equal
deleted
inserted
replaced
635 \end{figure} |
635 \end{figure} |
636 We notice that a certain subset of the disk is a product region and remove it. |
636 We notice that a certain subset of the disk is a product region and remove it. |
637 |
637 |
638 Given 2-morphisms $f$ and $g$, we define the horizontal composition $f *_h g$ to be any of the four |
638 Given 2-morphisms $f$ and $g$, we define the horizontal composition $f *_h g$ to be any of the four |
639 equal 2-morphisms in Figure \ref{fzo5}. |
639 equal 2-morphisms in Figure \ref{fzo5}. |
640 Figure \ref{fig:horizontal-compositions-equal}illustrates part of the proof that these four 2-morphisms are equal. |
640 Figure \ref{fig:horizontal-compositions-equal} illustrates part of the proof that these four 2-morphisms are equal. |
641 Similar arguments show that horizontal composition is associative. |
641 Similar arguments show that horizontal composition is associative. |
642 \begin{figure}[t] |
642 \begin{figure}[t] |
643 \begin{align*} |
643 \begin{align*} |
644 \raisebox{-.9cm}{ |
644 \raisebox{-.9cm}{ |
645 \begin{tikzpicture} |
645 \begin{tikzpicture} |
924 structure maps $u:a\bullet \id_y\to a$ and $v:\id_y\bullet b\to b$, as well as an associator |
924 structure maps $u:a\bullet \id_y\to a$ and $v:\id_y\bullet b\to b$, as well as an associator |
925 $\alpha: (a\bullet \id_y)\bullet b\to a\bullet(\id_y\bullet b)$, as shown in |
925 $\alpha: (a\bullet \id_y)\bullet b\to a\bullet(\id_y\bullet b)$, as shown in |
926 Figure \ref{fig:ingredients-triangle-axiom}. |
926 Figure \ref{fig:ingredients-triangle-axiom}. |
927 (See also Figures \ref{fzo2} and \ref{fig:associator}.) |
927 (See also Figures \ref{fzo2} and \ref{fig:associator}.) |
928 We now show that $D$ satisfies the triangle axiom, which states that $u\bullet\id_b$ |
928 We now show that $D$ satisfies the triangle axiom, which states that $u\bullet\id_b$ |
929 is equal to the composition of $\alpha$ and $\id_a\bullet v$. |
929 is equal to the vertical composition of $\alpha$ and $\id_a\bullet v$. |
930 (Both are 2-morphisms from $(a\bullet \id_y)\bullet b$ to $a\bullet b$.) |
930 (Both are 2-morphisms from $(a\bullet \id_y)\bullet b$ to $a\bullet b$.) |
931 \begin{figure}[t] |
931 \begin{figure}[t] |
932 %\begin{align*} |
932 %\begin{align*} |
933 %\mathfig{0.4}{triangle/triangle4a} \\ |
933 %\mathfig{0.4}{triangle/triangle4a} \\ |
934 %\mathfig{0.4}{triangle/triangle4b} \\ |
934 %\mathfig{0.4}{triangle/triangle4b} \\ |
1014 \end{align*} |
1014 \end{align*} |
1015 \caption{Ingredients for the triangle axiom.} |
1015 \caption{Ingredients for the triangle axiom.} |
1016 \label{fig:ingredients-triangle-axiom} |
1016 \label{fig:ingredients-triangle-axiom} |
1017 \end{figure} |
1017 \end{figure} |
1018 |
1018 |
1019 The horizontal compositions $u\bullet\id_b$ and $\id_a\bullet v$ are shown in Figure \ref{fig:horizontal-composition} |
1019 The horizontal compositions $u *_h \id_b$ and $\id_a *_h v$ are shown in Figure \ref{fig:horizontal-composition} |
1020 (see also Figure \ref{fzo5}). |
1020 (see also Figure \ref{fzo5}). |
1021 The vertical composition of $\alpha$ and $\id_a\bullet v$ is shown in Figure \ref{fig:vertical-composition}. |
1021 The vertical composition of $\alpha$ and $\id_a *_h v$ is shown in Figure \ref{fig:vertical-composition}. |
1022 Figure \ref{fig:adding-a-collar} shows that we can add a collar to $u\bullet\id_b$ so that the result differs from |
1022 Figure \ref{fig:adding-a-collar} shows that we can add a collar to $u *_h \id_b$ so that the result differs from |
1023 Figure \ref{fig:vertical-composition} by an isotopy rel boundary. |
1023 Figure \ref{fig:vertical-composition} by an isotopy rel boundary. |
1024 Note that here we have used in an essential way the associativity of product morphisms (Axiom \ref{axiom:product}.3) |
1024 Note that here we have used in an essential way the associativity of product morphisms (Axiom \ref{axiom:product}.3) |
1025 as well as compatibility of product morphisms with fiber-preserving maps (Axiom \ref{axiom:product}.1). |
1025 as well as compatibility of product morphisms with fiber-preserving maps (Axiom \ref{axiom:product}.1). |
1026 \begin{figure}[t] |
1026 \begin{figure}[t] |
1027 %\begin{align*} |
1027 %\begin{align*} |