Merge pull request #263 from FlynnFehre/monomorphism_function

Added ButterflyFaithfulMonomorphismIntoSimplicialSurface and AllButterflyFaithfulMonomorphismsIntoSimplicialSurface

doc/tikz-files/Image_Butterfly_Faithful_Monomorphism_Hexagon.tex

@@ -0,0 +1,319 @@
+\documentclass{standalone}
+
+\pagestyle{empty}
+% This document contains the TikZ-header for all our LaTeX-computations.
+% It especially contains all global graphic parameters.
+
+\usepackage{amsmath, amssymb, amsfonts} % Standard Math-stuff
+
+\usepackage{ifthen}
+
+\usepackage{tikz}
+\usetikzlibrary{calc}
+\usetikzlibrary{positioning}
+\usetikzlibrary{shapes}
+\usetikzlibrary{patterns}
+
+
+% Sometimes we want to implement different behaviour for the generated 
+% HTML-pictures (for example, shading is not supported in HTML).
+% For that we define a macro to check whether we run the code with
+% htlatex. The code comes from 
+% https://tex.stackexchange.com/questions/93852/what-is-the-correct-way-to-check-for-latex-pdflatex-and-html-in-the-same-latex
+\makeatletter
+\edef\texforht{TT\noexpand\fi
+  \@ifpackageloaded{tex4ht}
+    {\noexpand\iftrue}
+    {\noexpand\iffalse}}
+\makeatother
+
+
+% Define a text=none option for nodes that ignores the given text, from
+% https://tex.stackexchange.com/questions/59354/no-text-none-in-tikz
+\makeatletter
+\newif\iftikz@node@phantom
+\tikzset{
+  phantom/.is if=tikz@node@phantom,
+  text/.code=%
+    \edef\tikz@temp{#1}%
+    \ifx\tikz@temp\tikz@nonetext
+      \tikz@node@phantomtrue
+    \else
+      \tikz@node@phantomfalse
+      \let\tikz@textcolor\tikz@temp
+    \fi
+}
+\usepackage{etoolbox}
+\patchcmd\tikz@fig@continue{\tikz@node@transformations}{%
+  \iftikz@node@phantom
+    \setbox\pgfnodeparttextbox\hbox{}
+  \fi\tikz@node@transformations}{}{}
+\makeatother
+
+% Find the angle of a given line (within TikZ)
+\newcommand{\tikzAngleOfLine}{\tikz@AngleOfLine}
+\def\tikz@AngleOfLine(#1)(#2)#3{%
+  \pgfmathanglebetweenpoints{%
+    \pgfpointanchor{#1}{center}}{%
+    \pgfpointanchor{#2}{center}}
+  \pgfmathsetmacro{#3}{\pgfmathresult}%
+}
+
+% Now we define the global styles
+% The global styles are defined nestedly. You have to give your tikzpicture
+% the global options [vertexStyle, edgeStyle, faceStyle] to activate them.
+% 
+% You can disable labels by using the option nolabels, i.e. 
+% vertexStyle=nolabels to deactivate vertex labels.
+%
+% If you want to have a specific style for your picture, you can also use
+% this specific meta-style instead of the general style. For example if you
+% want to use double edges in one single picture - no matter the style of
+% the rest of the document - you can use edgeDouble instead of edgeStyle.
+%
+% To set the default style, modify the vertexStyle/.default entry.
+
+% Vertex styles
+\tikzset{ 
+    vertexNodePlain/.style = {fill=#1, shape=circle, inner sep=0pt, minimum size=2pt, text=none},
+    vertexNodePlain/.default=gray,
+    vertexPlain/labels/.style = {
+        vertexNode/.style={vertexNodePlain=##1},
+        vertexLabel/.style={gray}
+    },
+    vertexPlain/nolabels/.style = {
+        vertexNode/.style={vertexNodePlain=##1},
+        vertexLabel/.style={text=none}
+    },
+    vertexPlain/.style = vertexPlain/#1,
+    vertexPlain/.default=labels
+}
+\tikzset{
+    vertexNodeNormal/.style = {fill=#1, shape=circle, inner sep=0pt, minimum size=4pt, text=none},
+    vertexNodeNormal/.default = blue,
+    vertexNormal/labels/.style = {
+        vertexNode/.style={vertexNodeNormal=##1},
+        vertexLabel/.style={blue}
+    },
+    vertexNormal/nolabels/.style = {
+        vertexNode/.style={vertexNodeNormal=##1},
+        vertexLabel/.style={text=none}
+    },
+    vertexNormal/.style = vertexNormal/#1,
+    vertexNormal/.default=labels
+}
+\tikzset{
+    vertexNodeBallShading/pdf/.style = {ball color=#1},
+    vertexNodeBallShading/svg/.style = {fill=#1},
+    vertexNodeBallShading/.code = {% Conditional shading depending whether we want pdf or svg output
+        \if\texforht
+            \tikzset{vertexNodeBallShading/svg=#1!90!black}
+        \else
+            \tikzset{vertexNodeBallShading/pdf=#1}
+        \fi
+    },
+    vertexNodeBall/.style = {shape=circle, vertexNodeBallShading=#1, inner sep=2pt, outer sep=0pt, minimum size=3pt, font=\tiny},
+    vertexNodeBall/.default = white,
+    vertexBall/labels/.style = {
+        vertexNode/.style={vertexNodeBall=##1, text=black},
+        vertexLabel/.style={text=none}
+    },
+    vertexBall/nolabels/.style = {
+        vertexNode/.style={vertexNodeBall=##1, text=none},
+        vertexLabel/.style={text=none}
+    },
+    vertexBall/.style = vertexBall/#1,
+    vertexBall/.default=labels
+}
+\tikzset{ 
+    vertexStyle/.style={vertexNormal=#1},
+    vertexStyle/.default = labels
+}
+
+
+% 1) optional: colour of vertex
+% 2) position of the vertex
+% 3) relative position of the node
+% 4) name of the vertex
+\newcommand{\vertexLabelR}[4][]{
+    \ifthenelse{ \equal{#1}{} }
+        { \node[vertexNode] (#2 name) at (#2) {#4}; }
+        { \node[vertexNode=#1] (#2 name) at (#2) {#4}; }
+    \node[vertexLabel, #3] at (#2) {#4};
+}
+% 1) optional: colour of vertex
+% 2) position of the vertex
+% 3) absolute position of the node
+% 4) name of the vertex
+\newcommand{\vertexLabelA}[4][]{
+    \ifthenelse{ \equal{#1}{} }
+        { \node[vertexNode] (#2 name) at (#2) {#4}; }
+        { \node[vertexNode=#1] (#2 name) at (#2) {#4}; }
+    \node[vertexLabel] at (#3) {#4};
+}
+
+
+% Edge styles
+% If you have trouble with the double-lines overlapping, this might (?) help:
+% https://tex.stackexchange.com/questions/288159/closing-the-ends-of-double-line-in-tikz
+\newcommand{\edgeLabelColor}{blue!20!white}
+\tikzset{
+    edgeLineNone/.style = {draw=none},
+    edgeLineNone/.default=black,
+    edgeNone/labels/.style = {
+        edge/.style = {edgeLineNone=##1},
+        edgeLabel/.style = {fill=\edgeLabelColor,font=\small}
+    },
+    edgeNone/nolabels/.style = {
+        edge/.style = {edgeLineNone=##1},
+        edgeLabel/.style = {text=none}
+    },
+    edgeNone/.style = edgeNone/#1,
+    edgeNone/.default = labels
+}
+\tikzset{
+    edgeLinePlain/.style={line join=round, draw=#1},
+    edgeLinePlain/.default=black,
+    edgePlain/labels/.style = {
+        edge/.style={edgeLinePlain=##1},
+        edgeLabel/.style={fill=\edgeLabelColor,font=\small}
+    },
+    edgePlain/nolabels/.style = {
+        edge/.style={edgeLinePlain=##1},
+        edgeLabel/.style={text=none}
+    },
+    edgePlain/.style = edgePlain/#1,
+    edgePlain/.default = labels
+}
+\tikzset{
+    edgeLineDouble/.style = {very thin, double=#1, double distance=.8pt, line join=round},
+    edgeLineDouble/.default=gray!90!white,
+    edgeDouble/labels/.style = {
+        edge/.style = {edgeLineDouble=##1},
+        edgeLabel/.style = {fill=\edgeLabelColor,font=\small}
+    },
+    edgeDouble/nolabels/.style = {
+        edge/.style = {edgeLineDouble=##1},
+        edgeLabel/.style = {text=none}
+    },
+    edgeDouble/.style = edgeDouble/#1,
+    edgeDouble/.default = labels
+}
+\tikzset{
+    edgeStyle/.style = {edgePlain=#1},
+    edgeStyle/.default = labels
+}
+
+% Face styles
+% Here we have an exception - the style face is always defined.
+% 
+\newcommand{\faceColorY}{yellow!60!white}   % yellow
+\newcommand{\faceColorB}{blue!60!white}     % blue
+\newcommand{\faceColorC}{cyan!60}           % cyan
+\newcommand{\faceColorR}{red!60!white}      % red
+\newcommand{\faceColorG}{green!60!white}    % green
+\newcommand{\faceColorO}{orange!50!yellow!70!white} % orange
+
+% define default face colour (and default swap colour)
+\newcommand{\faceColor}{\faceColorY}
+\newcommand{\faceColorSwap}{\faceColorC}
+
+% define secondary default colours (to use in a single section)
+\newcommand{\faceColorFirst}{green!40!white}
+\newcommand{\faceColorSecond}{gray!15!white}
+\newcommand{\faceColorThird}{red!17!white}
+\newcommand{\faceColorFourth}{olive!20!white}
+
+\tikzset{
+    face/.style = {fill=#1},
+    face/.default = \faceColor,
+    faceY/.style = {face=\faceColorY},
+    faceB/.style = {face=\faceColorB},
+    faceC/.style = {face=\faceColorC},
+    faceR/.style = {face=\faceColorR},
+    faceG/.style = {face=\faceColorG},
+    faceO/.style = {face=\faceColorO}
+}
+\tikzset{
+    faceStyle/labels/.style = {
+        faceLabel/.style = {}
+    },
+    faceStyle/nolabels/.style = {
+        faceLabel/.style = {text=none}
+    },
+    faceStyle/.style = faceStyle/#1,
+    faceStyle/.default = labels
+}
+\tikzset{ face/.style={fill=#1} }
+\tikzset{ faceSwap/.code=
+    \ifdefined\swapColors
+        \tikzset{face=\faceColorSwap}
+    \else
+        \tikzset{face=\faceColor}
+    \fi
+}
+
+
+
+\usepackage{hyperref}
+
+
+\begin{document}
+
+
+
+\begin{tikzpicture}[vertexStyle, edgeStyle, faceStyle]
+
+    \newcommand{\setupCoord}{
+        \coordinate (Z) at (0,0);
+        \foreach \i in {1,...,6}{
+            \coordinate (A\i) at (180-60*\i:2.5);
+        }
+    }
+
+
+    \begin{scope}[shift={(3.5,0)}]
+        \setupCoord
+
+        \draw[edge,face]
+            (Z) -- (A1) -- node[edgeLabel]{1} (A2) -- cycle
+            (Z) -- node[edgeLabel]{9} (A2) -- node[edgeLabel]{2} (A3) -- cycle
+            (Z) -- node[edgeLabel]{10} (A3) -- node[edgeLabel]{3} (A4) -- cycle
+            (Z) -- node[edgeLabel]{11} (A4) -- node[edgeLabel]{4} (A5) -- cycle
+            (Z) -- node[edgeLabel]{12} (A5) -- node[edgeLabel]{5} (A6) -- cycle
+            (Z) -- node[edgeLabel]{13} (A6) -- node[edgeLabel]{6} (A1) -- node[edgeLabel]{8} (Z);
+
+        \foreach \p/\r/\n in {1/above left/1, 2/above right/2, 3/right/3, 4/below right/4, 5/below left/5, 6/left/6}{
+            \vertexLabelR{A\p}{\r}{\n}
+        }
+        \vertexLabelR{Z}{above}{8}
+
+        \foreach \i/\j/\n in {1/2/I, 2/3/II, 3/4/III, 4/5/IV, 5/6/V, 6/1/VI}{
+            \node[faceLabel] at (barycentric cs:A\i=1,A\j=1,Z=1) {$\n$};
+        }
+    \end{scope}
+
+
+    \node at (0,0) {$\rightarrow$};
+
+
+    \begin{scope}[shift={(-3.5,0)}]
+        \setupCoord
+
+        \draw[edge,face]
+            (Z) -- node[edgeLabel]{5} (A1) -- node[edgeLabel]{1} (A2) -- cycle
+            (Z) -- node[edgeLabel]{6} (A2) -- node[edgeLabel]{2} (A3) -- cycle
+            (Z) -- node[edgeLabel]{7} (A3) -- node[edgeLabel]{3} (A4) -- node[edgeLabel]{4} (Z);
+
+        \foreach \p/\r/\n in {1/above left/1, 2/above right/2, 3/right/3, 4/below right/4}{
+            \vertexLabelR{A\p}{\r}{\n}
+        }
+        \vertexLabelR{Z}{below left}{5}
+
+        \foreach \i/\j/\n in {1/2/I, 2/3/II, 3/4/III}{
+            \node[faceLabel] at (barycentric cs:A\i=1,A\j=1,Z=1) {$\n$};
+        }
+    \end{scope}
+\end{tikzpicture}
+
+\end{document}