Commit ba0a363b authored by Niles Johnson's avatar Niles Johnson

new demo

parent d1926ff7
\documentclass[10pt]{article}
\usepackage{fouriernc} % Fourier fonts instead of Computer Modern
\usepackage[margin=1cm]{geometry}
\usepackage{multicol}
\pagestyle{empty}
%%
%% commutative diagrams with tikz
%%
%% based on tikz-cd, but improved so that we
%% don't have to use the matrix format
%%
\usepackage{tikz, tikz-cd}
\usetikzlibrary{calc}
\usetikzlibrary{decorations.markings,decorations.pathmorphing}
\usetikzlibrary{arrows}
\usetikzlibrary{matrix}
%\tikzset{/tikz/commutative diagrams/arrow style=math font} % matching arrowheads
\tikzset{/tikz/commutative diagrams/arrow style=tikz,>=stealth} %almost matching arrowheads; look better in pdf
%\tikzset{mm/.style={execute at begin node=$\displaystyle, execute at end node=$}}
%\tikzset{tikzob/.style={commutative diagrams/every diagram, every cell}}
%\tikzset{tikzar/.style={commutative diagrams/.cd, every arrow, every label, font={\small}}}
\tikzset{0cell/.style={commutative diagrams/every diagram,
every cell,
execute at begin node=$\displaystyle, execute at end node=$,
font={\small},
}
}
\tikzset{1cell/.style={commutative diagrams/.cd,
every arrow,
every label,
execute at begin node=$\displaystyle, execute at end node=$,
}
}
\tikzset{2cell/.style={commutative diagrams/every diagram,
every cell,
execute at begin node=$\displaystyle, execute at end node=$,
}
}
\tikzset{equal/.style={commutative diagrams/equal}}
\tikzset{tikzsquiggle/.style={decorate, decoration={
snake,
segment length=8pt,
amplitude=.9pt,post=lineto,
post length=2pt}}}
\tikzset{cross line/.style={preaction={draw=white, -, line width=6pt}}}
\tikzset{
between/.style args={#1 and #2 at #3}{
at = ($(#1)!#3!(#2)$)
},
}
%%
%% macros for the demo
%%
\newcommand{\ol}[1]{\overline{#1}}
\newcommand{\wt}[1]{\widetilde{#1}}
\newcommand{\dar}{\downarrow}
\newcommand{\cn}{\colon}
\begin{document}
Given a 2-cell $\gamma\cn u \to u'$ in $B$, we have 1-cells in $F\dar y$
given by $(\wt{u},\theta_{\wt{u}})$ and $(\wt{u'},\theta_{\wt{u'}})$:
\[
\begin{tikzpicture}[x=15mm,y=15mm,baseline={(x).base}]
\draw[0cell] %objects
(0,0) node (y) {y}
(120:2) node (a) {F(\wt{x})}
(120:1) node (x) {x}
(60:2) node (b) {F(\wt{y})}
;
\path[1cell] %arrows
(a) edge[swap] node {f_{\wt{x}}} (x)
(x) edge[swap] node {u} (y)
(b) edge node {f_{\wt{y}}} (y)
(a) edge node (T) {F(\wt{u})} (b)
;
\draw[2cell] %2cells
node[between=y and T at .65, rotate=30, font=\Large] (A) {\Rightarrow}
(A) node[below right] {\theta_{\wt{u}}}
;
\end{tikzpicture}
\qquad \mathrm{ and } \qquad
\begin{tikzpicture}[x=15mm,y=15mm,baseline={(x).base}]
\draw[0cell] %objects
(0,0) node (y) {y.}
(120:2) node (a) {F(\wt{x})}
(120:1) node (x) {x}
(60:2) node (b) {F(\wt{y})}
;
\path[1cell] %arrows
(a) edge[swap] node {f_{\wt{x}}} (x)
(x) edge[swap] node {u'} (y)
(b) edge node {f_{\wt{y}}} (y)
(a) edge node (T) {F(\wt{u'})} (b)
;
\draw[2cell] %2cells
node[between=y and T at .65, rotate=30, font=\Large] (A) {\Rightarrow}
(A) node[below right] {\theta_{\wt{u'}}}
;
\end{tikzpicture}
\]
Moreover, we have a third 1-cell in $(1_{\wt{x}}, \ol{\gamma})$ where
$\ol{\gamma}$ denotes the 2-cell obtained from the pasting below:
\[
\begin{tikzpicture}[x=20mm,y=25mm]
\draw[0cell] %objects
(0,-.25) node (y) {y.}
(120:2) node (a) {F(\wt{x})}
(90:.6) node (x) {x}
(60:2) node (b) {F(\wt{x})}
% (60:1) node (x2) {x}
;
\path[1cell] %arrows
(a) edge[swap] node {f_{\wt{x}}} (x)
(b) edge node {f_{\wt{x}}} (x)
(x) edge[swap, bend right] node {u} (y)
(x) edge[bend left] node {u'} (y)
% (x) edge[swap] node (1) {1_x} (x2)
(a) edge[bend left] node (T) {F(1_{\wt{x}})} (b)
(a) edge[swap, bend right] node (B) {1_{F(\wt{x})}} (b)
;
\draw[2cell] %2cells
node[between=x and B at .6, rotate=30, font=\Large] (A) {\Rightarrow}
(A) ++(.02,0) node[below] {\ell}
node[between=y and x at .45, font=\Large] (B) {\Rightarrow}
(B) node[above] {\gamma}
node[between=a and b at .5, rotate=90, font=\Large] (C) {\Rightarrow}
(C) node[right] {F_{\wt{x}}}
;
\end{tikzpicture}
\]
These three 1-cells form a triangle in $F \dar y$ and therefore
there is a unique 2-cell filling the diagram below in $F \dar y$.
\[
\begin{tikzpicture}[x=15mm,y=15mm]
\draw[0cell] %objects
(0,0) node (y) {(\wt{y},f_{\wt{y}})}
(90:2) node (x) {(\wt{x}, u f_{\wt{x}})}
(30:2) node (x') {(\wt{x}, u' f_{\wt{x}})}
;
\path[1cell] %arrows
(x) edge[swap] node (L) {(\wt{u},\theta_{\wt{u}})} (y)
(x) edge node {(1_{\wt{x}}, \ol{\gamma})} (x')
(x') edge node {(\wt{u'},\theta_{\wt{u'}})} (y)
;
\draw[2cell] %2cells
node[between=L and x' at .5, font=\Large,shift={(0,-.1)}] (A) {\Rightarrow}
(A) node[above] {(\wt{\gamma})}
;
\end{tikzpicture}
\]
Thus we have the following equality in $B$.
\[
\begin{tikzpicture}[x=25mm,y=25mm,baseline={(eq).base}]
\def\w{1.5} % horizontal shift between pastings
\def\h{-1.5} % vertical shift of pastings
\def\m{.5} % margin around pastings
\draw[font=\Large] (\w/2+\m,0) node (eq) {=};
\newcommand{\boundary}{
\draw[0cell] %objects
(0,-.25) node (y) {y}
(120:2) node (a) {F(\wt{x})}
(135:.6) node (x) {x}
(60:2) node (b) {F(\wt{y})}
;
\path[1cell] %arrows
(a) edge[swap] node {f_{\wt{x}}} (x)
(b) edge node (fy) {f_{\wt{y}}} (y)
(x) edge[swap] node {u} (y)
% (x) edge[swap] node (1) {1_x} (x2)
(a) edge[bend left] node (T) {F(\wt{u'} 1_{\wt{x}})} (b)
;
}
\begin{scope}[shift={(0,\h/2)}]
\boundary
\path[1cell] %arrows
(a) edge[swap, bend right] node (B) {F(\wt{u})} (b)
;
\draw[2cell] %objects
node[between=a and b at .5, rotate=90, font=\Large] (C) {\Rightarrow}
(C) node[right] {F(\wt{\gamma})}
;
\end{scope}
\begin{scope}[shift={(\w+\m+\m,\h/2)}]
\boundary
\draw[0cell] %objects
(90:1.25) node (a2) {F(\wt{x})}
;
\path[1cell] %arrows
% (x) edge[bend left] node {u'} (y)
(a) edge[swap, bend left] node {F(1_{\wt{x}})} (a2)
(a2) edge[swap, bend left] node {F(\wt{u'})} (b)
(a2) edge node {} (y)
;
\draw[2cell] %objects
node[between=x and a2 at .5, shift={(-.15,.175)}, rotate=60, font=\Large] (AA) {\Rightarrow}
(AA) node[below, shift={(.05,-.05)}] {\ol{\gamma}}
node[between=fy and a2 at .5, shift={(.05,.1)}, rotate=60, font=\Large] (BB) {\Rightarrow}
(BB) node[below, shift={(.05,-.05)}] {\theta_{\wt{u'}}}
node[between=a2 and T at .55, shift={(-.075,0)}, rotate=90, font=\Large] (CC) {\Rightarrow}
(CC) node[right] {F_{\wt{u'},1_{\wt{x}}}}
;
\end{scope}
\end{tikzpicture}
\]
\end{document}
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