13 One must then show that the axioms of \S\ref{ss:n-cat-def} imply the traditional $n$-category axioms. |
13 One must then show that the axioms of \S\ref{ss:n-cat-def} imply the traditional $n$-category axioms. |
14 One should also show that composing the two arrows (between traditional and topological $n$-categories) |
14 One should also show that composing the two arrows (between traditional and topological $n$-categories) |
15 yields the appropriate sort of equivalence on each side. |
15 yields the appropriate sort of equivalence on each side. |
16 Since we haven't given a definition for functors between topological $n$-categories |
16 Since we haven't given a definition for functors between topological $n$-categories |
17 (the paper is already too long!), we do not pursue this here. |
17 (the paper is already too long!), we do not pursue this here. |
18 \nn{say something about modules and tensor products?} |
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19 |
18 |
20 We emphasize that we are just sketching some of the main ideas in this appendix --- |
19 We emphasize that we are just sketching some of the main ideas in this appendix --- |
21 it falls well short of proving the definitions are equivalent. |
20 it falls well short of proving the definitions are equivalent. |
22 |
21 |
23 %\nn{cases to cover: (a) plain $n$-cats for $n=1,2$; (b) $n$-cat modules for $n=1$, also 2?; |
22 %\nn{cases to cover: (a) plain $n$-cats for $n=1,2$; (b) $n$-cat modules for $n=1$, also 2?; |
159 |
158 |
160 Let $a: y\to x$ be a 1-morphism. |
159 Let $a: y\to x$ be a 1-morphism. |
161 Define 2-morphsims $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ |
160 Define 2-morphsims $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ |
162 as shown in Figure \ref{fzo2}. |
161 as shown in Figure \ref{fzo2}. |
163 \begin{figure}[t] |
162 \begin{figure}[t] |
164 \begin{equation*} |
163 \begin{tikzpicture} |
165 \mathfig{.73}{tempkw/zo2} |
164 \newcommand{\rr}{6} |
166 \end{equation*} |
165 \newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
167 \caption{blah blah} |
166 |
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167 \node(A) at (0,0) { |
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168 \begin{tikzpicture} |
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169 \node[red,left] at (0,0) {$y$}; |
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170 \draw (0,0) \vertex arc (-120:-105:\rr) node[red,below] {$a$} arc(-105:-90:\rr) \vertex node[red,below](x2) {$x$}; |
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171 \draw (0,0) \vertex arc (120:105:\rr) node[red,above] {$a$} arc (105:90:\rr) \vertex node[red,above](x1) {$x$} -- (x2); |
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172 \begin{scope} |
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173 \path[clip] (0,0) arc (-120:-60:\rr) arc (60:120:\rr); |
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174 \foreach \x in {0,0.24,...,3} { |
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175 \draw[green!50!brown] (\x,1) -- (\x,-1); |
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176 } |
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177 \end{scope} |
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178 \draw[red, decorate,decoration={brace,amplitude=5pt}] ($(x1)+(0.2,-0.2)$) -- ($(x2)+(0.2,0.2)$) node[midway, xshift=0.7cm] {$x \times I$}; |
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179 \end{tikzpicture} |
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180 }; |
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181 |
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182 \node(B) at (-4,-4) { |
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183 \begin{tikzpicture} |
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184 \node[red,left] at (0,0) {$y$}; |
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185 \draw (0,0) \vertex |
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186 arc (120:105:\rr) node[red,above] {$a$} |
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187 arc (105:90:\rr) node[red,above] {$x$} \vertex |
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188 arc (90:75:\rr) node[red,above] {$x \times I$} |
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189 arc (75:60:\rr) \vertex node[red,right] {$x$} |
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190 arc (-60:-90:\rr) node[red,below] {$a$} |
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191 arc (-90:-120:\rr); |
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192 \begin{scope} |
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193 \path[clip] (0,0) arc (-120:-60:\rr) arc (60:120:\rr); |
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194 \foreach \x in {0,0.48,...,9} { |
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195 \draw[green!50!brown] (\x/4,1) -- (\x,-1); |
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196 } |
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197 \end{scope} |
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198 \end{tikzpicture} |
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199 }; |
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200 |
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201 \node(C) at (4,-4) { |
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202 \begin{tikzpicture}[y=-1cm] |
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203 \node[red,left] at (0,0) {$y$}; |
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204 \draw (0,0) \vertex |
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205 arc (120:105:\rr) node[red,below] {$a$} |
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206 arc (105:90:\rr) node[red,below] {$x$} \vertex |
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207 arc (90:75:\rr) node[red,below] {$x \times I$} |
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208 arc (75:60:\rr) \vertex node[red,right] {$x$} |
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209 arc (-60:-90:\rr) node[red,above] {$a$} |
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210 arc (-90:-120:\rr); |
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211 \begin{scope} |
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212 \path[clip] (0,0) arc (-120:-60:\rr) arc (60:120:\rr); |
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213 \foreach \x in {0,0.48,...,9} { |
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214 \draw[green!50!brown] (\x/4,1) -- (\x,-1); |
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215 } |
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216 \end{scope} |
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217 \end{tikzpicture} |
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218 }; |
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219 |
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220 \draw[->] (A) -- (B); |
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221 \draw[->] (A) -- (C); |
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222 \end{tikzpicture} |
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223 \caption{Producing weak identities from half pinched products} |
168 \label{fzo2} |
224 \label{fzo2} |
169 \end{figure} |
225 \end{figure} |
170 As suggested by the figure, these are two different reparameterizations |
226 As suggested by the figure, these are two different reparameterizations |
171 of a half-pinched version of $a\times I$. |
227 of a half-pinched version of $a\times I$. |
172 We must show that the two compositions of these two maps give the identity 2-morphisms |
228 We must show that the two compositions of these two maps give the identity 2-morphisms |
174 Figure \ref{fzo3} shows one case. |
230 Figure \ref{fzo3} shows one case. |
175 \begin{figure}[t] |
231 \begin{figure}[t] |
176 \begin{equation*} |
232 \begin{equation*} |
177 \mathfig{.83}{tempkw/zo3} |
233 \mathfig{.83}{tempkw/zo3} |
178 \end{equation*} |
234 \end{equation*} |
179 \caption{blah blah} |
235 \caption{Composition of weak identities, 1} |
180 \label{fzo3} |
236 \label{fzo3} |
181 \end{figure} |
237 \end{figure} |
182 In the first step we have inserted a copy of $(x\times I)\times I$. |
238 In the first step we have inserted a copy of $(x\times I)\times I$. |
183 Figure \ref{fzo4} shows the other case. |
239 Figure \ref{fzo4} shows the other case. |
184 \begin{figure}[t] |
240 \begin{figure}[t] |
185 \begin{equation*} |
241 \begin{equation*} |
186 \mathfig{.83}{tempkw/zo4} |
242 \mathfig{.83}{tempkw/zo4} |
187 \end{equation*} |
243 \end{equation*} |
188 \caption{blah blah} |
244 \caption{Composition of weak identities, 2} |
189 \label{fzo4} |
245 \label{fzo4} |
190 \end{figure} |
246 \end{figure} |
191 We identify a product region and remove it. |
247 We identify a product region and remove it. |
192 |
248 |
193 We define horizontal composition of 2-morphisms as shown in Figure \ref{fzo5}. |
249 We define horizontal composition of 2-morphisms as shown in Figure \ref{fzo5}. |