# HG changeset patch # User Scott Morrison # Date 1323327654 28800 # Node ID d73a88d7849860fb432836bfac666eb714bce897 # Parent d85867a9954592dad780e82a5ef14b2f42236749 chopping up hand-drawn figures, and getting all the figure environments and references right diff -r d85867a99545 -r d73a88d78498 diagrams/triangle/triangle.pdf Binary file diagrams/triangle/triangle.pdf has changed diff -r d85867a99545 -r d73a88d78498 diagrams/triangle/triangle2.pdf Binary file diagrams/triangle/triangle2.pdf has changed diff -r d85867a99545 -r d73a88d78498 diagrams/triangle/triangle3b.pdf Binary file diagrams/triangle/triangle3b.pdf has changed diff -r d85867a99545 -r d73a88d78498 diagrams/triangle/triangle3c.pdf Binary file diagrams/triangle/triangle3c.pdf has changed diff -r d85867a99545 -r d73a88d78498 diagrams/triangle/triangle4a.pdf Binary file diagrams/triangle/triangle4a.pdf has changed diff -r d85867a99545 -r d73a88d78498 diagrams/triangle/triangle4b.pdf Binary file diagrams/triangle/triangle4b.pdf has changed diff -r d85867a99545 -r d73a88d78498 diagrams/triangle/triangle4c.pdf Binary file diagrams/triangle/triangle4c.pdf has changed diff -r d85867a99545 -r d73a88d78498 diagrams/triangle/triangle4d.pdf Binary file diagrams/triangle/triangle4d.pdf has changed diff -r d85867a99545 -r d73a88d78498 diagrams/triangle/triangle4e.pdf Binary file diagrams/triangle/triangle4e.pdf has changed diff -r d85867a99545 -r d73a88d78498 diagrams/triangle/triangle4f.pdf Binary file diagrams/triangle/triangle4f.pdf has changed diff -r d85867a99545 -r d73a88d78498 diagrams/triangle/triangle5.pdf Binary file diagrams/triangle/triangle5.pdf has changed diff -r d85867a99545 -r d73a88d78498 text/appendixes/comparing_defs.tex --- a/text/appendixes/comparing_defs.tex Wed Dec 07 18:40:50 2011 -0800 +++ b/text/appendixes/comparing_defs.tex Wed Dec 07 23:00:54 2011 -0800 @@ -527,8 +527,7 @@ Given 2-morphisms $f$ and $g$, we define the horizontal composition $f *_h g$ to be any of the four equal 2-morphisms in Figure \ref{fzo5}. -\nn{add three remaining cases of triangle.pdf 3.b to fzo5} -Figure \nn{triangle 3.c, but not necessarily crooked} illustrates part of the proof that these four 2-morphisms are equal. +Figure \ref{fig:horizontal-compositions-equal}illustrates part of the proof that these four 2-morphisms are equal. Similar arguments show that horizontal composition is associative. \begin{figure}[t] \begin{equation*} @@ -572,44 +571,81 @@ \draw[->, thick, blue!75!yellow] (1.5,.78) node[black, above] {$(b\cdot c)\times I$} -- (2.5,0); \end{tikzpicture}} \end{equation*} +\begin{equation*} +\mathfig{0.6}{triangle/triangle3b} +\end{equation*} \caption{Horizontal composition of 2-morphisms} \label{fzo5} \end{figure} +\begin{figure}[t] +$$ +\mathfig{0.6}{triangle/triangle3c} +$$ +\caption{Part of the proof that the four different horizontal compositions of 2-morphisms are equal.} +\label{fig:horizontal-compositions-equal} +\end{figure} Given 1-morphisms $a$, $b$ and $c$ of $D$, we define the associator from $(a\bullet b)\bullet c$ to $a\bullet(b\bullet c)$ as in Figure \nn{like triangle 4.a, but more general; use three colors as in that fig}. This is just a reparameterization of the pinched product $(a\bullet b\bullet c)\times I$ of $\cC$. +\begin{figure}[t] +$$ +\mathfig{0.4}{triangle/triangle4a} +$$ +\nn{remember to change the labels} +\caption{An associator.} +\label{fig:associator} +\end{figure} Let $x,y,z$ be objects of $D$ and let $a:x\to y$ and $b:y\to z$ be 1-morphisms of $D$. We have already defined above structure maps $u:a\bullet \id_y\to a$ and $v:\id_y\bullet b\to b$, as well as an associator $\alpha: (a\bullet \id_y)\bullet b\to a\bullet(\id_y\bullet b)$, as shown in -Figure \nn{triangle 4.b, 4.c and 4.a; note change from ``assoc" to ``$\alpha$"}. -(See also Figures \ref{fzo2} and \nn{previous associator fig}.) +Figure \ref{fig:ingredients-triangle-axiom}. +(See also Figures \ref{fzo2} and \ref{fig:associator}.) We now show that $D$ satisfies the triangle axiom, which states that $u\bullet\id_b$ is equal to the composition of $\alpha$ and $\id_a\bullet v$. (Both are 2-morphisms from $(a\bullet \id_y)\bullet b$ to $a\bullet b$.) +\begin{figure}[t] +\begin{align*} +\mathfig{0.4}{triangle/triangle4a} \\ +\mathfig{0.4}{triangle/triangle4b} \\ +\mathfig{0.4}{triangle/triangle4c} +\end{align*} +\nn{remember to change `assoc' to $\alpha$} +\caption{Ingredients for the triangle axiom.} +\label{fig:ingredients-triangle-axiom} +\end{figure} -The horizontal compositions $u\bullet\id_b$ and $\id_a\bullet v$ are shown in Figure \nn{triangle 4.d and 4.e} +The horizontal compositions $u\bullet\id_b$ and $\id_a\bullet v$ are shown in Figure \ref{fig:horizontal-composition} (see also Figure \ref{fzo5}). -The vertical composition of $\alpha$ and $\id_a\bullet v$ is shown in Figure \nn{triangle 4.f}. -Figure \nn{triangle 5} shows that we can add a collar to $u\bullet\id_b$ so that the result differs from -Figure \nn{ref to 4.f above} by an isotopy rel boundary. +The vertical composition of $\alpha$ and $\id_a\bullet v$ is shown in Figure \ref{fig:vertical-composition}. +Figure \ref{fig:adding-a-collar} shows that we can add a collar to $u\bullet\id_b$ so that the result differs from +Figure \ref{fig:vertical-composition} by an isotopy rel boundary. Note that here we have used in an essential way the associativity of product morphisms (Axiom \ref{axiom:product}.3) as well as compatibility of product morphisms with fiber-preserving maps (Axiom \ref{axiom:product}.1). - - - - - - -%\nn{need to find a list of axioms for pivotal 2-cats to check} - - - - - - +\begin{figure}[t] +\begin{align*} +\mathfig{0.4}{triangle/triangle4d} +\mathfig{0.4}{triangle/triangle4e} +\end{align*} +\caption{Horizontal compositions in the triangle axiom.} +\label{fig:horizontal-composition} +\end{figure} +\begin{figure}[t] +\begin{align*} +\mathfig{0.4}{triangle/triangle4f} +\end{align*} +\caption{Vertical composition in the triangle axiom.} +\label{fig:vertical-composition} +\end{figure} +\begin{figure}[t] +\begin{align*} +\mathfig{0.4}{triangle/triangle5} +\end{align*} +\caption{Adding a collar in the proof of the triangle axiom.} +\label{fig:adding-a-collar} +\end{figure} diff -r d85867a99545 -r d73a88d78498 triangle.pdf Binary file triangle.pdf has changed