equal
deleted
inserted
replaced
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] |
140 \begin{center} |
140 \centering |
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} |
186 \end{center} |
|
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] |
207 \begin{center} |
206 \centering |
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} |
277 \end{center} |
|
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] |
287 \begin{center} |
285 \centering |
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} |
403 \end{center} |
|
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] |
410 \begin{center} |
407 \centering |
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} |
505 \end{center} |
|
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 |