop[3]{\qunitob}\qbball}
+ =\cdots
+ \]
+\end{CODEBOX}
+
+\subsection{特殊ボックス}
+
+\begin{itemize}
+ \item \texttt{\textbackslash qstate}: \(\q[scale=0.5]{\qstate}\) (\(0\)入力).
+ \item \texttt{\textbackslash qeffect}: \(\q[scale=0.5]{\qeffect}\) (\(0\)出力).
+ \item \texttt{\textbackslash qscalar}: \(\q[scale=0.5]{\qscalar}\) (\(0\)入力, \(0\)出力).
+ \item \texttt{\textbackslash qasym}: \(\q[scale=0.5]{\qasym}\) (非対称ボックス).
+ \item \texttt{\textbackslash qtrans}: \(\q[scale=0.5]{\qtrans}\) (転置ボックス). \pageref{subsub:transpose}ページの\ref{subsub:transpose}項.
+ \item \texttt{\textbackslash qadj}: \(\q[scale=0.5]{\qadj}\) (随伴ボックス). \pageref{subsub:adjoint}ページの\ref{subsub:adjoint}項.
+ \item \texttt{\textbackslash qconj}: \(\q[scale=0.5]{\qconj}\) (共役ボックス). \pageref{subsub:conj}ページの\ref{subsub:conj}項.
+\end{itemize}
+\subsubsection{\(0\)入力}
+\label{subsub:zeroinput}
+\begin{codebox}{}{def:state}
+ \q{\qstate}
+\end{codebox}
+\subsubsection{\(0\)出力}
+\label{subsub:zerooutput}
+\begin{codebox}{}{def:effect}
+ \q{\qeffect}
+\end{codebox}
+\subsubsection{\(0\)入力\(0\)出力}
+\label{subsub:zeroinputoutput}
+\begin{codebox}{}{def:scalar}
+ \q{\qscalar}
+\end{codebox}
+\paragraph{\(\q[scale=0.4]{\qeffect\qstate\qwire[dom={O-1-2}, cod={1,2}]\qwire[dom={1,-1}, cod={I-1-1}]}=\q[scale=0.4]{\qwire[vlen=0.5]\n\qstate\n\qeffect\n[0.5]\qwire[vlen=0.5]}=\q[scale=0.4]{\qstate\qeffect\qwire[dom={O-1-1}, cod={1,2}]\qwire[dom={1,-1}, cod={I-1-2}]}\)}
+\begin{CODEBOX}{}{coherence:unitor}
+ \[
+ \q{
+ \qeffect\qstate
+ \qwire[dom={O-1-2}, cod={1,2}]
+ \qwire[dom={1,-1}, cod={I-1-1}]
+ }=\q{
+ \qwire[arrowtype=dotted]\qwire\n
+ \qeffect\qstate\n
+ \qwire\qwire[arrowtype=dotted]
+ }\stackrel{\text{(\ref{law:interchange}, \ref{law:identity})}}{=}\q{
+ \qwire[arrowtype=dotted]\qstate\n
+ \qeffect\qwire[arrowtype=dotted]
+ }\stackrel{\text{(\ref{law:interchange}, \ref{cond:unitob})}}{=}\q{\qstate\n\qeffect}
+ \]
+\end{CODEBOX}
+\begin{CODEBOX}{表現行列}{matrixrepre}
+ \begin{align*}
+ \q{\qbox[name={\(f\)}]}
+ &=\sum\limits_{i,j}
+ \left(
+ \q{
+ \qeffect[name={\(w_j^{\dagger}\)}]\n
+ \qbox[name={\(f\)}]\n
+ \qstate[name={\(v_i\)}]\n
+ }\q{
+ \qstate[name={\(w_j\)}]\n
+ \qeffect[name={\(v_i^{\dagger}\)}]
+ }
+ \right)
+ =\sum\limits_{i,j}\left(
+ \q{
+ \qscalar[name={\(w_j^{\dagger}\circ f\circ v_i\)}]
+ }
+ \q{
+ \qstate[name={\(w_j\)}]\n
+ \qeffect[name={\(v_i^{\dagger}\)}]
+ }
+ \right)\\
+ &=\begin{pmatrix}
+ \q{\qscalar[name={\(w_1^{\dagger}\circ f\circ v_1\)}]}&&\cdots&&\q{\qscalar[name={\(w_1^{\dagger}\circ f\circ v_{m}\)}]}\\
+ &\ddots&&&\\
+ \vdots& &\q{\qscalar[name={\(w_j^{\dagger}\circ f\circ v_i\)}]}& &\vdots\\
+ &&&\ddots&\\
+ \q{\qscalar[name={\(w_n^{\dagger}\circ f\circ v_1\)}]}&&\cdots&&\q{\qscalar[name={\(w_n^{\dagger}\circ f\circ v_m\)}]}
+ \end{pmatrix}
+ \end{align*}
+\end{CODEBOX}
+
+\subsection{モノイド対象}
+\label{sub:monoid}
+\begin{itemize}
+ \item \texttt{\textbackslash qmul}: \(\q[scale=0.5]{\qmul}\)
+ \iitem \texttt{hlen=2}: \(\q[scale=0.5]{\qmul[hlen=2]}\)
+ \item \texttt{\textbackslash qcomul}: \(\q[scale=0.5]{\qcomul}\)
+ \iitem \texttt{hlen=2}: \(\q[scale=0.5]{\qcomul[hlen=2]}\)
+ \item \texttt{\textbackslash qunit}: \(\q[scale=0.5]{\qunit}\)
+ \item \texttt{\textbackslash qcounit}: \(\q[scale=0.5]{\qcounit}\)
+ \item \texttt{\textbackslash qspider}: \(\q[scale=0.5]{\qspider}\)
+\end{itemize}
+
+\begin{CODEBOX}{結合律}{AssociateLaw}
+ \begin{equation}
+ \label{law:associate}
+ \q{
+ \qspace[0.5]\qmul[hlen=2]\n
+ \qmul[hlen=2]\qspace[-0.5]\qwire
+ }=\q{
+ \qmul[hlen=2]\n
+ \qwire\qspace[-0.5]\qmul[hlen=2]
+ }
+ \end{equation}
+\end{CODEBOX}
+ \begin{CODEBOX}{単位律}{UnitLaw}
+ \begin{equation}
+ \label{law:unit}
+ \q{
+ \qmul[hlen=2]\n
+ \qwire\qunit
+ }=\q{\qwire[vlen=2]}
+ =\q
+ {
+ \qmul[hlen=2]\n
+ \qunit\qwire
+ }
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{余結合律}{CoassociateLaw}
+ \begin{equation}
+ \label{law:coassociate}
+ \q{
+ \qcomul[hlen=2]\qspace[-0.5]\qwire\n
+ \qspace[0.5]\qcomul[hlen=2]
+ }
+ =\q{
+ \qwire\qspace[-0.5]\qcomul[hlen=2]\n
+ \qcomul[hlen=2]
+ }
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{余単位律}{CounitLaw}
+ \begin{equation}
+ \label{law:counit}
+ \q{
+ \qcounit\qwire\n
+ \qcomul[hlen=2]
+ }
+ =\q{\qwire[vlen=2]}
+ =\q{
+ \qwire\qcounit\n
+ \qcomul[hlen=2]
+ }
+ \end{equation}
+\end{CODEBOX}
+
+\subsubsection{フロベニウス則}
+\begin{FCODEBOX}{フロベニウス則}{FrobeniusLaw}
+ \[
+ \q{
+ \qwire\qmul[color=black]\n
+ \qcomul\qwire
+ }=\q{
+ \qcomul\n
+ \qmul[color=black]
+ }=\q{
+ \qmul[color=black]\qwire\n
+ \qwire\qcomul
+ }
+ \]
+ \end{FCODEBOX}
+ 失敗例として挙げられているが, 実際にはフロベニウス則が正確に記述されている.
+ むしろ, 全てのワイヤーの間隔が整数値に保たれているため, 垂直合成時の調整が容易であり, 場合によっては調整自体が不要となる可能性がある.
+ これは単なる好みの問題に過ぎない.
+ \verb|\qmul|や\verb|\qcomul|のデフォルトの描画結果が\(\q[scale=0.5]{\qmul}\), \(\q[scale=0.5]{\qcomul}\)であるのは,
+ \verb|\qbox[s=2]|や\verb|\qbox[n=2]|の描画結果が\(\q[scale=0.5]{\qbox[s=2]}\), \(\q[scale=0.5]{\qbox[n=2]}\)であることの名残である.
+ このような場合の調整キーとして\verb|hlen|がある:
+ \begin{CODEBOX}{フロベニウス則}{FrobeniusLaw2}
+ \[
+ \q{
+ \qwire\qmul[hlen=2,color=black]\n
+ \qcomul[hlen=2]\qwire
+ }=\q{
+ \qcomul[hlen=2]\n
+ \qmul[hlen=2,color=black]
+ }=\q{
+ \qmul[hlen=2,color=black]\qwire\n
+ \qwire\qcomul[hlen=2]
+ }
+ \]
+ \end{CODEBOX}
+
+ \verb|\qspider|は, ノードがボックスではなく円であることを除けば, \verb|\qbox|と同じである.
+ \begin{codebox}{}{spider202503240521}
+ \[\q{\qspider}\]
+ \[\q{\qspider[n=0]}\]
+ \[\q{\qspider[n=0,s=4]}\]
+ \[\q{\qspider[n=2,s=4]}\]
+ \[\q{\qspider[N={1,4},s=4]}\]
+ \[\q{\qspider[N={2,3},s=4]}\]
+ \[\q{\qspider[p=3,N={2},S={1,3}]}\]
+ \[\q{\qspider[p=3,N={2},S={1,3},hlen=2]}\]
+ \[\q{\qspider[n=1,s=2,hlen=2]}\]
+ \end{codebox}
+
+ \begin{CODEBOX}{}{spider202503270738}
+ \[
+ \q{
+ \qspider[n=2, s=4, hlen=4]\qwire\qwire\n
+ \qwire\qwire\qspider[n=4, s=2, hlen=4]
+ }
+ =\q{\qspider[p=4]}\]
+ \end{CODEBOX}
+
+\subsection{組紐}
+\label{sub:braidedcategory}
+\begin{itemize}
+ \item \texttt{\textbackslash qbraid}: \(\q[scale=0.5]{\qbraid}\)
+ \item \texttt{\textbackslash qbraidinv}: \(\q[scale=0.5]{\qbraidinv}\)
+ \item \texttt{\textbackslash qsym}: \(\q[scale=0.5]{\qsym}\)
+\end{itemize}
+
+\paragraph{\texttt{num L} と \texttt{num R} キー}
+ワイヤーを交叉させた図式を描画するコマンド\verb|\qbraid|, \verb|\qbraidinv|, \verb|\qsym|に対し,
+\texttt{num L}は左側のワイヤーの本数を, \texttt{num R}は右側のワイヤーの本数を指定するキーである.
+例えば, \texttt{num L=3}, \texttt{num R=2} とすれば, \(\q{\qbraid[num L=3,num R=2]}\)が描画される.
+双方デフォルト値はともに\(1\)である.
+
+\begin{CODEBOX}{六角形等式}{hexagon}
+ \begin{equation}
+ \label{eq:hexagon}
+ \q
+ {
+ \qbraid[num R=2,,hlen=2,vlen=2]
+ }
+ =
+ \q
+ {
+ \qwire\qbraid\n
+ \qbraid\qwire
+ }
+ \qquad
+ \q
+ {
+ \qbraid[num L=2,,hlen=2,vlen=2]
+ }
+ =
+ \q
+ {
+ \qbraid\qwire\n
+ \qwire\qbraid
+ }
+ \end{equation}
+\end{CODEBOX}
+\paragraph{\({\q[scale=0.5]{\qbraid}}^{-1}=\q[scale=0.5]{\qbraidinv}\)}
+\begin{CODEBOX}{}{braidisom2}
+ \begin{equation}
+ \label{eq:inverse:braid}
+ \q
+ {
+ \qbraidinv\n
+ \qbraid
+ }
+ =
+ \q
+ {
+ \qwire[vlen=2]\qwire[vlen=2]
+ }
+ \qquad
+ \q
+ {
+ \qbraid\n
+ \qbraidinv
+ }
+ =
+ \q
+ {
+ \qwire[vlen=2]\qwire[vlen=2]
+ }
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{組紐の自然性}{braidnatural}
+ \begin{equation}
+ \label{cond:naturality:braid}
+ \q
+ {
+ \qbraid\n
+ \qbox[name={\(f\)}]\qbox[name={\(g\)}]
+ }
+ =
+ \q
+ {
+ \qbox[name={\(g\)}]\qbox[name={\(f\)}]\n
+ \qbraid
+ }
+ \qquad
+ \q
+ {
+ \qbraidinv\n
+ \qbox[name={\(f\)}]\qbox[name={\(g\)}]
+ }
+ =
+ \q
+ {
+ \qbox[name={\(g\)}]\qbox[name={\(f\)}]\n
+ \qbraidinv
+ }
+ \end{equation}
+\end{CODEBOX}
+
+
+コマンド\verb|\qbraid|, \verb|\qbraidinv|, \verb|\qsym|に対し,
+\texttt{num R=0}と指定すると, \(\q[scale=0.65]{\qbraid[num R=0]}\)のように右側のワイヤーが点線となる.
+\texttt{num L=0}と指定すると, \(\q[scale=0.65]{\qbraid[num L=0]}\)のように左側のワイヤーが点線となる.
+これは, 単位対象を空図式, すなわち\(0\)本のワイヤーとして表すことに由来している.
+\begin{CODEBOX}{}{stateeffect3dimbraid}
+ \begin{gather}
+ \q{
+ \qwire\qstate\n
+ \qwire
+ }
+ =\q{
+ \qwire\qstate\n
+ \qbraid[num L=0]
+ }\stackrel{\text{(\ref{cond:naturality:braid})}}{=}\q{
+ \qbraid\n
+ \qstate\qwire
+ }\label{iso:u:state}\\
+ \q{
+ \qunitob\qwire\n
+ \qeffect\qwire
+ }
+ =\q{
+ \qbraid[num L=0]\n
+ \qeffect\qwire
+ }\stackrel{\text{(\ref{cond:naturality:braid})}}{=}\q{
+ \qwire\qeffect\n
+ \qbraid
+ }\label{iso:u:effect}
+ \end{gather}
+\end{CODEBOX}
+\begin{CODEBOX}{}{iso:spatial:scalar}
+ \begin{equation}
+ \label{iso:spatial}
+ \q{\qscalar\qmv[0,-1]\qwire[vlen=3]}
+ =\q{\qbraid[num L=0]\n\qscalar\qmv[0,-1]\qwire[vlen=2]}
+ \stackrel{\text{(\ref{cond:naturality:braid})}}{=}
+ \q{\qwire[vlen=2]\qscalar\n\qbraid[num L=0]}
+ =\q{\qscalar\qmv[-2,-1]\qwire[vlen=3]}
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{}{iso:spatial:state.effect}
+ \[
+ \q{\qscalar\qmv[0,-1]\qwire[vlen=3]}
+ =\q{\qeffect\n\qstate\qmv[0,-0.5]\qwire[vlen=3]}
+ \stackrel{\text{(\ref{iso:u:effect})}}{=}
+ \q{\qwire\qeffect\n\qbraid\n\qstate\qwire}
+ \stackrel{\text{(\ref{iso:u:state})}}{=}
+ \q{\qeffect\n\qstate\qmv[-2,-0.5]\qwire[vlen=3]}
+ =\q{\qscalar\qmv[-2,-1]\qwire[vlen=3]}
+ \]
+\end{CODEBOX}
+\paragraph{\(\q[scale=0.5]{\qbraid\qwire\n\qwire\qstate[n=2]}=\q[scale=0.5]{\qwire\qbraidinv\n\qstate[n=2]\qwire}\)}
+\begin{CODEBOX}{}{dashedbraid}
+ \begin{align}
+ \q
+ {
+ \qbraid\qwire\n
+ \qwire\qstate[n=2]
+ }
+ &=\q
+ {
+ \qbraid\qwire\n
+ \qwire\qstate[n=2]\n\n
+ \qbraidinv[num L=0,vlen=2,hlen=2.5]
+ }
+ \stackrel{\text{(\ref{cond:naturality:braid})}}{=}\q
+ {
+ \qbraid\qwire\n\n
+ \qbraidinv[num L=2,vlen=2]\n
+ \qstate[n=2]\qwire\n
+ \qspace[0.5]\qwire[arrowtype={densely dotted}]\qspace[0.5]\qwire
+ }\\
+ &=\q
+ {
+ \qbraid\qwire\n\n
+ \qbraidinv[num L=2,vlen=2]\n
+ \qstate[n=2]\qwire\n
+ \qspace[2]\qwire
+ \qbb[green][3]
+ }\stackrel{\text{(\ref{eq:hexagon})}}{=}\q
+ {
+ \qbraid\qwire\n
+ \qbraidinv\qwire\n
+ \qwire\qbraidinv\n
+ \qstate[n=2]\qwire\n
+ \qspace[2]\qwire
+ \qbb[green][3,...,6]
+ }\\
+ &=\q
+ {
+ \qbraid\qwire\n
+ \qbraidinv\qwire\n
+ \qwire\qbraidinv\n
+ \qstate[n=2]\qwire\n
+ \qspace[2]\qwire
+ \qbb[orange][1,3]
+ }
+ \stackrel{\text{(\ref{eq:inverse:braid})}}{=}\q
+ {
+ \qloop[2]{\qloop[3]{\qwire}\n}
+ \qwire\qbraidinv\n
+ \qstate[n=2]\qwire\n
+ \qspace[2]\qwire
+ \qbb[orange][1,2,4,5]
+ }
+ \end{align}
+\end{CODEBOX}
+
+\begin{CODEBOX}{ヤン-バクスター方程式}{YangBaxer}
+ \[
+ \q
+ {
+ \qbraid\qwire\n
+ \qwire\qbraid\n
+ \qbraid\qwire
+ }=\q
+ {
+ \qwire\qbraid\n
+ \qbraid\qwire\n
+ \qwire\qbraid
+ }
+ \]
+\end{CODEBOX}
+
+\begin{codebox}[center lower,listing side text,righthand width=5cm,bicolor,colbacklower=white]{可換性}{def:cmt}
+ \begin{equation}
+ \label{cond:cmt}
+ \q{\qmul[hlen=2]}=\q{\qmul[hlen=2]\n\qbraid}
+ \end{equation}
+\end{codebox}
+\begin{codebox}[center lower,listing side text,righthand width=5cm,bicolor,colbacklower=white]{余可換性}{def:ccmt}
+ \begin{equation}
+ \label{cond:ccmt}
+ \q{\qcomul[hlen=2]}=\q{\qbraid\n\qcomul[hlen=2]}
+ \end{equation}
+\end{codebox}
+\subsubsection{双代数}
+\begin{CODEBOX}{双代数則}{law:bialgebra}
+ \begin{gather}
+ \q{\qcomul[hlen=2]\qcirc\qmul[hlen=2, color=black]}
+ =\q{
+ \qmul[hlen=2, color=black]\qmul[hlen=2, color=black]\n
+ \qwire\qbraid\qwire\n\qcomul[hlen=2]\qcomul[hlen=2]
+ }\\
+ \q{\qspace[0.5]\qcounit\qcirc\qmul[hlen=2, color=black]}
+ =\q{\qcounit\qcounit}\\
+ \q{\qcomul[hlen=2]\qcirc\qspace[0.5]\qunit[color=black]}
+ =\q{\qunit[color=black]\qunit[color=black]}\\
+ \q{\qcounit\n\qunit[color=black]}
+ =\q{\qunitob}
+ \end{gather}
+\end{CODEBOX}
+\subsubsection{ホップ代数}
+\begin{CODEBOX}{対合射とホップ則}{law:hopf}
+ \begin{equation}
+ \q{
+ \qmul[hlen=2, color=black]\n
+ \qwire\qbox[name={\(s\)}]\n
+ \qcomul[hlen=2]
+ }
+ =\q{\qunit[color=black]\qcirc\qcounit}
+ =\q{
+ \qmul[hlen=2, color=black]\n
+ \qbox[name={\(s\)}]\qwire\n
+ \qcomul[hlen=2]
+ }
+ \end{equation}
+\end{CODEBOX}
+\subsubsection{平衡圏}
+\label{subsub:balancedcategory}
+\begin{CODEBOX}{ツイスト}{def:twist}
+ \begin{gather}
+ \q{
+ \qbox[name={\(\theta_{X\otimes Y}\)}, p=2]
+ \symbolI{\(X\)}\symbolI[2]{\(Y\)}
+ \symbolO{\(X\)}\symbolO[2]{\(Y\)}
+ }
+ =\q{
+ \qbox[name={\(\theta_X\)}]\qbox[name={\(\theta_Y\)}]\n
+ \qbraid\n\qbraid
+ }\label{cond:twist}\\
+ \q{\qbox[name={\(\theta_I\)}]}=\q{\qunitob}\label{cond:i:twist}\\
+ \forall\q{\qbox[name={\(f\)}]}\colon
+ \q{
+ \qbox[name={\(\theta_{\operatorname{cod}(f)}\)}, hlen=1.5]\n
+ \qbox[name={\(f\)}, hlen=1.5]
+ }=\q{
+ \qbox[name={\(f\)}, hlen=1.5]\n
+ \qbox[name={\(\theta_{\operatorname{dom}(f)}\)}, hlen=1.5]
+ }\label{cond:naturality:twist}
+ \end{gather}
+\end{CODEBOX}
+\subsubsection{自明な平衡圏, 対称モノイダル圏}
+\label{subsub:symmetry}
+\begin{codebox}[center lower,listing side text,righthand width=5cm,bicolor,colbacklower=white]{対称}{defeq:sym}
+ \begin{equation}
+ \label{cond:symmetry}
+ \q
+ {
+ \qbraid\n
+ \qbraid
+ }
+ =\q{\qloop[2]{\qwire[vlen=2]}}
+ \end{equation}
+\end{codebox}
+\begin{codebox}{}{wowsym}
+ \[\q{\qbraid}=\q{\qbraidinv}\]
+\end{codebox}
+\begin{codebox}{シンメトリー}{def:sym}
+ \q{\qsym}
+\end{codebox}
+
+\subsection{双対性}
+\label{subsub:duality}
+\begin{itemize}
+ \item \texttt{\textbackslash qcap}: \(\q[scale=0.6]{\qcap}\)
+ \item \texttt{\textbackslash qcup}: \(\q[scale=0.6]{\qcup}\)
+ \item \texttt{\textbackslash qcaprev}: \(\q[scale=0.6]{\qcaprev}\)
+ \item \texttt{\textbackslash qcuprev}: \(\q[scale=0.6]{\qcuprev}\)
+\end{itemize}
+
+\begin{CODEBOX}{ジグザグ等式}{Snake}
+ \begin{equation}
+ \label{eq:snake}
+ \q{
+ \qcap\qwireup\n
+ \qwireup\qcup
+ }
+ =\q{
+ \qwireup[vlen=2]
+ }
+ \qquad\text{と}\qquad
+ \q{
+ \qwiredown\qcap\n
+ \qcup\qwiredown
+ }
+ =\q{
+ \qwiredown[vlen=2]
+ }
+ \end{equation}
+\end{CODEBOX}
+
+\subsubsection{転置ボックス}
+\label{subsub:transpose}
+\begin{CODEBOX}{転置}{Transpose}
+ \[
+ \left(\q{\qwire\n\qasym[name={\(f\)}]\n\qwire}\right)^T
+ =\q{
+ \qwire\n
+ \qtrans[name={\(f\)}]\n
+ \qwire
+ }\coloneq\q{
+ \qwire\qcap\n
+ \qwire\qasym[name={\(f\)}]\qwire\n
+ \qcup\qwire
+ }
+ \]
+\end{CODEBOX}
+\begin{CODEBOX}{}{ex:202504021826}
+ \begin{equation}
+ \label{eq:wirerev}
+ \left(\q{\qwireup}\right)^T=\q{\qwiredown}
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{スライディング}{Sliding}
+ \begin{equation}
+ \label{eq:sliding}
+ \q{
+ \qcap \n
+ \qasym \qwire
+ }
+ =\q{
+ \qcap \n
+ \qwire \qtrans
+ }
+ \qquad
+ \q{
+ \qwire \qasym \n
+ \qcup
+ }
+ =\q{
+ \qtrans \qwire \n
+ \qcup
+ }
+ \end{equation}
+\end{CODEBOX}
+
+\begin{CODEBOX}{}{MapStateDuality}
+ \begin{equation}
+ \begin{array}{rccc}
+ \operatorname{NAME}\colon & \operatorname{Hom}(X,Y)
+ & \to & \operatorname{Hom}(I,X^T\otimes Y)\\
+ & \rotatebox{90}{\(\in\)} && \rotatebox{90}{\(\in\)}\\
+ & \q{
+ \qasym
+ \symbolI{\(X\)}
+ \symbolO{\(Y\)}
+ } & \mapsto
+ & \q{
+ \qwire[label={\(X^T\)}, label at={at end}, label side={above}] \qasym \n
+ \qcup
+ \symbolO{\(Y\)}[id=2]
+ }
+ \end{array}
+ \end{equation}
+\end{CODEBOX}
+
+\begin{CODEBOX}{}{MapEffectDuality}
+ \begin{equation}
+ \begin{array}{rccc}
+ \operatorname{CONAME}\colon & \operatorname{Hom}(X,Y)
+ & \to & \operatorname{Hom}(X\otimes Y^T,I)\\
+ & \rotatebox{90}{\(\in\)}&&\rotatebox{90}{\(\in\)}\\
+ & \q{
+ \qasym
+ \symbolI{\(X\)}
+ \symbolO{\(Y\)}
+ } & \mapsto
+ & \q{
+ \qcap \n
+ \qasym \qwire[label={\(Y^T\)}, label at={at start}, label side={below}]
+ \symbolI{\(X\)}[id=2]
+ }
+ \end{array}
+ \end{equation}
+\end{CODEBOX}
+
+\subsubsection{コピー禁止定理}
+\label{subsub:nocopy}
+\begin{CODEBOX}{一様コピー}{def:copy}
+ \begin{gather}
+ \q{
+ \qwire\qbraid\qwire\n
+ \qcomul[hlen=2, name={\(\text{copy}_X\)}]\qcomul[hlen=2, name={\(\text{copy}_Y\)}]
+ \symbolI{\(X\)}[id=4]\symbolI{\(Y\)}[id=5]
+ \symbolO{\(X\)}[id=1]\symbolO{\(Y\)}[id=2]
+ \symbolO[2]{\(X\)}[id=2]\symbolO{\(Y\)}[id=3]
+ }=\q{
+ \qcomul[hlen=3, vlen=2, name={\(\text{copy}_{X\otimes Y}\)}]\symbolI{\(X\otimes Y\)}
+ \symbolO{\(X\otimes Y\)}\symbolO[2]{\(X\otimes Y\)}
+ }\label{cond:copy}\\
+ \forall\q{\qbox[name={\(f\)}]\symbolI{\(X\)}\symbolO{\(Y\)}}\colon
+ \q{
+ \qbox[name={\(f\)}]\qbox[name={\(f\)}]\n
+ \qcomul[hlen=2, name={\(\text{copy}_X\)}]
+ }=\q{
+ \qcomul[hlen=2, name={\(\text{copy}_Y\)}]\n
+ \qspace[0.5]\qbox[name={\(f\)}]
+ }\label{cond:morcopy}\\
+ \q{
+ \qcomul[hlen=2, name={\(\text{copy}_I\)}]\symbolI{\(I\)}
+ \symbolO{\(I\)}\symbolO[2]{\(I\)}
+ }=\q{\qunitob}\label{cond:i:copy}
+ \end{gather}
+\end{CODEBOX}
+
+\begin{CODEBOX}{コピー可能状態}{def:copyablestate}
+ \[
+ \q{
+ \qcomul[hlen=2, name={\(\operatorname{copy}\)}]\n
+ \qspace[0.5]\qstate[name={\(a\)}]
+ }=\q{
+ \qstate[name={\(a\)}]\qstate[name={\(a\)}]
+ }
+ \]
+\end{CODEBOX}
+\begin{CODEBOX}{}{len:copycupcup}
+ \begin{align*}
+ \q{\qcup\qcup}&\stackrel{\text{(\ref{cond:i:copy})}}{=}
+ \q{
+ \qcup\qcup\n\qmv[0.5,0.4]
+ \qcomul[dom arrowtype=dotted, cod arrowtype=dotted, hlen=3, name={\(\operatorname{copy}\)}]
+ }\stackrel{\text{(\ref{cond:morcopy})}}{=}
+ \q{
+ \qspider[
+ name={\(\operatorname{copy}\)},
+ N={1,2,4,5},
+ s=2,
+ hlen=2
+ ]\n\qcup
+ }\\
+ &\stackrel{\text{(\ref{cond:copy})}}{=}\q{
+ \qwire\qbraid\qwire\n
+ \qloop[4]{\qwire}\n
+ \qloop[2]{\qcomul[hlen=2, name={\(\operatorname{copy}\)}]}\n
+ \qspace[0.5]\qcup[hlen=3]
+ \qbb[red][4,8]
+ }\stackrel{\text{(\ref{cond:ccmt})}}{=}\q{
+ \qwire\qbraid\qwire\n
+ \qbraidinv\qwire\qwire\n
+ \qloop[2]{\qcomul[hlen=2, name={\(\operatorname{copy}\)}]}\n
+ \qspace[0.5]\qcup[hlen=3]
+ \qbb[red][4,7]
+ }\\
+ &=\q{
+ \qwire\qbraid\qwire\n
+ \qbraidinv\qwire\qwire\n[2]
+ \qloop[4]{\qwire[vlen=2]}\n
+ \qloop[2]{\qcomul[hlen=2, name={\(\operatorname{copy}\)}]}\n
+ \qspace[0.5]\qcup[hlen=3]
+ \qbb[blue][8,9]
+ }
+ \stackrel{\text{(\ref{eq:inverse:braid})}}{=}
+ \q{
+ \qwire\qbraid\qwire\n
+ \qbraidinv\qwire\qwire\n
+ \qwire\qbraidinv\qwire\n
+ \qwire\qbraid\qwire\n
+ \qloop[2]{\qcomul[hlen=2, name={\(\operatorname{copy}\)}]}\n
+ \qspace[0.5]\qcup[hlen=3]
+ \qbb[blue][8,11]
+ }\\
+ &=\q{
+ \qwire\qbraid\qwire\n
+ \qbraidinv\qwire\qwire\n
+ \qwire\qbraidinv\qwire\n
+ \qwire\qbraid\qwire\n
+ \qloop[2]{\qcomul[hlen=2, name={\(\operatorname{copy}\)}]}\n
+ \qspace[0.5]\qcup[hlen=3]
+ \qbb[yellow][10,14,15]
+ }\stackrel{\text{(\ref{cond:copy}, \ref{cond:morcopy}, \ref{cond:i:copy})}}{=}
+ \q{
+ \qwire\qbraid\qwire\n
+ \qbraidinv\qwire\qwire\n
+ \qwire\qbraidinv\qwire\n
+ \qcup\qcup
+ \qbb[yellow][10,11]
+ }\\
+ &=\q{
+ \qwire\qbraid\qwire\n
+ \qcup[n=4]
+ }
+ \end{align*}
+\end{CODEBOX}
+\begin{CODEBOX}{}{len:copybraid}
+ \begin{align*}
+ \q{
+ \qbraid
+ \symbolI{\(X\)}\symbolO{\(X\)}
+ \symbolI[2]{\(X\)}\symbolO[2]{\(X\)}
+ }&\stackrel{\text{(\ref{eq:snake})}}{=}\q{
+ \qwire\qspace[2]\qcap\qwire\n
+ \qbraid[hlen=4]\qwire\qwire\n
+ \qwire\qcap\qwire\qcup\n
+ \qwire\qwire\qcup
+ \qbb[blue][1,5]
+ }=\q{
+ \qcap[hlen=4]\qwire\n
+ \qwire\qspace[2]\qbraidinv\n
+ \qwire\qcap\qwire\qwire\qwire[vlen=3]\n
+ \qwire\qwire\qcup\qcup
+ \qbb[blue][1,4]
+ \qbb[yellow][12,13]
+ }\\
+ &\stackrel{\text{例 \ref{len:copycupcup}}}{=}\q{
+ \qcap[hlen=4]\qwire\n
+ \qwire\qspace[2]\qbraidinv\n
+ \qwire\qcap\qwire\qwire\qwire[vlen=3]\n
+ \qwire\qwire\qwire\qbraid\qwire\n
+ \qwire\qwire\qcup[n=4]
+ \qbb[yellow][12,14,17]
+ }\\
+ &=\q{
+ \qcap[s=4]\qwire\n
+ \qloop[3]{\qwire}\qbraidinv\n
+ \qloop[3]{\qwire}\qbraid\qwire[vlen=3]\n
+ \qwire\qwire\qcup[n=4]
+ \qbb[green][6,10]
+ }\stackrel{\text{(\ref{eq:inverse:braid})}}{=}\q{
+ \qcap[s=4]\qwire\qwire\n[2]
+ \qloop[6]{\qwire[vlen=2]}\n
+ \qwire\qwire\qcup[n=4]
+ \qbb[green][7,8]
+ }\\
+ &\stackrel{\text{(\ref{eq:snake})}}{=}\q{
+ \qwire\qwire
+ \symbolI{\(X\)}[id=1]\symbolO{\(X\)}[id=1]
+ \symbolI{\(X\)}\symbolO{\(X\)}
+ }
+ \end{align*}
+\end{CODEBOX}
+
+\begin{CODEBOX}{クローン禁止定理}{thm:nocopy}
+ \begin{align*}
+ \q{
+ \qasym[name={\(f\)}]
+ \symbolI{\(X\)}\symbolO{\(X\)}
+ }&\stackrel{\text{(\ref{eq:snake})}}{=}
+ \q{
+ \qcap\qasym[name={\(f\)}]\n
+ \qwire\qcup
+ }\stackrel{\text{(\ref{eq:inverse:braid})}}{=}\q{
+ \qcap\qasym[name={\(f\)}]\n
+ \qwire\qbraidinv\n
+ \qwire\qbraid\n
+ \qwire\qcup
+ }\stackrel{\text{(\ref{cond:naturality:braid})}}{=}\q{
+ \qcap\qwire\n
+ \qwire\qbraidinv\n
+ \qwire\qasym[name={\(f\)}]\qwire\n
+ \qwire\qbraid\n
+ \qwire\qcup
+ \qbb[red][1,4]
+ }\\
+ &=\q{
+ \qwire\qcap\n
+ \qbraid\qwire\n
+ \qwire\qasym[name={\(f\)}]\qwire\n
+ \qwire\qbraid\n
+ \qwire\qcup
+ \qbb[red][1,4]
+ }=\q{
+ \qwire\qcap\n
+ \qbraid\qwire\n
+ \qwire\qasym[name={\(f\)}]\qwire\n
+ \qwire\qbraid\n
+ \qwire\qcup
+ \qbb[blue][3]
+ }\stackrel{\text{例 \ref{len:copybraid}}}{=}\q{
+ \qwire\qcap\n
+ \qwire\qwire\qwire\n
+ \qwire\qasym[name={\(f\)}]\qwire\n
+ \qwire\qbraid\n
+ \qwire\qcup
+ \qbb[blue][3,4]
+ }
+ \end{align*}
+\end{CODEBOX}
+\subsection{ピボタル圏}
+\label{sub:pivotal}
+\begin{CODEBOX}{}{struct:Pivotal}
+ \begin{equation}
+ \label{cond:naturality:pivotal}
+ \forall\q{\qasym[name={\(f\)}]\symbolI{\(X\)}\symbolO{\(Y\)}}\colon
+ \q[scale=0.5]{
+ \qwire\qcap[s=4]\n
+ \qwire\qwire\qasym[name={\(f\)}]\qwire\qwire\n
+ \qcup[n=4]\qwire\qmv[-1,-1]\qasym[name={\(\pi_X\)}]
+ }=\q{
+ \qasym[name={\(\pi_Y\)}]\n
+ \qasym[name={\(f\)}]
+ }
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{}{dualrev}
+ \begin{equation}
+ \label{def:dualrev}
+ \q{
+ \qcaprev\n
+ \symbolI{\(X^T\)}\symbolI[2]{\(X\)}
+ }\coloneq\q
+ {
+ \qcap \n
+ \qwire \qwireupup \n
+ \qwire \qasym[name={\(\pi_x\)}] \n
+ }\qquad\q{
+ \qcuprev
+ \symbolO{\(X\)}\symbolO[2]{\(X^T\)}
+ }\coloneq\q
+ {
+ \qasym[name={\(\pi_X^{-1}\)}]\qwire\n
+ \qwireupup\qwire\n
+ \qcup
+ }
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{}{revSliding}
+ \begin{equation}
+ \label{eq:sliding:rev}
+ \q{
+ \qcaprev\n
+ \qtrans\qwire\n
+ }=\q{
+ \qcaprev\n
+ \qwire\qasym
+ }\qquad\q{
+ \qwire\qtrans\n
+ \qcuprev
+ }=\q{
+ \qasym\qwire\n
+ \qcuprev
+ }
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{}{corr:pivotal:twist}
+ \begin{gather}
+ \q{
+ \qwireuu[label={\(X^{**}\)}]\n
+ \qasym[name={\(\pi_X\)}]\n
+ \qwireup[label={\(X\)}]
+ }=\q{
+ \qwireuu[label={\(X^{**}\)}]\qcap\n
+ \qbraid\qwired[label={\(X^T\)}]\n
+ \qasym[name={\(\theta_X^{-1}\)}]\qcuprev\n
+ \qwireup[label={\(X\)}]
+ }\label{twist:pivotal}\\
+ \q{
+ \qwireup[label={\(X\)}]\n
+ \qasym[name={\(\theta_X\)}]\n
+ \qwireup[label={\(X\)}]
+ }=\q{
+ \qwireup[label={\(X\)}]\n
+ \qasym[name={\(\pi_X^{-1}\)}]\n
+ \qwireuu[label={\(X^{**}\)}]\qcap\n
+ \qbraid\qwired[label={\(X^T\)}]\n
+ \qwire[label={\(X\)}]\qcuprev
+ }\label{pivotal:twist}
+ \end{gather}
+\end{CODEBOX}
+
+\subsubsection{トレース}
+\label{subsub:trace}
+\begin{CODEBOX}{トレース}{Trdef}
+ \[
+ \begin{array}{rccc}
+ \operatorname{Tr}\colon & \operatorname{Hom}(X,X) & \to &\operatorname{Hom}(I,I)\\
+ &\rotatebox{90}{\(\in\)}&&\rotatebox{90}{\(\in\)}\\
+ &\q{
+ \qasym[name={\(f\)}]
+ } & \mapsto & \q{
+ \qcap\n
+ \qasym[name={\(f\)}]\qwire\n
+ \qcuprev
+ }
+ \end{array}
+ \]
+\end{CODEBOX}
+\begin{CODEBOX}{トレースの可換性}{Cotr}
+ \begin{equation}
+ \label{cmt:trace}
+ \operatorname{Tr}\left(\q{
+ \qasym[name={\(g\)}]\n
+ \qasym[name={\(f\)}]
+ }\right)
+ =\q{
+ \qcap\n
+ \qasym[name={\(g\)}]\qwire\n
+ \qasym[name={\(f\)}]\qwire\n
+ \qcuprev
+ }
+ \stackrel{\text{(\ref{eq:sliding})}}{=}\q{
+ \qcap\n
+ \qasym[name={\(f\)}]\qtrans[name={\(g\)}]\n
+ \qcuprev
+ }\stackrel{\text{(\ref{eq:sliding:rev})}}{=}\q{
+ \qcap\n
+ \qasym[name={\(f\)}]\qwire\n
+ \qasym[name={\(g\)}]\qwire\n
+ \qcuprev
+ }=\operatorname{Tr}\left(\q{
+ \qasym[name={\(f\)}]\n
+ \qasym[name={\(g\)}]
+ }\right)
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{対象の次元}{def:dim}
+ \[
+ \q{\qscalar[name={\(\operatorname{dim}(X)\)}]}
+ \coloneq\q{\qscalar[name={\(\operatorname{Tr}(\operatorname{id}_X)\)}]}
+ =\q{\qcap\n\qcuprev}
+ \]
+\end{CODEBOX}
+\subsection{ダガー}
+\label{sub:dagger}
+\subsubsection{随伴ボックス}
+\label{subsub:adjoint}
+\begin{CODEBOX}{随伴}{def:adj}
+ \begin{gather}
+ \left(
+ \q{
+ \qasym[name={\(f\)}]
+ \symbolI{\(X\)}\symbolO{\(Y\)}
+ }
+ \right)^{\dagger}
+ \coloneq
+ \q{
+ \qadj[name={\(f\)}]
+ \symbolI{\(Y\)}\symbolO{\(X\)}
+ }\\
+ \left(\q{\qasym[name={\(f\)}]}\right)^{\dagger\dagger}
+ =\left(\q{\qadj[name={\(f\)}]}\right)^{\dagger}
+ =\q{\qasym[name={\(f\)}]}\\
+ \left(\q{\qwire}\right)^{\dagger}=\q{\qwire}\\
+ \q{
+ \qadj[name={\(f\)}]\n
+ \qadj[name={\(g\)}]
+ }=\q{
+ \qadj[name={\(g\circ f\)}]
+ }
+\end{gather}
+\end{CODEBOX}
+\subsubsection{共役ボックス}
+\label{subsub:conj}
+\begin{CODEBOX}{}{prep:conj}
+ \begin{align*}
+ \left(\q{\qadj}\right)^T
+ &=\q{
+ \qwire\qcap\n
+ \qwire\qadj\qwire\n
+ \qcup\qwire
+ }=\q{
+ \qcaprev\qwire\n
+ \qwire\qadj\qwire\n
+ \qwire\qcuprev
+ }=\left(
+ \q{
+ \qwire\qcap\n
+ \qwire\qasym\qwire\n
+ \qcup\qwire
+ }
+ \right)^{\dagger}=\left(\q{\qtrans}\right)^{\dagger}
+ \end{align*}
+\end{CODEBOX}
+\section{素子一覧}
+\label{sec:fcommands}
+ \begin{tcbraster}[raster columns=4,raster equal height=rows,fonttitle=\bfseries,
+ colframe=cyan!90,coltitle=cyan!50,colbacktitle=black,
+ subtitle style={boxrule=0.4pt,
+ colback=black!70,
+ colupper=cyan!50}
+ ]
+ \def\mortype{射}
+ \def\AoE{舞台}
+ \def\RMK{備考}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qwire}]
+ \[\q{\qwire}\]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qbox}]
+ \[
+ \q{
+ \qbox
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qstate}]
+ \[
+ \q{
+ \qstate
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qeffect}]
+ \[
+ \q{
+ \qeffect
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qscalar}]
+ \[
+ \q{
+ \qscalar
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qunitob}]
+ \[
+ \q{
+ \qunitob
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qmul}]
+ \[
+ \q{
+ \qmul[hlen=2]
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qcomul}]
+ \[
+ \q{
+ \qcomul[hlen=2]
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qunit}]
+ \[
+ \q{
+ \qunit
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qcounit}]
+ \[
+ \q{
+ \qcounit
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qspider}]
+ \[
+ \q{
+ \qspider[p=4,hlen=2]
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qbraid}]
+ \[
+ \q{
+ \qbraid
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qbraidinv}]
+ \[
+ \q{
+ \qbraidinv
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qsym}]
+ \[
+ \q{
+ \qsym
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qcap}]
+ \[
+ \q{
+ \qcap
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qcup}]
+ \[
+ \q{
+ \qcup
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qwireu}]
+ \[\q{\qwireup}\]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qwired}]
+ \[\q{\qwiredown}\]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qwireuu}]
+ \[\q{\qwireuu}\]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qwiredd}]
+ \[\q{\qwiredd}\]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qasym}]
+ \[
+ \q{
+ \qasym
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qtrans}]
+ \[
+ \q{
+ \qtrans
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash adj}]
+ \[
+ \q{
+ \qadj
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qconj}]
+ \[
+ \q{
+ \qconj
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qcaprev}]
+ \[
+ \q{
+ \qcaprev
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qcuprev}]
+ \[
+ \q{
+ \qcuprev
+ }
+ \]
+ \end{tcolorbox}
+ \end{tcbraster}
+ \addcontentsline{toc}{section}{参考文献}
+ \begin{thebibliography}{99}
+ \bibitem{tikz-manual}
+ Till Tantau,
+ \textit{The TikZ and PGF Packages Manual for Version 3.1.10},
+ Institut für Theoretische Informatik, Universität zu Lübeck,
+ n.d.,
+ \url{https://github.com/pgf-tikz/pgf}
+ (retrieved March 22, 2025).
+ \bibitem{geniussorting}
+ Anonymous,
+ \textit{Genius Sorting Algorithm},
+ 4chan Programming Board,
+ January 20, 2011,
+ \url{https://dis.4chan.org/read/prog/1295544154}
+ (accessed January 20, 2011; link currently inactive).
+ \bibitem{gfxhatena}
+ gfx,
+ \textit{常識を覆すソートアルゴリズム！その名も"sleep sort"！},
+ Hatena Diary,
+ May 19, 2011,
+ \url{https://gfx.hatenadiary.org/entry/20110519/1305810786}.
+ \bibitem{literateprograms}
+ LiteratePrograms,
+ \textit{Turing Machine Simulator in \LaTeX},
+ \url{https://literateprograms.org/turing_machine_simulator__latex_.html},
+ n.d.,
+ retrieved March 22, 2025.
+ \bibitem{overleaf}
+ Overleaf,
+ \textit{\LaTeX\ is More Powerful than You Think: Computing the Fibonacci Numbers and Turing Completeness},
+ \url{https://www.overleaf.com/learn/latex/Articles/LaTeX_is_More_Powerful_than_you_Think_-_Computing_the_Fibonacci_Numbers_and_Turing_Completeness},
+ n.d.,
+ retrieved March 22, 2025.
+ \end{thebibliography}
+ \section*{変更履歴}
+ \subsection*{Version 1.1.0 (2025-04-03)}
+ \paragraph{以下のコマンドが更新された:}
+ \begin{itemize}
+ \item \texttt{\textbackslash symbolI}
+ \item \texttt{\textbackslash symbols}
+ \item \texttt{\textbackslash symbolS}
+ \item \texttt{\textbackslash symbolO}
+ \item \texttt{\textbackslash symboln}
+ \item \texttt{\textbackslash symbolN}
+ \end{itemize}
+ これらのコマンドの\texttt{ID}は, 従来の必須引数からオプション引数に変更され, ユーザーは\texttt{ID}をオプションでキーで指定できるようになり, コマンド使用時の柔軟性が向上した.
+ \begin{itemize}
+ \item 旧: \texttt{\textbackslash symbolI[i]\{ID\}\{text\}[above right]}
+ \item 新: \texttt{\textbackslash symbolI[i]\{text\}[above right, id=ID]}
+ \end{itemize}
+ \paragraph{英語のドキュメントが追加された.}
+ \subsection*{Version 1.0.0 (2025-04-01)}
+ \paragraph{初回リリース.}
+ \section*{License and Copyright}
+
+ This package is distributed under the \textbf{LaTeX Project Public License (LPPL)}, version 1.3c or later.
+ For details, see \url{https://www.latex-project.org/lppl.txt}.
+
+ Copyright \textcopyright 2025 Niina Ryota
+
+ You are free to distribute and modify this package under the terms of LPPL.
+ However, any modified versions \textbf{must not} be distributed under the original name, and the original author must be credited.
+\end{document}
Property changes on: trunk/Master/texmf-dist/doc/latex/qworld/qworld.tex
___________________________________________________________________
Added: svn:eol-style
## -0,0 +1 ##
+native
\ No newline at end of property
Added: trunk/Master/texmf-dist/doc/latex/qworld/qworld_eng.pdf
===================================================================
(Binary files differ)
Index: trunk/Master/texmf-dist/doc/latex/qworld/qworld_eng.pdf
===================================================================
--- trunk/Master/texmf-dist/doc/latex/qworld/qworld_eng.pdf 2025-04-03 19:52:19 UTC (rev 74842)
+++ trunk/Master/texmf-dist/doc/latex/qworld/qworld_eng.pdf 2025-04-03 19:52:38 UTC (rev 74843)
Property changes on: trunk/Master/texmf-dist/doc/latex/qworld/qworld_eng.pdf
___________________________________________________________________
Added: svn:mime-type
## -0,0 +1 ##
+application/pdf
\ No newline at end of property
Added: trunk/Master/texmf-dist/doc/latex/qworld/qworld_eng.tex
===================================================================
--- trunk/Master/texmf-dist/doc/latex/qworld/qworld_eng.tex (rev 0)
+++ trunk/Master/texmf-dist/doc/latex/qworld/qworld_eng.tex 2025-04-03 19:52:38 UTC (rev 74843)
@@ -0,0 +1,2384 @@
+\documentclass{article}
+\usepackage[a4paper, textwidth=150mm]{geometry}
+
+\usepackage{qworld}
+
+\usepackage{bookmark}
+\usepackage{fontspec}
+\usepackage{pdfpages}
+
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{amsfonts}
+\usepackage{mathtools}
+
+\usepackage{ascmac}
+\usepackage{fancybox}
+\usepackage{float}
+
+\usepackage{bbm}
+\usepackage{mathrsfs}
+%\numberwithin{equation}{subsection}
+
+\usepackage{framed}
+\newenvironment{longshadowbox}{%
+ \def\FrameCommand{\fboxsep=\FrameSep \shadowbox}%
+ \MakeFramed {\FrameRestore}}%
+{\endMakeFramed}
+
+\usepackage{url}
+\usepackage{hyperref}
+\hypersetup{colorlinks=true, linkcolor=blue}
+
+\usepackage{iitem}
+
+\usepackage{xspace}
+\newcommand{\qworldname}{\texttt{QWorld}\xspace}
+\newcommand{\tikzname}{Ti\emph{k}Z\xspace}
+\newcommand{\texlivename}{\TeX{} \textsf{Live}\xspace}
+
+\setcounter{tocdepth}{3}
+
+\usepackage[most]{tcolorbox}
+\newtcblisting{mybox}[2][]{
+ colback=red!5!white,
+ colframe=red!75!black,
+ fonttitle=\bfseries,
+ title=#2,
+ #1
+}
+
+\tcbset
+{
+ codeinfo/.style=
+ {
+ colframe=cyan!90,
+ coltitle=cyan!60,
+ colbacktitle=black,
+ fonttitle=\small\sffamily\bfseries,
+ fontupper=\small,
+ fontlower=\small
+ },
+ exampletitleinfo/.style 2 args=
+ {
+ codeinfo,
+ title={Example \thetcbcounter : #1},
+ label={#2}
+ },
+ failurecodeinfo/.style=
+ {
+ colframe=red!90,
+ coltitle=red!60,
+ colbacktitle=black,
+ fonttitle=\small\sffamily\bfseries,
+ fontupper=\small,
+ fontlower=\small
+ },
+ failureexampletitleinfo/.style 2 args=
+ {
+ failurecodeinfo,
+ title={Failure Example \thetcbcounter : #1},
+ label={#2}
+ }
+}
+\newtcblisting{texexp}[1]
+{
+ codeinfo,
+ #1
+}
+\newtcblisting[auto counter,number within=section]{codebox}[3][center lower,listing side text,righthand width=3.5cm,bicolor,colbacklower=white]
+{
+ exampletitleinfo={#2}{#3},
+ breakable,
+ #1
+}
+
+\newtcblisting[use counter from=codebox,number within=section]{CODEBOX}[3][center lower,bicolor,colbacklower=white]
+{
+ exampletitleinfo={#2}{#3},
+ breakable,
+ #1
+}
+\newtcolorbox[use counter from=codebox]{codeboxspec}[3][]
+{
+ exampletitleinfo={#2}{#3},
+ breakable,
+ #1
+}
+
+\newtcblisting{texfexp}[1]
+{
+ failurecodeinfo,
+ #1
+}
+\newtcblisting[auto counter,number within=section]{fcodebox}[3][center lower,listing side text,righthand width=3.5cm,bicolor,colbacklower=white]
+{
+ failureexampletitleinfo={#2}{#3},
+ breakable,
+ #1
+}
+
+\newtcblisting[use counter from=fcodebox,number within=section]{FCODEBOX}[3][center lower,bicolor,colbacklower=white]
+{
+ failureexampletitleinfo={#2}{#3},
+ breakable,
+ #1
+}
+\newtcolorbox[use counter from=fcodebox]{fcodeboxspec}[3][]
+{
+ failureexampletitleinfo={#2}{#3},
+ breakable,
+ #1
+}
+
+\begin{document}
+
+\title{
+ \Huge\textbf{The QWorld Package}\\[4pt]
+ \scriptsize\texttt{Version 1.1.0} \\[4pt]
+}
+\author{
+ \textbf{Niina Ryota} \\[4pt]
+}
+\date{\today}
+\maketitle
+
+\begin{abstract}
+ \qworldname is a \LaTeX\ package designed for efficiently rendering complex graphical calculus in monoidal categories.
+ In particular, it supports the typesetting of diagrammatic languages in category theory, quantum theory, and related fields, where diagrammatic reasoning plays a crucial role.
+\end{abstract}
+
+\tableofcontents
+\section{Introduction}
+This document aims to provide a step-by-step guide to installing and using the \qworldname package, from basic commands to advanced applications.
+\qworldname is a \LaTeX\ package specifically designed for the typesetting of diagrammatic languages, offering an intuitive command set for describing diagrams related to monoidal categories, Frobenius structures, braiding, Hopf algebras, symmetries, dualities, pivotal structures, and dagger categories using \tikzname\cite{tikz-manual}.
+\section{Basic Usage}
+\subsection{Fundamental Elements}
+The following are the most fundamental commands in the \qworldname package, forming the basis for drawing diagrams:
+\begin{itemize}
+ \item \texttt{\textbackslash q}: A canvas for placing diagram elements.
+ \item \texttt{\textbackslash qbox}: Draws a box\(\q[scale=0.5]{\qbox}\).
+ \item \texttt{\textbackslash qwire}: Draws a wire\(\q[scale=0.5]{\qwire}\).
+ \item \texttt{\textbackslash qcirc}: Compositon.
+\end{itemize}
+These commands are used within the canvas environment, denoted by \verb|\q{...}|.
+For example, writing \verb|\q{\qbox}| results in the rendering of a box\(\q[scale=0.5]{\qbox}\).\par
+ To compose \(\q[scale=0.5]{\qbox}\)and\(\q[scale=0.5]{\qbox}\), use \verb|\qcirc|.
+ This is analogous to function composition in mathematics, such as \(g\circ f\) (\verb|\(g\circ f\)|).
+ \begin{codebox}{}{Ex:20250317}
+ \q{ \qbox \qcirc \qbox }
+ \end{codebox}
+ Additionally, placing multiple \verb|\qbox| commands in sequence results in boxes\(\q[scale=0.5]{\qbox}\)being arranged horizontally:
+ \begin{codebox}{}{boxs2}
+ \q{ \qbox \qbox \qbox }
+ \end{codebox}
+ It is also possible to compose multiple boxes using\verb|\qcirc|:
+ \begin{codebox}{}{boxsbr}
+ \q
+ {
+ \qbox \qbox \qbox
+ \qcirc
+ \qbox \qbox \qbox
+ }
+ \end{codebox}
+ Note that \verb|\n| is synonymous with\verb|\qcirc|
+ \begin{codebox}{}{boxsn}
+ \q
+ {
+ \qbox \qbox \qbox \n
+ \qbox \qbox \qbox
+ }
+\end{codebox}
+\subsubsection{Identification Number}
+All commands placed within \verb|\q{...}| are automatically assigned unique identification numbers based on their execution order.
+For example, in the previous example \ref{boxsn}, each \verb|\qbox| is numbered as follows:
+\[\q{\qloop[2]{\qloop[3]{\qbox[show id={true}]}\n}}\]
+For more details, see Section \ref{subsub:show id} on page \pageref{subsub:show id}.
+\subsubsection{Bounding Box}
+\[\q{
+ \qcaprev \qcap \n
+ \qwire \qbraid \qwire \n
+ \qwire \qbraid \qwire \n
+ \qcup \qcuprev
+ \qbb[red][1,2]
+ \bbsymbol{Measure}
+ \bbsymbol[W]{Output}[left]
+ \bbsymbol[NW]{Computation}[above left]
+ \bbsymbol[NE]{Physics}[above right]
+ \qbb[yellow][3,5,6,8]
+ \bbsymbol{Braid}
+ \bbsymbol[W]{Compute}[left]
+ \qbb[cyan][9,10]
+ \bbsymbol{Create}
+ \bbsymbol[W]{Initialize}[left]
+}\]
+See the following example \ref{ex:hopflink}:
+\begin{codebox}{}{ex:hopflink}
+ \q{
+ \qcaprev \qcap \n
+ \qwire \qbraid \qwire \n
+ \qwire \qbraid \qwire \n
+ \qcup \qcuprev
+ }
+\end{codebox}
+\begin{itemize}
+ \item \verb|\qcaprev| draws \(\q[scale=0.6]{\qcaprev}\).
+ \item \verb|\qcap| draws \(\q[scale=0.6]{\qcap}\).
+ \item \verb|\qwire| draws \(\q[scale=0.5]{\qwire}\).
+ \item \verb|\qbraid| draws \(\q[scale=0.5]{\qbraid}\).
+ \item \verb|\qcup| draws\(\q[scale=0.6]{\qcup}\).
+ \item \verb|\qcuprev| draws \(\q[scale=0.6]{\qcuprev}\).
+\end{itemize}
+As a whole, the elements are arranged in the diagram according to the order in which the commands are written.\par
+Each element has an associated bounding box. The following are examples:
+ \begin{itemize}
+ \item The bounding box of\q{\qcaprev}is visualized as\q{\qcaprev\qbball}.
+ \item The bounding box of\q{\qbraid}is visualized as\q{\qbraid\qbball}.
+ \item The bounding box of\q{\qwire}is visualized as\q{\qwire\qbball}.
+ \item The bounding box of\q{\qcuprev}is visualized as\q{\qcuprev\qbball}.
+ \end{itemize}
+ The following commands are available for visualizing bounding boxes and placing text near them:
+ \begin{itemize}
+ \item Visualization
+ \iitem \texttt{\textbackslash qbb}: Displays the bounding box of a specific element.
+ \iitem \texttt{\textbackslash qbball}: Displays the bounding boxes of all elements at once.
+ \iitem \texttt{\textbackslash qBBall}: Displays the bounding box surrounding the entire canvas.
+ \item Placing text
+ \iitem \texttt{\textbackslash bbsymbol}: Places text within a bounding box.
+ \end{itemize}
+
+ \begin{verbatim}
+ \qbb[<color>][<ID list>]
+ \end{verbatim}
+ \begin{itemize}
+ \item First argument (optional): Specifies the color. If omitted, no color is applied.
+ \item Second argument: Specifies a list of element IDs to display their bounding boxes.
+ \end{itemize}
+ \begin{codebox}{}{Ex:20250320}
+ \q{\qbox\qbb[1]}
+ \end{codebox}
+ \begin{codebox}{}{ex:202503270454}
+ \q{
+ \qcaprev \qcap \n
+ \qwire \qbraid \qwire \n
+ \qwire \qbraid \qwire \n
+ \qcup \qcuprev
+ \qbb[4,7]
+ }
+ \end{codebox}
+ \begin{CODEBOX}{}{Ex:20250321}
+ \q{\qbox\qbox\qbox\qbox\qbox\qbox\qbb[yellow][2,...,5]}
+ \end{CODEBOX}
+ \begin{codebox}{}{Ex:20250222}
+ \q
+ {
+ \qbox\qbox\qbox\n
+ \qbox\qbox\qbox
+ \qbb[green][2,5]
+ }
+ \end{codebox}
+ \begin{codebox}{}{Ex:20250318}
+ \q{
+ \qcaprev \qcap \n
+ \qwire \qbraid \qwire \n
+ \qwire \qbraid \qwire \n
+ \qcup \qcuprev
+ \qbb[blue][4,7]
+ }
+ \end{codebox}
+ \begin{verbatim}
+ \qbball[<color>]
+ \end{verbatim}
+ \begin{itemize}
+ \item Displays multiple bounding boxes at once (even for a single element).
+ \item Ensures that the bounding boxes of different elements are properly aligned without overlap.
+ \item The \qworldname package \textbf{minimizes the need for tedious manual coordinate adjustments} in diagram drawing.
+ \end{itemize}
+ \begin{codebox}[center lower,listing side text,righthand width=5cm,bicolor,colbacklower=white]{}{Ex:202503270511}
+ \q{
+ \qcaprev \qcap \n
+ \qwire \qbraid \qwire \n
+ \qwire \qbraid \qwire \n
+ \qcup \qcuprev
+ \qbball
+ }
+ \end{codebox}
+
+ \begin{verbatim}
+ \qBBall[<color>]
+ \end{verbatim}
+ \begin{itemize}
+ \item Displays the bounding box surrounding all elements within the canvas.
+ \end{itemize}
+ \begin{codebox}[center lower,listing side text,righthand width=5cm,bicolor,colbacklower=white]{}{Ex:20250316}
+ \q{
+ \qcaprev \qcap \n
+ \qwire \qbraid \qwire \n
+ \qwire \qbraid \qwire \n
+ \qcup \qcuprev
+ \qBBall
+ }
+ \end{codebox}
+ \begin{verbatim}
+ \bbsymbol[<direction>]{<text>}[<options>]
+ \end{verbatim}
+ \begin{itemize}
+ \item First argument (optional): Specifies placement direction. Options:
+ \iitem N (North), S (South), E (East), W (West),
+ \iitem NW (Northwest), NE (Northeast), SE (Southeast), SW (Southwest),
+ \iitem C (Center).
+ \iitem Default: E (East).
+ \item Second argument: Specifies the text to be placed.
+ \item Third argument (optional): Adjusts position (e.g., above, below, left, right).
+ \iitem Default: right.
+ \end{itemize}
+ \begin{CODEBOX}{}{Ex:bbsymbol}
+ \q[scale=5]{
+ \qbox
+ \qbb[1]
+ \bbsymbol{East (Default)}
+ \bbsymbol[W]{West}[left]
+ \bbsymbol[N]{North}[above]
+ \bbsymbol[S]{South}[below]
+ \bbsymbol[NE]{NorthEast}[above right]
+ \bbsymbol[NW]{NorthWest}[above left]
+ \bbsymbol[SE]{SouthWest}[below right]
+ \bbsymbol[SW]{SouthEast}[below left]
+ \bbsymbol[C]{Center}[]
+ }
+ \end{CODEBOX}
+ \begin{CODEBOX}{}{Ex:20250315}
+ \q
+ {
+ \qcaprev \qcap \n
+ \qwire \qbraid \qwire \n
+ \qwire \qbraid \qwire \n
+ \qcup \qcuprev
+ \qbb[red][1,2]
+ \bbsymbol{Measure}
+ \bbsymbol[W]{Output}[left]
+ \bbsymbol[NW]{Computation}[above left]
+ \bbsymbol[NE]{Physics}[above right]
+ \qbb[yellow][3,5,6,8]
+ \bbsymbol{Braid}
+ \bbsymbol[W]{Compute}[left]
+ \qbb[cyan][9,10]
+ \bbsymbol{Create}
+ \bbsymbol[W]{Initialize}[left]
+ }
+ \end{CODEBOX}
+
+ \subsection{Mathematical Mode}
+ When writing equations that include diagrammatic language, use the mathematical mode.
+ \begin{CODEBOX}{}{IntroEx}
+ \[
+ \operatorname{Tr}\left(\q{\qbraid\qcirc\qbraid}\right)
+ =\q{
+ \qcap \qcap \n
+ \qwire \qbraidinv \qwire \n
+ \qwire \qbraidinv \qwire \n
+ \qcuprev \qcuprev
+ }
+ \]
+ \end{CODEBOX}
+
+ \subsection{Installation}
+ The style file for the \qworldname package, \texttt{qworld.sty}, is available for download from CTAN.
+ To use it, place the file in your \TeX\ package directory and include the following line in the preamble:
+ \begin{verbatim}
+ \usepackage{qworld}
+ \end{verbatim}
+ \paragraph{Dependencies}
+ The package depends on the following:
+ \begin{itemize}
+ \item \tikzname, with the following libraries:
+ \iitem \texttt{cd}
+ \iitem \texttt{positioning}
+ \iitem \texttt{arrows}
+ \iitem \texttt{arrows.meta}
+ \iitem \texttt{calc}
+ \iitem \texttt{intersections}
+ \iitem \texttt{shapes.symbols}
+ \iitem \texttt{shapes.geometric}
+ \iitem \texttt{shapes.misc}
+ \iitem \texttt{decorations.pathreplacing}
+ \iitem \texttt{decorations.markings}
+ \iitem \texttt{decorations.pathmorphing}
+ \item \texttt{pgffor}
+ \item \texttt{ifthen}
+ \item \texttt{xparse}
+ \item \texttt{xfp}
+ \item \texttt{xstring}
+ \end{itemize}
+ These dependencies are included by default in the \texlivename distribution, so no additional setup is required.
+\subsection{Basic Customization}
+This section explains how to customize commands in the \qworldname package.
+ \subsubsection{Box Customization}
+ \label{subsubsec:boxcustomization}
+ Keys can be used to adjust the number and arrangement of the box’s boundary lines.
+For example, the following boxes can be drawn:
+\begin{itemize}
+ \item \(\q[scale=0.5]{\qbox[p=20,hlen=3]}\)
+ \item \(\q[scale=0.5]{\qbox[p=10,N={1,2,3,8,9,10},S={4,5,6,7},hlen=3]}\)
+\end{itemize}
+These keys are applicable to many elements other than wires.
+The main keys are as follows:
+\begin{itemize}
+ \item \texttt{n}
+ \iitem The number of points on the north edge of the box\(\q[scale=0.5]{\qbox[p=2, P={}]}\)where wires\(\q[scale=0.31]{\qwire}\)can be generated (these points will be referred to as \texttt{n} terminals).
+ \iitem For example, if \texttt{n=3}, three points can be placed at equal intervals on the north edge of the box.
+ \iitem The default value is \texttt{n=1}.
+ \iitem The point obtained by vertically moving an \texttt{n} terminal to the northernmost position is called an \texttt{N} terminal.
+ \iitem Both \texttt{n} terminals and \texttt{N} terminals are indexed as \(1,2,...\) from the west (left).
+ \iitem Generating a wire from the \(i\)-th \texttt{n} terminal means connecting the \(i\)-th \texttt{n} terminal and the \(i\)-th \texttt{N} terminal with a line segment.
+ \item \texttt{N}
+ \iitem The set of indices of the \texttt{n} terminals from which wires are actually generated.
+ \iitem For example, if \texttt{n=3} and \texttt{N=\{2\}}, a wire can be generated from the \(2\)nd of the three \texttt{n} terminals.
+ \iitem If \texttt{n} is unspecified and \texttt{N} is specified, \texttt{n} is automatically set to \(\texttt{n}\coloneq\max\texttt{N}\).
+ \iitem If \texttt{N} is unspecified and \texttt{n} is specified, \texttt{N} is automatically set to \(\texttt{N}\coloneq\{1,...,\texttt{n}\}\).
+ \iitem The northernmost end of a wire actually generated from an \texttt{n} terminal is called the output point, or \texttt{O} terminal.
+ \item \texttt{s}
+ \iitem The number of points on the south edge of the box where wires can be generated (these points will be referred to as \texttt{s} terminals).
+ \iitem For example, if \texttt{s=4}, four points can be placed at equal intervals on the south edge of the box.
+ \iitem The default value is \texttt{s=1}.
+ \iitem The point obtained by vertically moving an \texttt{s} terminal to the southernmost position is called an \texttt{S} terminal.
+ \iitem Both \texttt{s} terminals and \texttt{S} terminals are indexed as \(1,2,...\) from the west (left).
+ \iitem Generating a wire from the \(i\)-th \texttt{s} terminal means connecting the \(i\)-th \texttt{s} terminal and the \(i\)-th \texttt{S} terminal with a line segment.
+ \item \texttt{S}
+ \iitem The set of indices of the \texttt{s} terminals from which wires are actually generated.
+ \iitem For example, if \texttt{s=4} and \texttt{S=\{1,4\}}, wires can be generated from the 1st and 4th of the four \texttt{s} terminals.
+ \iitem If \texttt{s} is unspecified and \texttt{S} is specified, \texttt{s} is automatically set to \(\texttt{s}\coloneq\max\texttt{S}\).
+ \iitem If \texttt{S} is unspecified and \texttt{s} is specified, \texttt{S} is automatically set to \(\texttt{S}\coloneq\{1,...,\texttt{s}\}\).
+ \iitem The southernmost end of a wire actually generated from an \texttt{s} terminal is called the input point, or \texttt{I} terminal.
+ \item \texttt{hlen}
+ \iitem The horizontal length of the bounding box.
+ \iitem For example, if \texttt{hlen=2}, the horizontal length of the bounding box becomes \texttt{2}.
+ \iitem With one exception, the distance between the westernmost wire and the bounding box's boundary, and the distance between the easternmost wire and the bounding box's boundary, are both \(\fpeval{0.5*\qwireinterval}\).
+ \iitem When \texttt{hlen} is unspecified, the \texttt{n} terminals and \texttt{s} terminals are arranged with an interval of \(1\).
+ \iitem When \texttt{hlen} is specified, the \texttt{n} terminals and \texttt{s} terminals are arranged at equal intervals in a section of length \(\texttt{hlen}-\fpeval{\qwireinterval}\). The spacing depends on \texttt{hlen} and the number of terminals.
+ \iiitem If there is only one terminal, it is placed at the center of the section. In this case, if \(\texttt{hlen}>1\), the distance between the westernmost wire and the bounding box's boundary, and the distance between the easternmost wire and the bounding box's boundary, are both \(\frac{\texttt{hlen}}{2}>\fpeval{0.5*\qwireinterval}\).
+ \item \texttt{vlen}
+ \iitem The vertical length (height) of the bounding box.
+ \iitem The default value is \texttt{vlen=1}.
+ \item \texttt{id}
+ \iitem The identifier of the box.
+ \iitem For example, if \texttt{id=ID}, the following names are automatically assigned to the various points:
+ \iiitem The \(i\)-th \texttt{n} terminal (counting from the west): (n-\(i\)-ID).
+ \iiitem The \(i\)-th \texttt{s} terminal: (s-\(i\)-ID).
+ \iiitem The point obtained by vertically moving the \(i\)-th \texttt{n} terminal to the north edge of the box: (N-\(i\)-ID).
+ \iiitem The point obtained by vertically moving the \(i\)-th \texttt{s} terminal to the south edge of the box: (S-\(i\)-ID).
+ \iiitem The northwest corner of the bounding box: (NW-ID)
+ \iiitem The northeast corner of the bounding box: (NE-ID)
+ \iiitem The southwest corner of the bounding box: (SW-ID)
+ \iiitem The southeast corner of the bounding box: (SE-ID)
+ \iiitem The northernmost point of the bounding box: (N-ID)
+ \iiitem The southernmost point of the bounding box: (S-ID)
+ \iiitem The westernmost point of the bounding box: (W-ID)
+ \iiitem The easternmost point of the bounding box: (E-ID)
+ \iiitem The center of the bounding box: (C-ID)
+ \iiitem The northwest corner of the box: (nw-ID)
+ \iiitem The northeast corner of the box: (ne-ID)
+ \iiitem The southwest corner of the box: (sw-ID)
+ \iiitem The southeast corner of the box: (se-ID)
+ \iiitem The northernmost point of the box: (n-ID)
+ \iiitem The southernmost point of the box: (s-ID)
+ \iiitem The westernmost point of the box: (w-ID)
+ \iiitem The easternmost point of the box: (e-ID)
+ \iiitem The \(i\)-th output point: (O-\(i\)-ID)
+ \iiitem The \(i\)-th input point: (I-\(i\)-ID)
+ \iitem Regardless of whether it is specified, each command placed inside \texttt{\textbackslash q\{...\}} is assigned an identification number. Replacing the ID part with that number refers to the same point. See section \ref{subsub:show id} for details.
+ \item \texttt{p}, \texttt{P}
+ \iitem For a positive integer \texttt{k}, \texttt{p=k} means \texttt{n=k} and \texttt{s=k}.
+ \iitem For a finite set of positive integers \texttt{K}, \texttt{P=K} means \texttt{N=K} and \texttt{S=K}.
+ \item \texttt{name}
+ \iitem With \texttt{name=<text>}, \text{<text>} is placed at the center of the box.
+\end{itemize}
+\begin{CODEBOX}{}{ex:boxpoints}
+ \def\pradius{0.4pt}
+ \q[scale=4]
+ {
+ \qbox
+ [
+ n=3,N={2},
+ s=4,S={1,4},
+ hlen=2,
+ vlen=2,
+ id={ID},
+ name={Center (C-ID)}
+ ]\n
+ \qbb[ID]
+ \node [above] at (NE-ID) {NE-ID};
+ \node [above] at (NW-ID) {NW-ID};
+ \node [below] at (SE-ID) {SE-ID};
+ \node [below] at (SW-ID) {SW-ID};
+ \fill (NE-ID) circle (\pradius);
+ \fill (NW-ID) circle (\pradius);
+ \fill (SE-ID) circle (\pradius);
+ \fill (SW-ID) circle (\pradius);
+ \node [above] at (ne-ID) {ne-ID};
+ \node [above] at (nw-ID) {nw-ID};
+ \node [below] at (se-ID) {se-ID};
+ \node [below] at (sw-ID) {sw-ID};
+ \fill (ne-ID) circle (\pradius);
+ \fill (nw-ID) circle (\pradius);
+ \fill (se-ID) circle (\pradius);
+ \fill (sw-ID) circle (\pradius);
+ \foreach\i in {1,...,3}
+ {
+ \symboln[\i]{n-\i-ID}[below]
+ \fill (n-\i-ID) circle (\pradius);
+ \symbolN[\i]{N-\i-ID}
+ \fill (N-\i-ID) circle (\pradius);
+ }
+ \foreach\i in {1,...,4}
+ {
+ \symbols[\i]{s-\i-ID}[above]
+ \fill (s-\i-ID) circle (\pradius);
+ \symbolS[\i]{S-\i-ID}[below]
+ \fill (S-\i-ID) circle (\pradius);
+ }
+ \draw[line width=0.2pt] (NW-ID)
+ to [bend left=60] node [fill=white, midway] {hlen} (NE-ID);
+ \draw[line width=0.2pt] (NW-ID)
+ to [bend right=60] node [fill=white, midway] {vlen} (SW-ID);
+ \draw [red, <->] (C-ID -| W-ID)
+ -- node [midway, auto, font=\footnotesize] {0.5} (C-ID -| S-1-ID);
+ \draw [red, <->] (C-ID -| S-4-ID)
+ -- node [midway, auto, font=\footnotesize] {0.5} (C-ID -| E-ID);
+ \qwire[arrowtype={loosely dotted}, dom={s-1-ID}, cod={n-1-ID}]
+ \qwire[arrowtype={loosely dotted}, dom={s-4-ID}, cod={n-3-ID}]
+ \node (O1prime) at ([shift={(0.25,0.25)}]O-1-ID){O-1-ID};
+ \draw[->,red] (O1prime) to ([shift={(0.05,0.05)}]O-1-ID);
+ \node (I1prime) at ([shift={(0.5,-0.5)}]I-1-ID){I-1-ID};
+ \node (I2prime) at ([shift={(0.5,-0.5)}]I-2-ID) {I-2-ID};
+ \draw[->,red](I1prime) to ([shift={(0.05,-0.05)}]I-1-ID);
+ \draw[->,red](I2prime) to ([shift={(0.05,-0.05)}]I-2-ID);
+ }
+\end{CODEBOX}
+The \verb|\qwire| \verb|arrowtype| key, as well as the \verb|dom| and \verb|cod| keys, are explained in section \ref{subsubsec:wirecustomization} on page \pageref{subsubsec:wirecustomization}.
+
+\paragraph{Label (\texttt{\textbackslash symbol})}
+\begin{verbatim}
+ \symbolI[<number>]{<label>}[<options>]
+\end{verbatim}
+For example, \verb|\symbolI[2]{text}[below right, id=ID]| places the label "text" at the bottom right of the second input point (I-2-ID) of \verb|\q{\qbox[p=3, id=ID]}| (\(\q[scale=0.5]{\qbox[p=3, id=ID]}\)).
+When no number or options are specified, the default values \(1\), below, and \verb|id=<previous>| are applied, respectively.
+Other label display commands include:
+\begin{verbatim}
+ \symbolS[<number>]{<label>}[<options>]
+\end{verbatim}
+\begin{verbatim}
+ \symbols[<number>]{<label>}[<options>]
+\end{verbatim}
+\begin{verbatim}
+ \symbolO[<number>]{<label>}[<options>]
+\end{verbatim}
+\begin{verbatim}
+ \symbolN[<number>]{<label>}[<options>]
+\end{verbatim}
+\begin{verbatim}
+ \symboln[<number>]{<label>}[<options>]
+\end{verbatim}
+\verb|\symbolS| and \verb|\symbols| place the label \textless label\textgreater at points (S-\(i\)-ID) and (s-\(i\)-ID), respectively.
+The default number \(i\) is \(1\), and the default option is below.
+\verb|\symbolO|, \verb|\symbolN|, and \verb|\symboln| place the label \textless label\textgreater at points (O-\(i\)-ID), (N-\(i\)-ID), and (n-\(i\)-ID), respectively.
+The default number \(i\) is \(1\), and the default option is above.
+\begin{codebox}[center lower,listing side text,righthand width=5cm,bicolor,colbacklower=white]{}{Ex:20250311}
+ \q{
+ \qbox[p=5, S={1,4,5}, id=ID]
+ \symbolO{A}
+ \symbolN[3]{B}
+ \symboln[5]{C}[below left]
+ \symbolS[2]{D}
+ \symbols[3]{E}
+ \symbolI[2]{F}
+ \symbolI[3]{G}[below right]
+ }
+\end{codebox}
+\begin{CODEBOX}{}{Ex:20250310}
+ \q{
+ \qbox[p=10,N={1,2,5,6,9,10}, id=ID]
+ \foreach\i in {1,...,10}
+ {
+ \symbolI[\i]{\(x_{\i}\)}
+ }
+ \foreach\j in {1,...,6}
+ {
+ \symbolO[\j]{\(y_{\j}\)}
+ }
+ }
+\end{CODEBOX}
+
+\subsubsection{Repetition}
+\label{subsub:forloop}
+\begin{verbatim}
+ \qloop[<number of repetitions>]{<command>}
+\end{verbatim}
+By using the \verb|\qloop| command, you can repeat a command the specified number of times:
+\begin{CODEBOX}{}{Ex:20250305}
+ \q{ \qloop[10]{\qbox} }
+\end{CODEBOX}
+The \verb|\qloop| command can also be used outside of \verb|\q|:
+\begin{codebox}[center lower,listing side text,righthand width=5.5cm,bicolor,colbacklower=white]{}{Ex:202503220930}
+ \qloop[30]{I AM BOB}
+\end{codebox}
+
+\subsubsection{Displaying Identification Numbers}
+\label{subsub:show id}
+Commands inside \verb|\q{...}| are automatically assigned identification numbers.
+If \texttt{show id=true} is specified, those identification numbers will be displayed.
+\begin{CODEBOX}{}{Ex:20250303}
+ \q{
+ \qbox[show id={true}] \qbox[show id={true}] \qbox[show id={true}]\n
+ \qbox[show id={true}] \qbox[show id={true}] \qbox[show id={true}]
+ }
+\end{CODEBOX}
+\begin{CODEBOX}{}{Ex:20250302}
+ \q
+ {
+ \qloop[10]
+ {
+ \qloop[10]
+ {
+ \qbox[show id={true}]
+ }\n
+ }\n
+ }
+\end{CODEBOX}
+
+\subsubsection{Color}
+You can specify colors for the boxes:
+\begin{itemize}
+ \item \texttt{color}
+ \iitem The color inside the box, for example, \texttt{color=green} will produce \q[scale=0.5]{\qbox[color={green}]}.
+ \item \texttt{frame color}
+ \iitem The color of the box's frame, for example, \texttt{frame color=cyan}will produce \q[scale=0.5]{\qbox[frame color={cyan}]}.
+ \item \texttt{morphism color}
+ \iitem The color of the morphisms inside the box, for example, \texttt{morphism color=red} will produce \q[scale=0.5]{\qbox[morphism color={red}, name={\(f\)}]}.
+\end{itemize}
+\begin{CODEBOX}{}{Ex:20250259}
+ \q
+ {
+ \qbox
+ [
+ name={\(f\)},
+ color=green,
+ frame color=cyan,
+ morphism color=red
+ ]
+ }
+\end{CODEBOX}
+
+\paragraph{\texttt{N}, \texttt{S}, Arithmetic Sequence}
+\qworldname loads the \texttt{pgffor} package, so the shorthand notation using \verb|...| can be used for lists:
+\begin{itemize}
+ \item \(1,...,10\): An arithmetic sequence with a common difference of \(1\)
+ \item \(1,3,...,11\): An arithmetic sequence based on the first two terms with a common difference
+\end{itemize}
+\begin{CODEBOX}{}{Ex:20250258}
+ \q{
+ \qbox[N={1,...,10}, S={1,3,...,11}]
+ }
+\end{CODEBOX}
+\begin{CODEBOX}{}{Ex:20250257}
+ \q{
+ \qbox[N={1,...,4,6,8,...,20}, S={1,...,20}, hlen=10]
+ }
+\end{CODEBOX}
+\begin{CODEBOX}{}{Ex:20250256}
+ \q{
+ \qbox[N={1,...,4,7,10,...,19,20,21,...,25}, S={1,...,25}, hlen=10]
+ }
+\end{CODEBOX}
+\subsubsection{Relative Coordinates}
+This is a command for specifying the position of elements relative to a given point.
+Note that \(x\) and \(y\) do not have to be integers; expressions evaluable by \verb|\fpeval|,
+such as \verb|2^1.5|, \verb|cos(pi)|, and \verb|sprt(2)|, are valid.
+\begin{itemize}
+ \item \texttt{\textbackslash qmv[\(x\),\(y\)]}
+ \iitem Translates by (\(x\), \(y\)).
+ \iitem The default is \(x=y=0\).
+ \item \texttt{\textbackslash n[\(y\)]}
+ \iitem Moves vertically by \(y\) (\(Y\mapsto Y-y\)), and sets the X coordinate to \(0\).
+ \iitem The default is \(y=1\) , which behaves like a line break.
+ \item \texttt{\textbackslash qspace[\(x\)]}
+ \iitem Moves Horizontally by \(x\).
+ \iitem The default is \(x=1\), which behaves like the spacebar.
+ \iitem This is equivalent to \texttt{\textbackslash qmv[\(x\),0]}.
+ \item \texttt{\textbackslash qreset[\(x\),\(y\)]}
+ \iitem Moves to the point (\(x\),\(y\)).
+\end{itemize}
+ \q[scale=2.35]{
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (I) at (qworldat);
+ \qbox
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (II) at (qworldat);
+ \n
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (III) at (qworldat);
+ \qspace
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (IV) at (qworldat);
+ \qspace[0.5]
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (V) at (qworldat);
+ \n[2]
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (VI) at (qworldat);
+ \qmv[-2,-1]
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (VII) at (qworldat);
+ \n
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (VIII) at (qworldat);
+ \qspace[-3]
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (IX) at (qworldat);
+ \qreset
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (X) at (qworldat);
+ \qreset[-2,1]
+ \fill[blue] (qworldat) circle (1pt);
+ \coordinate (XI) at (qworldat);
+ \qbball
+ \qwireup[dom={I}, cod={II},
+ label={Auto-update}, label side={below},
+ label at={midway}, color=cyan]
+ \qwireup[dom={II},cod={III},
+ label={\texttt{\textbackslash n}},
+ label side={below}, color=cyan]
+ \qwireup[dom={III},cod={IV},
+ label={\texttt{\textbackslash qspace}},
+ label side={below}, color=cyan]
+ \qwireup[dom={IV},cod={V},
+ label={\texttt{\textbackslash qspace[0.5]}},
+ label at={at end}, label side={above right}, color=cyan]
+ \qwireup[dom={V},cod={VI},
+ label={\texttt{\textbackslash n[2]}},color=cyan]
+ \qwireup[dom={VI},cod={VII},
+ label={\texttt{\textbackslash qmv[-2,-1]}},color=cyan]
+ \qwireup[dom={VII},cod={VIII},
+ label={\texttt{\textbackslash n}},color=cyan]
+ \qwireup[dom={VIII},cod={IX},
+ label={\texttt{\textbackslash qspace[-3]}},
+ label side={below},color=cyan]
+ \qwireup[dom={IX},cod={X},
+ label={\texttt{\textbackslash qreset}},color=cyan]
+ \qwireup[dom={X},cod={XI},
+ label={\texttt{\textbackslash qreset[-2,1]}},color=cyan]
+ \draw[help lines] (-4,-6) grid (2,2);
+ }
+
+\subsubsection{Absolute Coordinates}
+By using the \texttt{at} key, positions can be specified with absolute coordinates.
+\verb|\q{\qbox[at={2,3}]}| is equivalent to \verb|\q{\qreset[2,3]\qbox}|.
+\begin{CODEBOX}{Absolute Coordinates}{Ex:20250254}
+ \q{
+ \coordinate (a) at (0,{\fpeval{2*sqrt(3)}});
+ \qbox[at=a]
+ \qbox[at={-2,0}]
+ \qbox[at={2,0}]
+ \fill[red] (a) circle (2pt);
+ \fill[red] (-2,0) circle (2pt);
+ \fill[red] (2,0) circle (2pt);
+ \node[below left] at (a) {a};
+ \node[below left] at (-2,0) {(-2,0)};
+ \node[below left] at (2,0) {(2,0)};
+ }
+\end{CODEBOX}
+The \TeX\ language is Turing complete\footnote{For example, the ability to simulate a Turing machine is demonstrated in the example from LiteratePrograms\cite{literateprograms}. Additionally, an article on Overleaf introduces a method to compute Fibonacci numbers using \TeX, illustrating the practical utility of its Turing completeness \cite{overleaf}}.
+In other words, every algorithm that exists in the world can be expressed in \TeX.
+Phisical simulations (such as weather forecasting or chemical reaction prediction),
+economic simulations (such as investment strategies),
+and even artificial intelligence learning algorithms (such as image recognition or natural language processing)------\par
+all of these can be implemented in \TeX.\par
+However, how many people would actually want to do that?
+I for one, wouldn't want to write even a sorting algorithm in \TeX.
+That's like tring to create Super Mario in spreadsheet.\par
+However, there is a twist.\par
+I had a ``revelation''\footnote{Original source: On January 20, 2011, an algorithm called "Sleep Sort" was posted on the anonymous message board 4chan \cite{geniussorting, gfxhatena}. This algorithm uses a \texttt{Bash} script to execute each input value as a thread asynchronously, sorting the list by the timing of each thread’s completion (the specified "sleep" time).}.
+And with the \texttt{at} key, I created a sorting algorithm.
+\begin{codebox}{}{Ex:genius}
+ \newcommand{\AtSort}[1][1]{
+ \q
+ {
+ \foreach \i in {#1}
+ {
+ \qbox[name={\i}, at={0, -\i}]\n
+ }
+ }
+ }
+ \AtSort[4,1,3,-1,6,2]
+\end{codebox}
+It works. Indeed, it works.\par
+But there is one problem.
+\textbf{If the list contains elements with large absolute values, the output will overflow the page}.
+\subsubsection{Wire Customization}
+\label{subsubsec:wirecustomization}
+Among the keys of the box \(\q[scale=0.5]{\qbox}\) introduced in section \ref{subsubsec:boxcustomization}, \texttt{id} and \texttt{vlen} are also valid for wires.
+The basic keys specific to wires are as follows:
+\begin{itemize}
+ \item \texttt{dom}: input point.
+ \item \texttt{cod}: output point.
+ \iitem Default value is the position obtained by moving \texttt{dom} \(1\) unit vertically to the north.
+ \item \texttt{label}: Label for the wire.
+ \item \texttt{label at}: Position of the label on the wire.
+ \iitem The default is midway.
+ \item \texttt{label side}: Relationship between the label and the wire.
+ \iitem The default is right.
+\end{itemize}
+\begin{codebox}{}{ex:202503260951}
+ \q{
+ \qbox[n=3,hlen=1.5]
+ \qbox[at={1,2}, s=2, hlen=1.5]
+ \qbox[at={-1,4}, s=2, hlen=1.5]
+ \qwire[dom={N-1-1}, cod={S-1-3}]
+ \qwire[dom={N-2-1}, cod={S-1-2}]
+ \qwire[dom={N-3-1}, cod={S-2-2}]
+ \qwire[dom={N-1-2}, cod={S-2-3}]
+ }
+\end{codebox}
+The bounding box of the wire \q[scale=0.6]{\qwire} is the same as that of \q[scale=0.5]{\qbox}:
+\begin{codebox}{}{ex:202503220926}
+ \q
+ {
+ \qwire\qbox\qwire\n
+ \qbox\qwire\qbox
+ \qbball
+ }
+\end{codebox}
+The bounding box of the wire matches that of a box with \texttt{dom} as the input point and \texttt{cod} as the output point:
+\begin{itemize}
+ \item \(\q[scale=0.5]{\qwire}\) and \(\q[scale=0.5]{\qbox}\)
+ \item \(\q[scale=0.5]{\qwire[dom={0,0},cod={1,1}]}\) and \(\q[scale=0.5]{\qbox[p=2,S={1},N={2}]}\)
+\end{itemize}
+
+\begin{CODEBOX}{}{Ex:20250309}
+ \q[scale=3]{
+ \coordinate (A) at (0,0);
+ \coordinate (B) at (2,3);
+ \qwire[dom=A, cod={1,1}, label={wire1}, label side={left}]
+ \qwire[dom={1,1}, cod=B, label={wire2}, label at={near end}]
+ \node[fill=red, circle] at (A) {A};
+ \node[fill=green, circle] at (B) {B};
+ \node[fill=yellow, circle] at (1,1) {(1,1)};
+ \qbball
+ }
+\end{CODEBOX}
+
+\begin{CODEBOX}{}{Ex:20250301}
+ \q{
+ \qloop[5]{
+ \qloop[10]{
+ \qbox[show id={true}]
+ }\n
+ }\n
+ \qspace[3]\qbox[s=2,show id={true},id={ego1}]\n\n
+ \qspace[6]\qbox[n=2,show id={true},id={ego2}]\n\n
+ \qloop[5]{
+ \qloop[10]{
+ \qbox[show id={true}]
+ }\n
+ }
+ \qwire[dom={O-1-55},cod={I-1-51}]
+ \qwire[dom={O-1-60},cod={I-1-52}]
+ \qwire[dom={O-1-51},cod={I-1-45}]
+ \qwire[dom={O-1-52},cod={I-2-51}]
+ \qwire[dom={O-2-52},cod={I-1-48}]
+ }
+\end{CODEBOX}
+
+\paragraph{arrowtype}
+The wire style is specified using the \texttt{arrowtype} key.
+\begin{codebox}[center lower,listing side text,righthand width=5cm,bicolor,colbacklower=white]{}{wirearrowtype}
+ \q{
+ \qwire[arrowtype=dotted]
+ \qwire[arrowtype=dashed]
+ \qwire[arrowtype={dash dot}]
+ \qwire[arrowtype={dash dot dot}]
+ \qwire[arrowtype={densely dotted}]
+ \qwire[arrowtype={loosely dotted}]
+ }
+\end{codebox}
+
+\begin{codebox}[center lower,listing side text,righthand width=5cm,bicolor,colbacklower=white]{}{boxarrowtype}
+ \q{
+ \qbox[dom arrowtype={dotted}]
+ \qbox[cod arrowtype={dotted}]
+ \qbox[arrowtype={dotted}]
+ }
+\end{codebox}
+
+\begin{codebox}[center lower,listing side text,righthand width=5cm,bicolor,colbacklower=white]{}{spiderarrowtype}
+ \q{
+ \qspider[dom arrowtype={dotted}]
+ \qspider[cod arrowtype={dotted}]
+ \qspider[arrowtype={dotted}]
+ }
+\end{codebox}
+
+\subsubsection{Canvas}
+\begin{verbatim}
+ \q[option]
+ {
+ command 1
+ command 2
+ ...
+ }
+\end{verbatim}
+The entire diagram can be transformed useing canvas options.
+\begin{codebox}{}{Ex:20250253}
+ \q[scale=2,rotate=30]{\qbox}
+\end{codebox}
+It is also possible to \textbf{slant} the canvas.
+\begin{CODEBOX}{}{Ex:20250252}
+ \def\anglerot{-55}
+ \tikzset
+ {
+ my slant style/.style={
+ yslant=-cot(\anglerot),
+ xscale=sin(-\anglerot)
+ }
+ }
+ \q[my slant style]
+ {
+ \qcaprev\qcap\n
+ \qwire\qbraidinv\qwiredown\n
+ \qwiredown\qbraidinv\qwire\n
+ \qcup\qcuprev
+ \qBBall
+ }
+\end{CODEBOX}
+\begin{CODEBOX}{}{Ex:20250251}
+ \def\anglerot{-60}
+ \tikzset
+ {
+ my slant style/.style={
+ yslant=-cot(\anglerot),
+ xscale=sin(-\anglerot)
+ }
+ }
+ \[\q[my slant style]
+ {
+ \qbraid\qwire\n
+ \qwire\qbraid\n
+ \qbraid\qwire
+ }
+ =\q[my slant style]
+ {
+ \qbraid\qwire\n
+ \qwire\qwire\qwire\n
+ \qbraid[num R=2]
+ \qbb[red][1]
+ \qbb[cyan][6]
+ }
+ =\q[my slant style]
+ {
+ \qbraid[ num R=2]\n
+ \qwire\qwire\qwire\n
+ \qwire\qbraid
+ \qbb[red][6]
+ \qbb[cyan][1]
+ }
+ =\q[my slant style]
+ {
+ \qwire\qbraid\n
+ \qbraid\qwire\n
+ \qwire\qbraid
+ }\]
+\end{CODEBOX}
+
+\section{Monoidal Categories and \qworldname}
+In this section, we explore how to typeset graphical calculus using the \qworldname package, examining its different applications and practical usage.\par
+While \qworldname can handle diagrams independently, it is often beneficial to combine it with packages like \texttt{amsmath}, \texttt{amssymb}, and \texttt{mathtools} for more flexible mathematical expressions.
+This section demonstrates how to typeset actual graphical calculus in \TeX\ while introducing the functionalities of \qworldname.\par
+To reproduce the examples in this section, please ensure that the following packages are loaded:
+\begin{verbatim}
+ \usepackage{amsmath, amssymb, mathtools}
+ \usepackage{qworld}
+\end{verbatim}
+
+\begin{codebox}{Domain}{ex:dom2}
+ \[
+ \operatorname{dom}\left(
+ \q{
+ \qbox[name={\(f\)}]
+ \symbolI{\(x\)}
+ \symbolO{\(y\)}
+ }
+ \right)=\q{\qwire[label={\(x\)}]}
+ \]
+\end{codebox}
+\begin{codebox}{Codomain}{ex:cod2}
+ \[
+ \operatorname{cod}\left(
+ \q{
+ \qbox[name={\(f\)}]
+ \symbolI{\(x\)}
+ \symbolO{\(y\)}
+ }
+ \right)=\q{\qwire[label={\(y\)}]}
+ \]
+\end{codebox}
+
+\begin{codebox}{(Vertical) Compositon}{def:composite}
+ \[
+ \q{
+ \qbox[name={\(f\)}]\qcirc\qbox[name={\(g\)}]
+ }=\q{\qbox[name={\(f\circ g\)}]}
+ \]
+\end{codebox}
+\begin{CODEBOX}{Associativity}{Ex:20250250}
+ \[
+ \forall\q{\qbox[name={\(h\)}] \symbolI{\(z\)} \symbolO{\(w\)}},
+ \q{\qbox[name={\(g\)}] \symbolI{\(y\)} \symbolO{\(z\)}},
+ \q{\qbox[name={\(f\)}] \symbolI{\(x\)} \symbolO{\(y\)}}:
+ \q{
+ \qbox[name={\(h\)}]\qcirc\qbox[name={\(g\)}]\qcirc\qbox[name={\(f\)}]
+ \qbb[2,3]
+ }
+ =
+ \q{
+ \qbox[name={\(h\)}]\qcirc\qbox[name={\(g\)}]\qcirc\qbox[name={\(f\)}]
+ \qbb[1,2]
+ }
+ \]
+\end{CODEBOX}
+\begin{CODEBOX}{Identity Law}{IdentityLaw}
+ \begin{equation}
+ \label{law:identity}
+ \forall\q
+ {
+ \qbox[name={\(f\)}]
+ \symbolS{\(x\)}
+ \symbolN{\(y\)}
+ }\colon
+ \q{
+ \qwire[label={\(y\)}]\n
+ \qbox[name={\(f\)}]
+ }=\q{
+ \qbox[name={\(f\)}]
+ }=\q{
+ \qbox[name={\(f\)}]\n
+ \qwire[label={\(x\)}]
+ }
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{Tensor Product, Horizontal (Parallel) Compositon}{monoidalproduct}
+ \[
+ \q
+ {
+ \qbox[name={\(f\)}] \qbox[name={\(g\)}]
+ \symbolO{\(Y_1\)}[id=1] \symbolO{\(Y_2\)}
+ \symbolI{\(X_1\)}[id=1] \symbolI{\(X_2\)}
+ }
+ =\q
+ {
+ \qbox[name={\(f\otimes g\)}, p=2]
+ \symbolO{\(Y_1\)} \symbolO[2]{\(Y_2\)}
+ \symbolI{\(X_1\)} \symbolI[2]{\(X_2\)}
+ }
+ =\q
+ {
+ \qbox[name={\(f\otimes g\)}]
+ \symbolO{\(Y_1\otimes Y_2\)}
+ \symbolI{\(X_1\otimes X_2\)}
+ }
+ \]
+\end{CODEBOX}
+
+\begin{CODEBOX}{Interchange Law (Covariant Functoriality of \(\bigotimes\))}{InterchangeLaw}
+ \begin{equation}
+ \label{law:interchange}
+ \q
+ {
+ \qbox[name={\(f_2\)}]\qbox[name={\(g_2\)}]\n
+ \qbox[name={\(f_1\)}]\qbox[name={\(g_1\)}]
+ \qbb[1,2]
+ \qbb[3,4]
+ }
+ =\q
+ {
+ \qbox[name={\(f_2\)}]\qbox[name={\(g_2\)}]\n
+ \qbox[name={\(f_1\)}]\qbox[name={\(g_1\)}]
+ \qbb[2,4]
+ \qbb[1,3]
+ }
+ \end{equation}
+\end{CODEBOX}
+
+\begin{CODEBOX}{Unit Object}{Ex:20250244}
+ \begin{gather}
+ \q{ \qwire[arrowtype=dotted]}
+ =\q{ \qunitob \qbball }\\
+ \q{ \qunitob \qbox[name={\(f\)}] \qbball }
+ =\q{ \qbox[name={\(f\)}] }
+ =\q{ \qbox[name={\(f\)}] \qunitob \qbball }\label{cond:unitob}
+ \end{gather}
+\end{CODEBOX}
+\begin{CODEBOX}{}{Ex:20250243}
+ \[
+ \q{\qunitob\qbball}
+ =\q{\qloop[2]{\qunitob}\qbball}
+ =\q{\qloop[3]{\qunitob}\qbball}
+ =\cdots
+ \]
+\end{CODEBOX}
+
+\subsection{Special Boxes}
+
+\begin{itemize}
+ \item \texttt{\textbackslash qstate}: \(\q[scale=0.5]{\qstate}\) (Zero Input).
+ \item \texttt{\textbackslash qeffect}: \(\q[scale=0.5]{\qeffect}\) (Zero Output).
+ \item \texttt{\textbackslash qscalar}: \(\q[scale=0.5]{\qscalar}\) (Zero Input, Zero Output).
+ \item \texttt{\textbackslash qasym}: \(\q[scale=0.5]{\qasym}\) (Asymmetric Box).
+ \item \texttt{\textbackslash qtrans}: \(\q[scale=0.5]{\qtrans}\) (Transpose Box). Section \ref{subsub:transpose} on page \pageref{subsub:transpose}.
+ \item \texttt{\textbackslash qadj}: \(\q[scale=0.5]{\qadj}\) (Adjoint Box). Section \ref{subsub:adjoint} on page \pageref{subsub:adjoint}.
+ \item \texttt{\textbackslash qconj}: \(\q[scale=0.5]{\qconj}\) (Conjugate Box). Section \ref{subsub:conj} on page \pageref{subsub:conj}.
+\end{itemize}
+\subsubsection{Zero Input}
+\label{subsub:zeroinput}
+\begin{codebox}{}{def:state}
+ \q{\qstate}
+\end{codebox}
+\subsubsection{Zero Output}
+\label{subsub:zerooutput}
+\begin{codebox}{}{def:effect}
+ \q{\qeffect}
+\end{codebox}
+\subsubsection{Zero Input, Zero Output}
+\label{subsub:zeroinputoutput}
+\begin{codebox}{}{def:scalar}
+ \q{\qscalar}
+\end{codebox}
+\paragraph{\(\q[scale=0.4]{\qeffect\qstate\qwire[dom={O-1-2}, cod={1,2}]\qwire[dom={1,-1}, cod={I-1-1}]}=\q[scale=0.4]{\qwire[vlen=0.5]\n\qstate\n\qeffect\n[0.5]\qwire[vlen=0.5]}=\q[scale=0.4]{\qstate\qeffect\qwire[dom={O-1-1}, cod={1,2}]\qwire[dom={1,-1}, cod={I-1-2}]}\)}
+\begin{CODEBOX}{}{coherence:unitor}
+ \[
+ \q{
+ \qeffect\qstate
+ \qwire[dom={O-1-2}, cod={1,2}]
+ \qwire[dom={1,-1}, cod={I-1-1}]
+ }=\q{
+ \qwire[arrowtype=dotted]\qwire\n
+ \qeffect\qstate\n
+ \qwire\qwire[arrowtype=dotted]
+ }\stackrel{\text{(\ref{law:interchange}, \ref{law:identity})}}{=}\q{
+ \qwire[arrowtype=dotted]\qstate\n
+ \qeffect\qwire[arrowtype=dotted]
+ }\stackrel{\text{(\ref{law:interchange}, \ref{cond:unitob})}}{=}\q{\qstate\n\qeffect}
+ \]
+\end{CODEBOX}
+\begin{CODEBOX}{Matrix Representation}{matrixrepre}
+ \begin{align*}
+ \q{\qbox[name={\(f\)}]}
+ &=\sum\limits_{i,j}
+ \left(
+ \q{
+ \qeffect[name={\(w_j^{\dagger}\)}]\n
+ \qbox[name={\(f\)}]\n
+ \qstate[name={\(v_i\)}]\n
+ }\q{
+ \qstate[name={\(w_j\)}]\n
+ \qeffect[name={\(v_i^{\dagger}\)}]
+ }
+ \right)
+ =\sum\limits_{i,j}\left(
+ \q{
+ \qscalar[name={\(w_j^{\dagger}\circ f\circ v_i\)}]
+ }
+ \q{
+ \qstate[name={\(w_j\)}]\n
+ \qeffect[name={\(v_i^{\dagger}\)}]
+ }
+ \right)\\
+ &=\begin{pmatrix}
+ \q{\qscalar[name={\(w_1^{\dagger}\circ f\circ v_1\)}]}&&\cdots&&\q{\qscalar[name={\(w_1^{\dagger}\circ f\circ v_{m}\)}]}\\
+ &\ddots&&&\\
+ \vdots& &\q{\qscalar[name={\(w_j^{\dagger}\circ f\circ v_i\)}]}& &\vdots\\
+ &&&\ddots&\\
+ \q{\qscalar[name={\(w_n^{\dagger}\circ f\circ v_1\)}]}&&\cdots&&\q{\qscalar[name={\(w_n^{\dagger}\circ f\circ v_m\)}]}
+ \end{pmatrix}
+ \end{align*}
+\end{CODEBOX}
+
+\subsection{Monoid Object}
+\label{sub:monoid}
+\begin{itemize}
+ \item \texttt{\textbackslash qmul}: \(\q[scale=0.5]{\qmul}\)
+ \iitem \texttt{hlen=2}: \(\q[scale=0.5]{\qmul[hlen=2]}\)
+ \item \texttt{\textbackslash qcomul}: \(\q[scale=0.5]{\qcomul}\)
+ \iitem \texttt{hlen=2}: \(\q[scale=0.5]{\qcomul[hlen=2]}\)
+ \item \texttt{\textbackslash qunit}: \(\q[scale=0.5]{\qunit}\)
+ \item \texttt{\textbackslash qcounit}: \(\q[scale=0.5]{\qcounit}\)
+ \item \texttt{\textbackslash qspider}: \(\q[scale=0.5]{\qspider}\)
+\end{itemize}
+
+\begin{CODEBOX}{Associativity}{AssociateLaw}
+ \begin{equation}
+ \label{law:associate}
+ \q{
+ \qspace[0.5]\qmul[hlen=2]\n
+ \qmul[hlen=2]\qspace[-0.5]\qwire
+ }=\q{
+ \qmul[hlen=2]\n
+ \qwire\qspace[-0.5]\qmul[hlen=2]
+ }
+ \end{equation}
+\end{CODEBOX}
+ \begin{CODEBOX}{Unitality}{UnitLaw}
+ \begin{equation}
+ \label{law:unit}
+ \q{
+ \qmul[hlen=2]\n
+ \qwire\qunit
+ }=\q{\qwire[vlen=2]}
+ =\q
+ {
+ \qmul[hlen=2]\n
+ \qunit\qwire
+ }
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{Coassociativity}{CoassociateLaw}
+ \begin{equation}
+ \label{law:coassociate}
+ \q{
+ \qcomul[hlen=2]\qspace[-0.5]\qwire\n
+ \qspace[0.5]\qcomul[hlen=2]
+ }
+ =\q{
+ \qwire\qspace[-0.5]\qcomul[hlen=2]\n
+ \qcomul[hlen=2]
+ }
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{Counitality}{CounitLaw}
+ \begin{equation}
+ \label{law:counit}
+ \q{
+ \qcounit\qwire\n
+ \qcomul[hlen=2]
+ }
+ =\q{\qwire[vlen=2]}
+ =\q{
+ \qwire\qcounit\n
+ \qcomul[hlen=2]
+ }
+ \end{equation}
+\end{CODEBOX}
+
+\subsubsection{Frobenius Law}
+\begin{FCODEBOX}{Frobenius Law}{FrobeniusLaw}
+ \[
+ \q{
+ \qwire\qmul[color=black]\n
+ \qcomul\qwire
+ }=\q{
+ \qcomul\n
+ \qmul[color=black]
+ }=\q{
+ \qmul[color=black]\qwire\n
+ \qwire\qcomul
+ }
+ \]
+ \end{FCODEBOX}
+ Although it is mentioned as a ``Failure Example'', in reality, the Frobenius law is accurately described.
+ Rather, since the spacing between all the wires is maintained as integer values, adjustment during vertical composition becomes easier, and in some cases, no adjustment may be necessary.
+ This is merely a matter of preference.
+ The default drawing results of \verb|\qmul| and \verb|\qcomul|, which are \q[scale=0.5]{\qmul} and \q[scale=0.5]{\qcomul}, are remnants of the default drawing results of \verb|\qbox[s=2]| and \verb|\qbox[n=2]|, which are \q[scale=0.5]{\qbox[s=2]} and \q[scale=0.5]{\qbox[n=2]}.
+ For adjustments in such cases, the adjustment key \verb|hlen| is available:
+ \begin{CODEBOX}{Frobenius Law}{FrobeniusLaw2}
+ \[
+ \q{
+ \qwire\qmul[hlen=2,color=black]\n
+ \qcomul[hlen=2]\qwire
+ }=\q{
+ \qcomul[hlen=2]\n
+ \qmul[hlen=2,color=black]
+ }=\q{
+ \qmul[hlen=2,color=black]\qwire\n
+ \qwire\qcomul[hlen=2]
+ }
+ \]
+ \end{CODEBOX}
+ The command \verb|\qspider| is the same as \verb|\qbox|, except that the nodes are circles rather than boxes.
+ \begin{codebox}{}{spider202503240521}
+ \[\q{\qspider}\]
+ \[\q{\qspider[n=0]}\]
+ \[\q{\qspider[n=0,s=4]}\]
+ \[\q{\qspider[n=2,s=4]}\]
+ \[\q{\qspider[N={1,4},s=4]}\]
+ \[\q{\qspider[N={2,3},s=4]}\]
+ \[\q{\qspider[p=3,N={2},S={1,3}]}\]
+ \[\q{\qspider[p=3,N={2},S={1,3},hlen=2]}\]
+ \[\q{\qspider[n=1,s=2,hlen=2]}\]
+ \end{codebox}
+
+ \begin{CODEBOX}{}{spider202503270738}
+ \[
+ \q{
+ \qspider[n=2, s=4, hlen=4]\qwire\qwire\n
+ \qwire\qwire\qspider[n=4, s=2, hlen=4]
+ }
+ =\q{\qspider[p=4]}\]
+ \end{CODEBOX}
+
+\subsection{Braiding}
+\label{sub:braidedcategory}
+\begin{itemize}
+ \item \texttt{\textbackslash qbraid}: \(\q[scale=0.5]{\qbraid}\)
+ \item \texttt{\textbackslash qbraidinv}: \(\q[scale=0.5]{\qbraidinv}\)
+ \item \texttt{\textbackslash qsym}: \(\q[scale=0.5]{\qsym}\)
+\end{itemize}
+
+\paragraph{\texttt{num L} and \texttt{num R}}
+
+For the commands \verb|\qbraid|, \verb|\qbraidinv|, and \verb|\qsym|, \texttt{num L} specifies the number of wires on the left side, and \texttt{num R} specifies the number of wires on the right side.
+For example, specifies \texttt{num L=3} and \texttt{num R=2} will result in the diagram \q{\qbraid[num L=3, num R=2]}.
+The default value for both is \(1\).
+
+\begin{CODEBOX}{Hexagon Equations}{hexagon}
+ \begin{equation}
+ \label{eq:hexagon}
+ \q
+ {
+ \qbraid[num R=2,,hlen=2,vlen=2]
+ }
+ =
+ \q
+ {
+ \qwire\qbraid\n
+ \qbraid\qwire
+ }
+ \qquad
+ \q
+ {
+ \qbraid[num L=2,,hlen=2,vlen=2]
+ }
+ =
+ \q
+ {
+ \qbraid\qwire\n
+ \qwire\qbraid
+ }
+ \end{equation}
+\end{CODEBOX}
+\paragraph{\({\q[scale=0.5]{\qbraid}}^{-1}=\q[scale=0.5]{\qbraidinv}\)}
+\begin{CODEBOX}{}{braidisom2}
+ \begin{equation}
+ \label{eq:inverse:braid}
+ \q
+ {
+ \qbraidinv\n
+ \qbraid
+ }
+ =
+ \q
+ {
+ \qwire[vlen=2]\qwire[vlen=2]
+ }
+ \qquad
+ \q
+ {
+ \qbraid\n
+ \qbraidinv
+ }
+ =
+ \q
+ {
+ \qwire[vlen=2]\qwire[vlen=2]
+ }
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{Naturality of the Braid (Being a Natural Transformation)}{braidnatural}
+ \begin{equation}
+ \label{cond:naturality:braid}
+ \q
+ {
+ \qbraid\n
+ \qbox[name={\(f\)}]\qbox[name={\(g\)}]
+ }
+ =
+ \q
+ {
+ \qbox[name={\(g\)}]\qbox[name={\(f\)}]\n
+ \qbraid
+ }
+ \qquad
+ \q
+ {
+ \qbraidinv\n
+ \qbox[name={\(f\)}]\qbox[name={\(g\)}]
+ }
+ =
+ \q
+ {
+ \qbox[name={\(g\)}]\qbox[name={\(f\)}]\n
+ \qbraidinv
+ }
+ \end{equation}
+\end{CODEBOX}
+
+For the commands \verb|\qbraid|, \verb|\qbraidinv|, and \verb|\qasym|, specifying \texttt{num R=0} results in a dashed line for the right wire, as shown in \q[scale=0.65]{\qbraid[num R=0]}.
+Similarly, specifying \texttt{num L=0} results in a dashed line for the left wire, as shown in \q[scale=0.65]{\qbraid[num L=0]}.
+This is due to representing the unit object as an empty diagrams, i.e., a diagram with \(0\) wires.
+\begin{CODEBOX}{}{stateeffect3dimbraid}
+ \begin{gather}
+ \q{
+ \qwire\qstate\n
+ \qwire
+ }
+ =\q{
+ \qwire\qstate\n
+ \qbraid[num L=0]
+ }\stackrel{\text{(\ref{cond:naturality:braid})}}{=}\q{
+ \qbraid\n
+ \qstate\qwire
+ }\label{iso:u:state}\\
+ \q{
+ \qunitob\qwire\n
+ \qeffect\qwire
+ }
+ =\q{
+ \qbraid[num L=0]\n
+ \qeffect\qwire
+ }\stackrel{\text{(\ref{cond:naturality:braid})}}{=}\q{
+ \qwire\qeffect\n
+ \qbraid
+ }\label{iso:u:effect}
+ \end{gather}
+\end{CODEBOX}
+\begin{CODEBOX}{}{iso:spatial:scalar}
+ \begin{equation}
+ \label{iso:spatial}
+ \q{\qscalar\qmv[0,-1]\qwire[vlen=3]}
+ =\q{\qbraid[num L=0]\n\qscalar\qmv[0,-1]\qwire[vlen=2]}
+ \stackrel{\text{(\ref{cond:naturality:braid})}}{=}
+ \q{\qwire[vlen=2]\qscalar\n\qbraid[num L=0]}
+ =\q{\qscalar\qmv[-2,-1]\qwire[vlen=3]}
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{}{iso:spatial:state.effect}
+ \[
+ \q{\qscalar\qmv[0,-1]\qwire[vlen=3]}
+ =\q{\qeffect\n\qstate\qmv[0,-0.5]\qwire[vlen=3]}
+ \stackrel{\text{(\ref{iso:u:effect})}}{=}
+ \q{\qwire\qeffect\n\qbraid\n\qstate\qwire}
+ \stackrel{\text{(\ref{iso:u:state})}}{=}
+ \q{\qeffect\n\qstate\qmv[-2,-0.5]\qwire[vlen=3]}
+ =\q{\qscalar\qmv[-2,-1]\qwire[vlen=3]}
+ \]
+\end{CODEBOX}
+\paragraph{\(\q[scale=0.5]{\qbraid\qwire\n\qwire\qstate[n=2]}=\q[scale=0.5]{\qwire\qbraidinv\n\qstate[n=2]\qwire}\)}
+\begin{CODEBOX}{}{dashedbraid}
+ \begin{align}
+ \q
+ {
+ \qbraid\qwire\n
+ \qwire\qstate[n=2]
+ }
+ &=\q
+ {
+ \qbraid\qwire\n
+ \qwire\qstate[n=2]\n\n
+ \qbraidinv[num L=0,vlen=2,hlen=2.5]
+ }
+ \stackrel{\text{(\ref{cond:naturality:braid})}}{=}\q
+ {
+ \qbraid\qwire\n\n
+ \qbraidinv[num L=2,vlen=2]\n
+ \qstate[n=2]\qwire\n
+ \qspace[0.5]\qwire[arrowtype={densely dotted}]\qspace[0.5]\qwire
+ }\\
+ &=\q
+ {
+ \qbraid\qwire\n\n
+ \qbraidinv[num L=2,vlen=2]\n
+ \qstate[n=2]\qwire\n
+ \qspace[2]\qwire
+ \qbb[green][3]
+ }\stackrel{\text{(\ref{eq:hexagon})}}{=}\q
+ {
+ \qbraid\qwire\n
+ \qbraidinv\qwire\n
+ \qwire\qbraidinv\n
+ \qstate[n=2]\qwire\n
+ \qspace[2]\qwire
+ \qbb[green][3,...,6]
+ }\\
+ &=\q
+ {
+ \qbraid\qwire\n
+ \qbraidinv\qwire\n
+ \qwire\qbraidinv\n
+ \qstate[n=2]\qwire\n
+ \qspace[2]\qwire
+ \qbb[orange][1,3]
+ }
+ \stackrel{\text{(\ref{eq:inverse:braid})}}{=}\q
+ {
+ \qloop[2]{\qloop[3]{\qwire}\n}
+ \qwire\qbraidinv\n
+ \qstate[n=2]\qwire\n
+ \qspace[2]\qwire
+ \qbb[orange][1,2,4,5]
+ }
+ \end{align}
+\end{CODEBOX}
+
+\begin{CODEBOX}{Yang-Baxer Equation}{YangBaxer}
+ \[
+ \q
+ {
+ \qbraid\qwire\n
+ \qwire\qbraid\n
+ \qbraid\qwire
+ }=\q
+ {
+ \qwire\qbraid\n
+ \qbraid\qwire\n
+ \qwire\qbraid
+ }
+ \]
+\end{CODEBOX}
+
+\begin{codebox}[center lower,listing side text,righthand width=5cm,bicolor,colbacklower=white]{Commutativity}{def:cmt}
+ \begin{equation}
+ \label{cond:cmt}
+ \q{\qmul[hlen=2]}=\q{\qmul[hlen=2]\n\qbraid}
+ \end{equation}
+\end{codebox}
+\begin{codebox}[center lower,listing side text,righthand width=5cm,bicolor,colbacklower=white]{Cocommutativity}{def:ccmt}
+ \begin{equation}
+ \label{cond:ccmt}
+ \q{\qcomul[hlen=2]}=\q{\qbraid\n\qcomul[hlen=2]}
+ \end{equation}
+\end{codebox}
+\subsubsection{Bialgebra}
+\begin{CODEBOX}{Bialgebra Law}{law:bialgebra}
+ \begin{gather}
+ \q{\qcomul[hlen=2]\qcirc\qmul[hlen=2, color=black]}
+ =\q{
+ \qmul[hlen=2, color=black]\qmul[hlen=2, color=black]\n
+ \qwire\qbraid\qwire\n\qcomul[hlen=2]\qcomul[hlen=2]
+ }\\
+ \q{\qspace[0.5]\qcounit\qcirc\qmul[hlen=2, color=black]}
+ =\q{\qcounit\qcounit}\\
+ \q{\qcomul[hlen=2]\qcirc\qspace[0.5]\qunit[color=black]}
+ =\q{\qunit[color=black]\qunit[color=black]}\\
+ \q{\qcounit\n\qunit[color=black]}
+ =\q{\qunitob}
+ \end{gather}
+\end{CODEBOX}
+\subsubsection{Hopf Algebra}
+\begin{CODEBOX}{Antipode and Hopf Algebra}{law:hopf}
+ \begin{equation}
+ \q{
+ \qmul[hlen=2, color=black]\n
+ \qwire\qbox[name={\(s\)}]\n
+ \qcomul[hlen=2]
+ }
+ =\q{\qunit[color=black]\qcirc\qcounit}
+ =\q{
+ \qmul[hlen=2, color=black]\n
+ \qbox[name={\(s\)}]\qwire\n
+ \qcomul[hlen=2]
+ }
+ \end{equation}
+\end{CODEBOX}
+\subsubsection{Balanced Structure}
+\label{subsub:balancedcategory}
+\begin{CODEBOX}{Twist}{def:twist}
+ \begin{gather}
+ \q{
+ \qbox[name={\(\theta_{X\otimes Y}\)}, p=2]
+ \symbolI{\(X\)}\symbolI[2]{\(Y\)}
+ \symbolO{\(X\)}\symbolO[2]{\(Y\)}
+ }
+ =\q{
+ \qbox[name={\(\theta_X\)}]\qbox[name={\(\theta_Y\)}]\n
+ \qbraid\n\qbraid
+ }\label{cond:twist}\\
+ \q{\qbox[name={\(\theta_I\)}]}=\q{\qunitob}\label{cond:i:twist}\\
+ \forall\q{\qbox[name={\(f\)}]}\colon
+ \q{
+ \qbox[name={\(\theta_{\operatorname{cod}(f)}\)}, hlen=1.5]\n
+ \qbox[name={\(f\)}, hlen=1.5]
+ }=\q{
+ \qbox[name={\(f\)}, hlen=1.5]\n
+ \qbox[name={\(\theta_{\operatorname{dom}(f)}\)}, hlen=1.5]
+ }\label{cond:naturality:twist}
+ \end{gather}
+\end{CODEBOX}
+\subsubsection{Trivial Balanced Structure, Symmetric Monoidal Category}
+\label{subsub:symmetry}
+\begin{codebox}[center lower,listing side text,righthand width=5cm,bicolor,colbacklower=white]{Symmetry}{defeq:sym}
+ \begin{equation}
+ \label{cond:symmetry}
+ \q
+ {
+ \qbraid\n
+ \qbraid
+ }
+ =\q{\qloop[2]{\qwire[vlen=2]}}
+ \end{equation}
+\end{codebox}
+\begin{codebox}{}{wowsym}
+ \[\q{\qbraid}=\q{\qbraidinv}\]
+\end{codebox}
+\begin{codebox}{Symmetry}{def:sym}
+ \q{\qsym}
+\end{codebox}
+
+\subsection{Duality}
+\label{subsub:duality}
+\begin{itemize}
+ \item \texttt{\textbackslash qcap}: \(\q[scale=0.6]{\qcap}\)
+ \item \texttt{\textbackslash qcup}: \(\q[scale=0.6]{\qcup}\)
+ \item \texttt{\textbackslash qcaprev}: \(\q[scale=0.6]{\qcaprev}\)
+ \item \texttt{\textbackslash qcuprev}: \(\q[scale=0.6]{\qcuprev}\)
+\end{itemize}
+
+\begin{CODEBOX}{Snake Equations}{Snake}
+ \begin{equation}
+ \label{eq:snake}
+ \q{
+ \qcap\qwireup\n
+ \qwireup\qcup
+ }
+ =\q{
+ \qwireup[vlen=2]
+ }
+ \qquad\land\qquad
+ \q{
+ \qwiredown\qcap\n
+ \qcup\qwiredown
+ }
+ =\q{
+ \qwiredown[vlen=2]
+ }
+ \end{equation}
+\end{CODEBOX}
+
+\subsubsection{Transpose Box}
+\label{subsub:transpose}
+\begin{CODEBOX}{Transpose}{Transpose}
+ \[
+ \left(\q{\qwire\n\qasym[name={\(f\)}]\n\qwire}\right)^T
+ =\q{
+ \qwire\n
+ \qtrans[name={\(f\)}]\n
+ \qwire
+ }\coloneq\q{
+ \qwire\qcap\n
+ \qwire\qasym[name={\(f\)}]\qwire\n
+ \qcup\qwire
+ }
+ \]
+\end{CODEBOX}
+
+\begin{CODEBOX}{}{ex:202504021826}
+ \begin{equation}
+ \label{eq:wirerev}
+ \left(\q{\qwireup}\right)^T=\q{\qwiredown}
+ \end{equation}
+\end{CODEBOX}
+
+\begin{CODEBOX}{Sliding}{Sliding}
+ \begin{equation}
+ \label{eq:sliding}
+ \q
+ {
+ \qcap \n
+ \qasym \qwire
+ }
+ =
+ \q
+ {
+ \qcap \n
+ \qwire \qtrans
+ }
+ \qquad
+ \q
+ {
+ \qwire\qasym \n
+ \qcup
+ }
+ =
+ \q
+ {
+ \qtrans \qwire \n
+ \qcup
+ }
+ \end{equation}
+\end{CODEBOX}
+
+\begin{CODEBOX}{}{MapStateDuality}
+ \begin{equation}
+ \begin{array}{rccc}
+ \operatorname{NAME}\colon & \operatorname{Hom}(X,Y)
+ & \to & \operatorname{Hom}(I,X^T\otimes Y)\\
+ & \rotatebox{90}{\(\in\)} && \rotatebox{90}{\(\in\)}\\
+ & \q{
+ \qasym
+ \symbolI{\(X\)}
+ \symbolO{\(Y\)}
+ } & \mapsto
+ & \q{
+ \qwire[label={\(X^T\)}, label at={at end}, label side={above}] \qasym \n
+ \qcup
+ \symbolO{\(Y\)}[id=2]
+ }
+ \end{array}
+ \end{equation}
+\end{CODEBOX}
+
+\begin{CODEBOX}{}{MapEffectDuality}
+ \begin{equation}
+ \begin{array}{rccc}
+ \operatorname{CONAME}\colon & \operatorname{Hom}(X,Y)
+ & \to & \operatorname{Hom}(X\otimes Y^T,I)\\
+ & \rotatebox{90}{\(\in\)}&&\rotatebox{90}{\(\in\)}\\
+ & \q{
+ \qasym
+ \symbolI{\(X\)}
+ \symbolO{\(Y\)}
+ } & \mapsto
+ & \q{
+ \qcap \n
+ \qasym \qwire[label={\(Y^T\)}, label at={at start}, label side={below}]
+ \symbolI{\(X\)}[id=2]
+ }
+ \end{array}
+ \end{equation}
+\end{CODEBOX}
+
+\subsubsection{No Cloning Theorem}
+\label{subsub:nocopy}
+\begin{CODEBOX}{Uniform Copying}{def:copy}
+ \begin{gather}
+ \q{
+ \qwire\qbraid\qwire\n
+ \qcomul[hlen=2, name={\(\operatorname{copy}_X\)}]\qcomul[hlen=2, name={\(\operatorname{copy}_Y\)}]
+ \symbolI{\(X\)}[id=4]\symbolI{\(Y\)}[id=5]
+ \symbolO{\(X\)}[id=1]\symbolO{\(Y\)}[id=2]
+ \symbolO[2]{\(X\)}[id=2]\symbolO{\(Y\)}[id=3]
+ }=\q{
+ \qcomul[hlen=3, vlen=2, name={\(\operatorname{copy}_{X\otimes Y}\)}]\symbolI{\(X\otimes Y\)}
+ \symbolO{\(X\otimes Y\)}\symbolO[2]{\(X\otimes Y\)}
+ }\label{cond:copy}\\
+ \forall\q{\qbox[name={\(f\)}]\symbolI{\(X\)}\symbolO{\(Y\)}}\colon
+ \q{
+ \qbox[name={\(f\)}]\qbox[name={\(f\)}]\n
+ \qcomul[hlen=2, name={\(\operatorname{copy}_X\)}]
+ }=\q{
+ \qcomul[hlen=2, name={\(\operatorname{copy}_Y\)}]\n
+ \qspace[0.5]\qbox[name={\(f\)}]
+ }\label{cond:morcopy}\\
+ \q{
+ \qcomul[hlen=2, name={\(\operatorname{copy}_I\)}]\symbolI{\(I\)}
+ \symbolO{\(I\)}\symbolO[2]{\(I\)}
+ }=\q{\qunitob}\label{cond:i:copy}
+ \end{gather}
+\end{CODEBOX}
+
+\begin{CODEBOX}{Copyable State}{def:copyablestate}
+ \[
+ \q{
+ \qcomul[hlen=2, name={\(\operatorname{copy}\)}]\n
+ \qspace[0.5]\qstate[name={\(a\)}]
+ }=\q{
+ \qstate[name={\(a\)}]\qstate[name={\(a\)}]
+ }
+ \]
+\end{CODEBOX}
+\begin{CODEBOX}{}{len:copycupcup}
+ \begin{align*}
+ \q{\qcup\qcup}&\stackrel{\text{(\ref{cond:i:copy})}}{=}
+ \q{
+ \qcup\qcup\n\qmv[0.5,0.4]
+ \qcomul[dom arrowtype=dotted, cod arrowtype=dotted, hlen=3, name={\(\operatorname{copy}\)}]
+ }\stackrel{\text{(\ref{cond:morcopy})}}{=}
+ \q{
+ \qspider[
+ name={\(\operatorname{copy}\)},
+ N={1,2,4,5},
+ s=2,
+ hlen=2
+ ]\n\qcup
+ }\\
+ &\stackrel{\text{(\ref{cond:copy})}}{=}\q{
+ \qwire\qbraid\qwire\n
+ \qloop[4]{\qwire}\n
+ \qcomul[hlen=2, name={\(\operatorname{copy}\)}]\qcomul[hlen=2]\n
+ \qspace[0.5]\qcup[hlen=3]
+ \qbb[red][4,8]
+ }\stackrel{\text{(\ref{cond:ccmt})}}{=}\q{
+ \qwire\qbraid\qwire\n
+ \qbraidinv\qwire\qwire\n
+ \qcomul[hlen=2]\qcomul[hlen=2]\n
+ \qspace[0.5]\qcup[hlen=3]
+ \qbb[red][4,7]
+ }\\
+ &=\q{
+ \qwire\qbraid\qwire\n
+ \qbraidinv\qwire\qwire\n[2]
+ \qloop[4]{\qwire[vlen=2]}\n
+ \qloop[2]{\qcomul[hlen=2, name={\(\operatorname{copy}\)}]}\n
+ \qspace[0.5]\qcup[hlen=3]
+ \qbb[blue][8,9]
+ }
+ \stackrel{\text{(\ref{eq:inverse:braid})}}{=}
+ \q{
+ \qwire\qbraid\qwire\n
+ \qbraidinv\qwire\qwire\n
+ \qwire\qbraidinv\qwire\n
+ \qwire\qbraid\qwire\n
+ \qloop[2]{\qcomul[hlen=2, name={\(\operatorname{copy}\)}]}\n
+ \qspace[0.5]\qcup[hlen=3]
+ \qbb[blue][8,11]
+ }\\
+ &=\q{
+ \qwire\qbraid\qwire\n
+ \qbraidinv\qwire\qwire\n
+ \qwire\qbraidinv\qwire\n
+ \qwire\qbraid\qwire\n
+ \qloop[2]{\qcomul[hlen=2, name={\(\operatorname{copy}\)}]}\n
+ \qspace[0.5]\qcup[hlen=3]
+ \qbb[yellow][10,14,15]
+ }\stackrel{\text{(\ref{cond:copy}, \ref{cond:morcopy}, \ref{cond:i:copy})}}{=}
+ \q{
+ \qwire\qbraid\qwire\n
+ \qbraidinv\qwire\qwire\n
+ \qwire\qbraidinv\qwire\n
+ \qcup\qcup
+ \qbb[yellow][10,11]
+ }\\
+ &=\q{
+ \qwire\qbraid\qwire\n
+ \qcup[n=4]
+ }
+ \end{align*}
+\end{CODEBOX}
+\begin{CODEBOX}{}{len:copybraid}
+ \begin{align*}
+ \q{
+ \qbraid
+ \symbolI{\(X\)}\symbolO{\(X\)}
+ \symbolI[2]{\(X\)}\symbolO[2]{\(X\)}
+ }&\stackrel{\text{(\ref{eq:snake})}}{=}\q{
+ \qwire\qspace[2]\qcap\qwire\n
+ \qbraid[hlen=4]\qwire\qwire\n
+ \qwire\qcap\qwire\qcup\n
+ \qwire\qwire\qcup
+ \qbb[blue][1,5]
+ }\\
+ &=\q{
+ \qcap[hlen=4]\qwire\n
+ \qwire\qspace[2]\qbraidinv\n
+ \qwire\qcap\qwire\qwire\qwire[vlen=3]\n
+ \qwire\qwire\qcup\qcup
+ \qbb[blue][1,4]
+ \qbb[yellow][12,13]
+ }\\
+ &\stackrel{\text{Example \ref{len:copycupcup}}}{=}\q{
+ \qcap[hlen=4]\qwire\n
+ \qwire\qspace[2]\qbraidinv\n
+ \qwire\qcap\qwire\qwire\qwire[vlen=3]\n
+ \qwire\qwire\qwire\qbraid\qwire\n
+ \qwire\qwire\qcup[n=4]
+ \qbb[yellow][12,14,17]
+ }\\
+ &=\q{
+ \qcap[s=4]\qwire\n
+ \qloop[3]{\qwire}\qbraidinv\n
+ \qloop[3]{\qwire}\qbraid\qwire[vlen=3]\n
+ \qwire\qwire\qcup[n=4]
+ \qbb[green][6,10]
+ }\\
+ &\stackrel{\text{(\ref{eq:inverse:braid})}}{=}\q{
+ \qcap[s=4]\qwire\qwire\n[2]
+ \qloop[6]{\qwire[vlen=2]}\n
+ \qwire\qwire\qcup[n=4]
+ \qbb[green][7,8]
+ }\\
+ &\stackrel{\text{(\ref{eq:snake})}}{=}\q{
+ \qwire\qwire
+ \symbolI{\(X\)}[id=1]\symbolO{\(X\)}[id=1]
+ \symbolI{\(X\)}\symbolO{\(X\)}
+ }
+ \end{align*}
+\end{CODEBOX}
+
+\begin{CODEBOX}{No Cloning Theorem}{thm:nocopy}
+ \begin{align*}
+ \q{
+ \qasym[name={\(f\)}]
+ \symbolI{\(X\)}\symbolO{\(X\)}
+ }&\stackrel{\text{(\ref{eq:snake})}}{=}
+ \q{
+ \qcap\qasym[name={\(f\)}]\n
+ \qwire\qcup
+ }\stackrel{\text{(\ref{eq:inverse:braid})}}{=}\q{
+ \qcap\qasym[name={\(f\)}]\n
+ \qwire\qbraidinv\n
+ \qwire\qbraid\n
+ \qwire\qcup
+ }\\
+ &\stackrel{\text{(\ref{cond:naturality:braid})}}{=}\q{
+ \qcap\qwire\n
+ \qwire\qbraidinv\n
+ \qwire\qasym[name={\(f\)}]\qwire\n
+ \qwire\qbraid\n
+ \qwire\qcup
+ \qbb[red][1,4]
+ }=\q{
+ \qwire\qcap\n
+ \qbraid\qwire\n
+ \qwire\qasym[name={\(f\)}]\qwire\n
+ \qwire\qbraid\n
+ \qwire\qcup
+ \qbb[red][1,4]
+ }\\
+ &=\q{
+ \qwire\qcap\n
+ \qbraid\qwire\n
+ \qwire\qasym[name={\(f\)}]\qwire\n
+ \qwire\qbraid\n
+ \qwire\qcup
+ \qbb[blue][3]
+ }\stackrel{\text{Example \ref{len:copybraid}}}{=}\q{
+ \qwire\qcap\n
+ \qwire\qwire\qwire\n
+ \qwire\qasym[name={\(f\)}]\qwire\n
+ \qwire\qbraid\n
+ \qwire\qcup
+ \qbb[blue][3,4]
+ }
+ \end{align*}
+\end{CODEBOX}
+\subsection{Pivotal Category}
+\label{sub:pivotal}
+\begin{CODEBOX}{}{struct:Pivotal}
+ \begin{equation}
+ \label{cond:naturality:pivotal}
+ \forall\q{\qasym[name={\(f\)}]\symbolI{\(X\)}\symbolO{\(Y\)}}\colon
+ \q[scale=0.5]{
+ \qwire\qcap[s=4]\n
+ \qwire\qwire\qasym[name={\(f\)}]\qwire\qwire\n
+ \qcup[n=4]\qwire\qmv[-1,-1]\qasym[name={\(\pi_X\)}]
+ }=\q{
+ \qasym[name={\(\pi_Y\)}]\n
+ \qasym[name={\(f\)}]
+ }
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{}{dualrev}
+ \begin{equation}
+ \label{def:dualrev}
+ \q{
+ \qcaprev\n
+ \symbolI{\(X^T\)}\symbolI[2]{\(X\)}
+ }\coloneq\q
+ {
+ \qcap \n
+ \qwire \qwireupup \n
+ \qwire \qasym[name={\(\pi_x\)}] \n
+ }\qquad\q{
+ \qcuprev
+ \symbolO{\(X\)}\symbolO[2]{\(X^T\)}
+ }\coloneq\q
+ {
+ \qasym[name={\(\pi_X^{-1}\)}]\qwire\n
+ \qwireupup\qwire\n
+ \qcup
+ }
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{}{revSliding}
+ \begin{equation}
+ \label{eq:sliding:rev}
+ \q{
+ \qcaprev\n
+ \qtrans\qwire\n
+ }=\q{
+ \qcaprev\n
+ \qwire\qasym
+ }\qquad\q{
+ \qwire\qtrans\n
+ \qcuprev
+ }=\q{
+ \qasym\qwire\n
+ \qcuprev
+ }
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{}{corr:pivotal:twist}
+ \begin{gather}
+ \q{
+ \qwireuu[label={\(X^{**}\)}]\n
+ \qasym[name={\(\pi_X\)}]\n
+ \qwireup[label={\(X\)}]
+ }=\q{
+ \qwireuu[label={\(X^{**}\)}]\qcap\n
+ \qbraid\qwired[label={\(X^T\)}]\n
+ \qasym[name={\(\theta_X^{-1}\)}]\qcuprev\n
+ \qwireup[label={\(X\)}]
+ }\label{twist:pivotal}\\
+ \q{
+ \qwireup[label={\(X\)}]\n
+ \qasym[name={\(\theta_X\)}]\n
+ \qwireup[label={\(X\)}]
+ }=\q{
+ \qwireup[label={\(X\)}]\n
+ \qasym[name={\(\pi_X^{-1}\)}]\n
+ \qwireuu[label={\(X^{**}\)}]\qcap\n
+ \qbraid\qwired[label={\(X^T\)}]\n
+ \qwire[label={\(X\)}]\qcuprev
+ }\label{pivotal:twist}
+ \end{gather}
+\end{CODEBOX}
+
+\subsubsection{Trace}
+\label{subsub:trace}
+\begin{CODEBOX}{Trace}{Trdef}
+ \[
+ \begin{array}{rccc}
+ \operatorname{Tr}\colon & \operatorname{Hom}(X,X) & \to &\operatorname{Hom}(I,I)\\
+ &\rotatebox{90}{\(\in\)}&&\rotatebox{90}{\(\in\)}\\
+ &\q{
+ \qasym[name={\(f\)}]
+ } & \mapsto & \q{
+ \qcap\n
+ \qasym[name={\(f\)}]\qwire\n
+ \qcuprev
+ }
+ \end{array}
+ \]
+\end{CODEBOX}
+\begin{CODEBOX}{Commutativity of the Trace}{Cotr}
+ \begin{equation}
+ \label{cmt:trace}
+ \operatorname{Tr}\left(\q{
+ \qasym[name={\(g\)}]\n
+ \qasym[name={\(f\)}]
+ }\right)
+ =\q{
+ \qcap\n
+ \qasym[name={\(g\)}]\qwire\n
+ \qasym[name={\(f\)}]\qwire\n
+ \qcuprev
+ }
+ \stackrel{\text{(\ref{eq:sliding})}}{=}\q{
+ \qcap\n
+ \qasym[name={\(f\)}]\qtrans[name={\(g\)}]\n
+ \qcuprev
+ }\stackrel{\text{(\ref{eq:sliding:rev})}}{=}\q{
+ \qcap\n
+ \qasym[name={\(f\)}]\qwire\n
+ \qasym[name={\(g\)}]\qwire\n
+ \qcuprev
+ }=\operatorname{Tr}\left(\q{
+ \qasym[name={\(f\)}]\n
+ \qasym[name={\(g\)}]
+ }\right)
+ \end{equation}
+\end{CODEBOX}
+\begin{CODEBOX}{Dimension}{def:dim}
+ \[
+ \q{\qscalar[name={\(\operatorname{dim}(X)\)}]}
+ \coloneq\q{\qscalar[name={\(\operatorname{Tr}(\operatorname{id}_X)\)}]}
+ =\q{\qcap\n\qcuprev}
+ \]
+\end{CODEBOX}
+\subsection{Dagger}
+\label{sub:dagger}
+\subsubsection{Adjoint Box}
+\label{subsub:adjoint}
+\begin{CODEBOX}{Adjoint}{def:adj}
+ \begin{gather}
+ \left(
+ \q{
+ \qasym[name={\(f\)}]
+ \symbolI{\(X\)}\symbolO{\(Y\)}
+ }
+ \right)^{\dagger}
+ \coloneq
+ \q{
+ \qadj[name={\(f\)}]
+ \symbolI{\(Y\)}\symbolO{\(X\)}
+ }\\
+ \left(\q{\qasym[name={\(f\)}]}\right)^{\dagger\dagger}
+ =\left(\q{\qadj[name={\(f\)}]}\right)^{\dagger}
+ =\q{\qasym[name={\(f\)}]}\\
+ \left(\q{\qwire}\right)^{\dagger}=\q{\qwire}\\
+ \q{
+ \qadj[name={\(f\)}]\n
+ \qadj[name={\(g\)}]
+ }=\q{
+ \qadj[name={\(g\circ f\)}]
+ }
+\end{gather}
+\end{CODEBOX}
+\subsubsection{Conjugate Box}
+\label{subsub:conj}
+\begin{CODEBOX}{}{prep:conj}
+ \begin{align*}
+ \left(\q{\qadj}\right)^T
+ &=\q{
+ \qwire\qcap\n
+ \qwire\qadj\qwire\n
+ \qcup\qwire
+ }=\q{
+ \qcaprev\qwire\n
+ \qwire\qadj\qwire\n
+ \qwire\qcuprev
+ }=\left(
+ \q{
+ \qwire\qcap\n
+ \qwire\qasym\qwire\n
+ \qcup\qwire
+ }
+ \right)^{\dagger}=\left(\q{\qtrans}\right)^{\dagger}
+ \end{align*}
+\end{CODEBOX}
+\section{List of Primitive Diagrams}
+\label{sec:fcommands}
+ \begin{tcbraster}[raster columns=4,raster equal height=rows,fonttitle=\bfseries,
+ colframe=cyan!90,coltitle=cyan!50,colbacktitle=black,
+ subtitle style={boxrule=0.4pt,
+ colback=black!70,
+ colupper=cyan!50}
+ ]
+ \def\mortype{射}
+ \def\AoE{舞台}
+ \def\RMK{備考}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qwire}]
+ \[\q{\qwire}\]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qbox}]
+ \[
+ \q{
+ \qbox
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qstate}]
+ \[
+ \q{
+ \qstate
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qeffect}]
+ \[
+ \q{
+ \qeffect
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qscalar}]
+ \[
+ \q{
+ \qscalar
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qunitob}]
+ \[
+ \q{
+ \qunitob
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qmul}]
+ \[
+ \q{
+ \qmul[hlen=2]
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qcomul}]
+ \[
+ \q{
+ \qcomul[hlen=2]
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qunit}]
+ \[
+ \q{
+ \qunit
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qcounit}]
+ \[
+ \q{
+ \qcounit
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qspider}]
+ \[
+ \q{
+ \qspider[p=4,hlen=2]
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qbraid}]
+ \[
+ \q{
+ \qbraid
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qbraidinv}]
+ \[
+ \q{
+ \qbraidinv
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qsym}]
+ \[
+ \q{
+ \qsym
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qcap}]
+ \[
+ \q{
+ \qcap
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qcup}]
+ \[
+ \q{
+ \qcup
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qwireu}]
+ \[\q{\qwireup}\]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qwired}]
+ \[\q{\qwiredown}\]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qwireuu}]
+ \[\q{\qwireuu}\]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qwiredd}]
+ \[\q{\qwiredd}\]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qasym}]
+ \[
+ \q{
+ \qasym
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qtrans}]
+ \[
+ \q{
+ \qtrans
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash adj}]
+ \[
+ \q{
+ \qadj
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qconj}]
+ \[
+ \q{
+ \qconj
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qcaprev}]
+ \[
+ \q{
+ \qcaprev
+ }
+ \]
+ \end{tcolorbox}
+ \begin{tcolorbox}[adjusted title=\texttt{\textbackslash qcuprev}]
+ \[
+ \q{
+ \qcuprev
+ }
+ \]
+ \end{tcolorbox}
+ \end{tcbraster}
+ \addcontentsline{toc}{section}{References}
+ \begin{thebibliography}{99}
+ \bibitem{tikz-manual}
+ Till Tantau,
+ \textit{The TikZ and PGF Packages Manual for Version 3.1.10},
+ Institut für Theoretische Informatik, Universität zu Lübeck,
+ n.d.,
+ \url{https://github.com/pgf-tikz/pgf}
+ (retrieved March 22, 2025).
+ \bibitem{geniussorting}
+ Anonymous,
+ \textit{Genius Sorting Algorithm},
+ 4chan Programming Board,
+ January 20, 2011,
+ \url{https://dis.4chan.org/read/prog/1295544154}
+ (accessed January 20, 2011; link currently inactive).
+ \bibitem{gfxhatena}
+ gfx,
+ \textit{Joushiki wo kutsugaesu so-to arugorizumu! Sono na mo "sleep sort"!},
+ Hatena Diary,
+ May 19, 2011,
+ \url{https://gfx.hatenadiary.org/entry/20110519/1305810786}.
+ \bibitem{literateprograms}
+ LiteratePrograms,
+ \textit{Turing Machine Simulator in \LaTeX},
+ \url{https://literateprograms.org/turing_machine_simulator__latex_.html},
+ n.d.,
+ retrieved March 22, 2025.
+ \bibitem{overleaf}
+ Overleaf,
+ \textit{\LaTeX\ is More Powerful than You Think: Computing the Fibonacci Numbers and Turing Completeness},
+ \url{https://www.overleaf.com/learn/latex/Articles/LaTeX_is_More_Powerful_than_you_Think_-_Computing_the_Fibonacci_Numbers_and_Turing_Completeness},
+ n.d.,
+ retrieved March 22, 2025.
+ \end{thebibliography}
+ \section*{Change Log}
+ \subsection*{1.1.0 (2025-04-03)}
+ \paragraph{The following commands have been updated.}
+ \begin{itemize}
+ \item \texttt{\textbackslash symbolI}
+ \item \texttt{\textbackslash symbols}
+ \item \texttt{\textbackslash symbolS}
+ \item \texttt{\textbackslash symbolO}
+ \item \texttt{\textbackslash symboln}
+ \item \texttt{\textbackslash symbolN}
+ \end{itemize}
+ The ID, which was previously a required argument for these commands, has been made optional.
+ Users can now specify the ID via an optional key, allowing for more flexible usage of the commands.
+ \begin{itemize}
+ \item Old format: \texttt{\textbackslash symbolI[i]\{ID\}\{text\}[above right]}
+ \item New format: \texttt{\textbackslash symbolI[i]\{text\}[above right, id=ID]}
+ \end{itemize}
+ \paragraph{English documentation has been added to the package.}
+ \subsection*{1.0.0 (2025-04-01)}
+ \paragraph{Initial release.}
+ \section*{License and Copyright}
+
+ This package is distributed under the \textbf{LaTeX Project Public License (LPPL)}, version 1.3c or later.
+ For details, see \url{https://www.latex-project.org/lppl.txt}.
+
+ Copyright \textcopyright 2025 Niina Ryota
+
+ You are free to distribute and modify this package under the terms of LPPL.
+ However, any modified versions \textbf{must not} be distributed under the original name, and the original author must be credited.
+\end{document}
Property changes on: trunk/Master/texmf-dist/doc/latex/qworld/qworld_eng.tex
___________________________________________________________________
Added: svn:eol-style
## -0,0 +1 ##
+native
\ No newline at end of property
Modified: trunk/Master/texmf-dist/tex/latex/qworld/qworld.sty
===================================================================
--- trunk/Master/texmf-dist/tex/latex/qworld/qworld.sty 2025-04-03 19:52:19 UTC (rev 74842)
+++ trunk/Master/texmf-dist/tex/latex/qworld/qworld.sty 2025-04-03 19:52:38 UTC (rev 74843)
@@ -1,5 +1,5 @@
%% A LaTeX package for typesetting string diagrams in various categorical structures, including monoidal, dagger compact closed, rigid, braided, symmetric, and compact categories.
-%% Copyright (C) 2025 Ryota
+%% Copyright (C) 2025 Niina Ryota
%%
%% This work may be distributed and/or modified under the conditions of the LaTeX Project Public License (LPPL),
%% version 1.3c or later. The latest version of this license is in:
@@ -8,13 +8,24 @@
%%
%% This work has the LPPL maintenance status "maintained".
%%
-%% The Current Maintainer of this work is Ryota.
+%% The Current Maintainer of this work is Niina Ryota.
%%
%% This work consists of the following files:
%% qworld.sty
%% qworld.pdf
-\ProvidesPackage{qworld}[2025/03/30 v1.0.0]
-\RequirePackage{tikz, pgffor, ifthen, xparse, xfp}
+%% Change Log
+%% -----------
+%%
+%% Version 1.1.0 (2025-04-03)
+%% - The following commands have been updated:
+%% - \symbolI, \symbols \symbolS, \symbolO, \symboln \symbolN.
+%% - The ID, which was previously a required argument for these commands, has been made optional.
+%% - Users can now specify the ID via an optional key, providing more flexibility when using the commands.
+%% - English documentation has been added to the package.
+%% Version 1.0.0 (2025-04-01)
+%% - Initial release of the package.
+\ProvidesPackage{qworld}[2025/04/03 Version 1.1.0]
+\RequirePackage{tikz, pgffor, ifthen, xparse, xfp, xstring}
\usetikzlibrary{cd,positioning,shapes.symbols,decorations.pathmorphing,arrows,calc,shapes.misc,intersections,decorations.pathreplacing,decorations.markings,arrows.meta, shapes.geometric}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newif\ifflowup
@@ -155,7 +166,6 @@
\qwboxprocess[braid]{#1}
}
-
\NewDocumentCommand{\qwbraidinv}{O{at=qworldat}}{
\qwboxprocess[braidinv]{#1}
}
@@ -1255,6 +1265,18 @@
}
}
\def\cardportsN{\theportNnum}
+ \def\maxN{0}
+ \ifthenelse{\equal{\PortsN}{}\or\equal{\PortsN}{{}}}
+ {
+ }{
+ \foreach \qwiii in\PortsN
+ {
+ \ifnum\maxN<\qwiii
+ \xdef\maxNtemp{\qwiii}
+ \fi
+ \xdef\maxN{\maxNtemp}
+ }
+ }
\ifthenelse{\equal{\numPortsN}{notentered}\or\equal{\numPortsN}{{notentered}}}
{
\ifthenelse{\equal{\PortsN}{}\or\equal{\PortsN}{{}}}
@@ -1261,17 +1283,21 @@
{
\pgfkeys{/qworld/box/.cd, n=1}
}{
- \xdef\maxN{0}
- \foreach \qwiii in\PortsN
- {
- \ifnum\maxN<\qwiii
- \xdef\maxNtemp{\qwiii}
- \fi
- \xdef\maxN{\maxNtemp}
- }
\pgfkeys{/qworld/box/.cd, n=\maxN}
}
}{}
+ \def\maxS{0}
+ \ifthenelse{\equal{\PortsS}{}\or\equal{\PortsS}{{}}}
+ {
+ }{
+ \foreach \qwiii in\PortsS
+ {
+ \ifnum\maxS<\qwiii
+ \xdef\maxStemp{\qwiii}
+ \fi
+ \xdef\maxS{\maxStemp}
+ }
+ }
\ifthenelse{\equal{\numPortsS}{notentered}\or\equal{\numPortsS}{{notentered}}}
{
\ifthenelse{\equal{\PortsS}{}\or\equal{\PortsS}{{}}}
@@ -1278,17 +1304,15 @@
{
\pgfkeys{/qworld/box/.cd, s=1}
}{
- \xdef\maxS{0}
- \foreach \qwiii in\PortsS
- {
- \ifnum\maxS<\qwiii
- \xdef\maxStemp{\qwiii}
- \fi
- \xdef\maxS{\maxStemp}
- }
\pgfkeys{/qworld/box/.cd, s=\maxS}
}
}{}
+ \ifnum\numPortsN<\maxN
+ \pgfkeys{/qworld/box/.cd, n=\maxN}
+ \fi
+ \ifnum\numPortsS<\maxS
+ \pgfkeys{/qworld/box/.cd, s=\maxS}
+ \fi
\def\qworldwireScorrection{0}
\def\qworldwireNcorrection{0}
\ifthenelse{\equal{\boxhorilen}{notentered}}
@@ -1424,7 +1448,7 @@
\def\yradius{\fpeval{0.5*\boxverlen-\qwfillrate}}
\def\radis{\fpeval{min(\xradius,\yradius)}}
\node[draw={\framecolor}, {\qwthickness}, fill={\boxcolor},
- circle, minimum width={\fpeval{2*\xradius}cm}, minimum height={\fpeval{2*\yradius}cm}
+ ellipse, minimum width={\fpeval{2*\xradius}cm}, minimum height={\fpeval{2*\yradius}cm}
\boxslantstyle, transform shape]
(SCALAR-\theqworldnumeros) at (C-\theqworldnumeros) {{\scalebox{\fpeval{min(\boxhorilen-\qwireinterval+2*\qwfillratex,\boxverlen-2*\qwfillrate)}}{\color{\morphismcolor}\boxname}}};
}{}
@@ -1598,8 +1622,15 @@
\ifx\boxname\empty
\draw[\qwthickness,draw=\framecolor,fill=\boxcolor] (C-\theqworldnumeros) circle [x radius={0.1},y radius={0.1}];
\else
- \draw[\qwthickness,draw=\framecolor,fill=\boxcolor] (C-\theqworldnumeros) circle [x radius={0.4},y radius={0.4}];
- \node[\boxslantstyle] at (C-\theqworldnumeros) [anchor=center]{{\scalebox{\fpeval{min(0.9*\boxhorilen,\boxverlen-2*\qwfillrate)}}{\color{\morphismcolor}\boxname}}};
+ % \draw[\qwthickness,draw=\framecolor,fill=\boxcolor] (C-\theqworldnumeros) circle [x radius={0.4},y radius={0.4}];
+ % \node[\boxslantstyle] at (C-\theqworldnumeros) [anchor=center]{{\scalebox{\fpeval{min(0.9*\boxhorilen,1-2*\qwfillrate)}}{\color{\morphismcolor}\boxname}}};
+ \def\xradius{\fpeval{0.4*\boxhorilen}}
+ \def\yradius{\fpeval{0.4*\boxverlen-\qwfillrate}}
+ \def\radis{\fpeval{min(\xradius,\yradius)}}
+ \node[draw={\framecolor}, {\qwthickness}, fill={\boxcolor},
+ ellipse, minimum width={\fpeval{\xradius}cm}, minimum height={\fpeval{\yradius}cm}
+ \boxslantstyle, transform shape]
+ (SCALAR-\theqworldnumeros) at (C-\theqworldnumeros) {{\scalebox{\fpeval{0.6*min(\boxhorilen-\qwireinterval+2*\qwfillratex,\boxverlen-2*\qwfillrate)}}{\color{\morphismcolor}\boxname}}};
\fi
}{}
\ifthenelse{\equal{\boxtype}{unitobject}}
@@ -1678,36 +1709,83 @@
\node[\codat] at (\codtype-\codnum-\codid) {\codlabel};
}
-\NewDocumentCommand{\qwI}{O{1} m m O{below}}
+\NewDocumentCommand{\qwI}{O{1} m O{below}}
{
- \node[#4] at (I-#1-#2) {#3};
+ \addtocounter{qworldnumeros}{-1}%
+ \qwforkeysanalyst{#3}
+ \node[below,\atinfo] at (I-#1-\qwforkeysid) {#2};%
+ \stepcounter{qworldnumeros}
}
-\NewDocumentCommand{\qws}{O{1} m m O{below}}
+\NewDocumentCommand{\qws}{O{1} m O{below}}
{
- \node[#4] at (s-#1-#2) {#3};
-}
+ \addtocounter{qworldnumeros}{-1}%
+ \qwforkeysanalyst{#3}
+ \node[below,\atinfo] at (s-#1-\qwforkeysid) {#2};%
+ \stepcounter{qworldnumeros}}
-\NewDocumentCommand{\qwS}{O{1} m m O{below}}
+\NewDocumentCommand{\qwS}{O{1} m O{below}}
{
- \node[#4] at (S-#1-#2) {#3};
-}
+ \addtocounter{qworldnumeros}{-1}%
+ \qwforkeysanalyst{#3}
+ \node[below,\atinfo] at (S-#1-\qwforkeysid) {#2};%
+ \stepcounter{qworldnumeros}}
-\NewDocumentCommand{\qwO}{O{1} m m O{above}}
+\NewDocumentCommand{\qwO}{O{1} m O{above}}
{
- \node[#4] at (O-#1-#2) {#3};
+ \addtocounter{qworldnumeros}{-1}%
+ \qwforkeysanalyst{#3}
+ \node[above,\atinfo] at (O-#1-\qwforkeysid) {#2};%
+ \stepcounter{qworldnumeros}
}
-\NewDocumentCommand{\qwn}{O{1} m m O{above}}
+\NewDocumentCommand{\qwn}{O{1} m O{above}}
{
- \node[#4] at (n-#1-#2) {#3};
+ \addtocounter{qworldnumeros}{-1}%
+ \qwforkeysanalyst{#3}
+ \node[above,\atinfo] at (n-#1-\qwforkeysid) {#2};%
+ \stepcounter{qworldnumeros}
}
-\NewDocumentCommand{\qwN}{O{1} m m O{above}}
+\NewDocumentCommand{\qwN}{O{1} m O{above}}
{
- \node[#4] at (N-#1-#2) {#3};
+ \addtocounter{qworldnumeros}{-1}%
+ \qwforkeysanalyst{#3}
+ \node[above,\atinfo] at (N-#1-\qwforkeysid) {#2};%
+ \stepcounter{qworldnumeros}
}
\newcommand{\qwunitobject}[1][at={qworldat}]{
\qwboxprocess[unitobject]{#1, P={}}
}
+\pgfkeys{
+ /qworld/forkeys/.cd,
+ id/.store in=\forkeysid,
+ id/.default=0
+}
+
+\newcommand{\qwforkeysanalyst}[1]
+{
+ \def\atinfo{}
+ \pgfkeys{
+ /qworld/forkeys/.cd,
+ id=\theqworldnumeros
+ }
+ \def\qwforkeysid{\forkeysid}
+ \foreach \kvornot in {#1}
+ {
+ \IfSubStr{\kvornot}{=}
+ {
+ \StrBehind{\kvornot}{id=}[\result]%
+ \pgfkeys{
+ /qworld/forkeys/.cd,id=\result
+ }
+ \xdef\qwforkeysid{\forkeysid}
+ }{
+ \xdef\atinfo{\kvornot}
+ }
+ }
+}
+\newcommand{\extractid}[1]{%
+ \StrBehind{#1}{id=}[\result]%
+}
Modified: trunk/Master/tlpkg/libexec/ctan2tds
===================================================================
--- trunk/Master/tlpkg/libexec/ctan2tds 2025-04-03 19:52:19 UTC (rev 74842)
+++ trunk/Master/tlpkg/libexec/ctan2tds 2025-04-03 19:52:38 UTC (rev 74843)
@@ -4175,7 +4175,6 @@
'pictex' => 'manual.pdf$', # no source
'piechartmp' => 'piechartmp.pdf', # no source
'ptptex' => '(overcite|cite|wrapfig).sty', # duplicated
- 'qworld' => 'qworld.pdf', # no source
'scientific-thesis-cover' => 'demo.pdf', # no source
'sectionbox' => 'example/lenna.*.jpg', # nonfree
'startex' => 'epsf.tex', # stale copy
More information about the tex-live-commits
mailing list.