135 on $C^2$ (Figure \ref{fzo1}).

   135 on $C^2$ (Figure \ref{fzo1}).

   136 Isotopy invariance implies that this is associative.

   136 Isotopy invariance implies that this is associative.

   137 We will define a ``horizontal" composition later.

   137 We will define a ``horizontal" composition later.

   138 

   138 

   139 \begin{figure}[t]

   139 \begin{figure}[t]

   141 \begin{tikzpicture}

   141 \begin{tikzpicture}

   142 

   142 

   143 \newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}}

   143 \newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}}

   144 \newcommand{\nsep}{1.8}

   144 \newcommand{\nsep}{1.8}

   145 

   145 

   181 };

   181 };

   182 

   182 

   183 \draw[->, thick, blue!50!green] (A) -- node[black, above] {$\cong$} (B);

   183 \draw[->, thick, blue!50!green] (A) -- node[black, above] {$\cong$} (B);

   184 

   184 

   185 \end{tikzpicture}

   185 \end{tikzpicture}

   187 \caption{Vertical composition of 2-morphisms}

   186 \caption{Vertical composition of 2-morphisms}

   188 \label{fzo1}

   187 \label{fzo1}

   189 \end{figure}

   188 \end{figure}

   190 

   189 

   191 Given $a\in C^1$, define $\id_a = a\times I \in C^2$ (pinched boundary).

   190 Given $a\in C^1$, define $\id_a = a\times I \in C^2$ (pinched boundary).

   202 

   201 

   203 Let $a: y\to x$ be a 1-morphism.

   202 Let $a: y\to x$ be a 1-morphism.

   204 Define 2-morphsims $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$

   203 Define 2-morphsims $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$

   205 as shown in Figure \ref{fzo2}.

   204 as shown in Figure \ref{fzo2}.

   206 \begin{figure}[t]

   205 \begin{figure}[t]

   208 \begin{tikzpicture}

   207 \begin{tikzpicture}

   209 \newcommand{\rr}{6}

   208 \newcommand{\rr}{6}

   210 \newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}}

   209 \newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}}

   211 \newcommand{\namedvertex}[1]{node[circle,fill=black,inner sep=1pt] (#1) {}}

   210 \newcommand{\namedvertex}[1]{node[circle,fill=black,inner sep=1pt] (#1) {}}

   212 

   211 

   272 };

   271 };

   273 

   272 

   274 \draw[->] (A) -- (B);

   273 \draw[->] (A) -- (B);

   275 \draw[->] (A) -- (C);

   274 \draw[->] (A) -- (C);

   276 \end{tikzpicture}

   275 \end{tikzpicture}

   278 \caption{Producing weak identities from half pinched products}

   276 \caption{Producing weak identities from half pinched products}

   279 \label{fzo2}

   277 \label{fzo2}

   280 \end{figure}

   278 \end{figure}

   281 As suggested by the figure, these are two different reparameterizations

   279 As suggested by the figure, these are two different reparameterizations

   282 of a half-pinched version of $a\times I$.

   280 of a half-pinched version of $a\times I$.

   283 We must show that the two compositions of these two maps give the identity 2-morphisms

   281 We must show that the two compositions of these two maps give the identity 2-morphisms

   284 on $a$ and $a\bullet \id_x$, as defined above.

   282 on $a$ and $a\bullet \id_x$, as defined above.

   285 Figure \ref{fzo3} shows one case.

   283 Figure \ref{fzo3} shows one case.

   286 \begin{figure}[t]

   284 \begin{figure}[t]

   288 \begin{tikzpicture}

   286 \begin{tikzpicture}

   289 

   287 

   290 \newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}}

   288 \newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}}

   291 \newcommand{\nsep}{1.8}

   289 \newcommand{\nsep}{1.8}

   292 

   290 

   398 

   396 

   399 \draw[->, thick, blue!50!green] (A) -- (B);

   397 \draw[->, thick, blue!50!green] (A) -- (B);

   400 \draw[->, thick, blue!50!green] (B) -- node[black, above] {$=$} (C);

   398 \draw[->, thick, blue!50!green] (B) -- node[black, above] {$=$} (C);

   401 

   399 

   402 \end{tikzpicture}

   400 \end{tikzpicture}

   404 \caption{Composition of weak identities, 1}

   401 \caption{Composition of weak identities, 1}

   405 \label{fzo3}

   402 \label{fzo3}

   406 \end{figure}

   403 \end{figure}

   407 In the first step we have inserted a copy of $(x\times I)\times I$.

   404 In the first step we have inserted a copy of $(x\times I)\times I$.

   408 Figure \ref{fzo4} shows the other case.

   405 Figure \ref{fzo4} shows the other case.

   409 \begin{figure}[t]

   406 \begin{figure}[t]

   411 \begin{tikzpicture}

   408 \begin{tikzpicture}

   412 

   409 

   413 \newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}}

   410 \newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}}

   414 \newcommand{\nsep}{1.8}

   411 \newcommand{\nsep}{1.8}

   415 

   412 

   500 

   497 

   501 \draw[->, thick, blue!50!green] (A) -- (B);

   498 \draw[->, thick, blue!50!green] (A) -- (B);

   502 \draw[->, thick, blue!50!green] ($(B) + (1,0)$) -- node[black, above] {$=$} (C);

   499 \draw[->, thick, blue!50!green] ($(B) + (1,0)$) -- node[black, above] {$=$} (C);

   503 

   500 

   504 \end{tikzpicture}

   501 \end{tikzpicture}

   506 \caption{Composition of weak identities, 2}

   502 \caption{Composition of weak identities, 2}

   507 \label{fzo4}

   503 \label{fzo4}

   508 \end{figure}

   504 \end{figure}

   509 We identify a product region and remove it.

   505 We identify a product region and remove it.

   510 

   506