213 |
213 |
214 In showing that identity 1-morphisms have the desired properties, we will |
214 In showing that identity 1-morphisms have the desired properties, we will |
215 rely heavily on the extended isotopy invariance of 2-morphisms in $\cC$. |
215 rely heavily on the extended isotopy invariance of 2-morphisms in $\cC$. |
216 Extended isotopy invariance implies that adding a product collar to a 2-morphism of $\cC$ has no effect, |
216 Extended isotopy invariance implies that adding a product collar to a 2-morphism of $\cC$ has no effect, |
217 and by cutting and regluing we can insert (or delete) product regions in the interior of 2-morphisms as well. |
217 and by cutting and regluing we can insert (or delete) product regions in the interior of 2-morphisms as well. |
218 Figure \nn{triangle.pdf 2.a through 2.d} shows some examples. |
218 Figure \ref{fig:product-regions} shows some examples. |
219 |
219 \begin{figure}[t] |
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220 $$ |
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221 \mathfig{0.5}{triangle/triangle2} |
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222 $$ |
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223 \begin{align*} |
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224 \begin{tikzpicture}[baseline] |
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225 \node[draw] (c) at (0,0) [circle through = {(1,0)}] {$f$}; |
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226 \node (d) at (c.east) [circle through = {(0.25,0)}] {}; |
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227 \foreach \n in {1,2} { |
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228 \node (p\n) at (intersection \n of c and d) {}; |
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229 \fill (p\n) circle (2pt); |
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230 } |
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231 \begin{scope}[decoration={brace,amplitude=10,aspect=0.5}] |
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232 \draw[decorate] (p2.east) -- node[right=2ex] {$a$} (p1.east); |
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233 \end{scope} |
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234 \end{tikzpicture} & = |
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235 \begin{tikzpicture}[baseline] |
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236 \node[draw] (c) at (0,0) [circle through = {(1,0)}] {}; |
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237 \begin{scope} |
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238 \path[clip] (c) circle (1); |
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239 \node[draw,dashed] (d) at (c.east) [circle through = {(0.25,0)}] {}; |
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240 \foreach \n in {1,2} { |
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241 \node (p\n) at (intersection \n of c and d) {}; |
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242 } |
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243 \node[left] at (c) {$f$}; |
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244 \path[clip] (d) circle (0.75); |
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245 \foreach \y in {1,0.86,...,-1} { |
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246 \draw[green!50!brown] (0,\y)--(1,\y); |
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247 } |
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248 \end{scope} |
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249 \draw[->,blue] (1.5,-1) node[below] {$a \times I$} -- (0.75,0); |
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250 \end{tikzpicture} \\ |
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251 \begin{tikzpicture}[baseline] |
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252 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {}; |
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253 \draw[dashed] (c.north) -- (c.south); |
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254 \node[right=6] at (c) {$g$}; |
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255 \node[left=6] at (c) {$f$}; |
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256 \end{tikzpicture} & = |
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257 \begin{tikzpicture}[baseline] |
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258 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {}; |
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259 \node[right=9] at (c) {$g$}; |
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260 \node[left=9] at (c) {$f$}; |
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261 \draw[dashed] (c.north) to[out=-115,in=115] (c.south) to[out=65,in=-65] (c.north); |
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262 \begin{scope} |
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263 \path[clip] (c.north) to[out=-115,in=115] (c.south) to[out=65,in=-65] (c.north); |
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264 \foreach \y in {1,0.86,...,-1} { |
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265 \draw[green!50!brown] (-1,\y)--(1,\y); |
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266 } |
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267 \end{scope} |
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268 \draw[->,blue] (.75,-1.25) node[below] {$a \times I$} -- (0,-0.25); |
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269 \end{tikzpicture} \\ |
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270 \begin{tikzpicture}[baseline] |
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271 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {}; |
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272 \draw[dashed] (c.north) -- (c.south); |
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273 \node[right=18] at (c) {$g$}; |
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274 \node[left=10] at (c) {$f$}; |
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275 \fill (0,0.4) node (p1) {} circle (2pt); |
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276 \fill (0,-0.4) node (p2) {} circle (2pt); |
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277 \begin{scope}[decoration={brace,amplitude=5,aspect=0.5}] |
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278 \draw[decorate] (p1.east) -- node[right=0.5ex] {\scriptsize $a$} (p2.east); |
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279 \end{scope} |
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280 \end{tikzpicture} & = |
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281 \begin{tikzpicture}[baseline] |
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282 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {}; |
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283 \node[draw,dashed] (d) at (0,0) [circle, minimum height=1cm,minimum width=1cm] {}; |
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284 \draw[dashed] (c.north) -- (d.north) (d.south) -- (c.south); |
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285 \node[right=18] at (c) {$g$}; |
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286 \node[left=18] at (c) {$f$}; |
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287 \clip (0,0) circle (0.5cm); |
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288 \foreach \y in {1,0.86,...,-1} { |
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289 \draw[green!50!brown] (-1,\y)--(1,\y); |
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290 } |
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291 \end{tikzpicture} |
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292 \end{align*} |
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293 \todo{fourth case} |
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294 \caption{Examples of inserting or deleting product regions.} |
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295 \label{fig:product-regions} |
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296 \end{figure} |
220 |
297 |
221 |
298 |
222 Let $a: y\to x$ be a 1-morphism. |
299 Let $a: y\to x$ be a 1-morphism. |
223 Define 2-morphsims $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ |
300 Define 2-morphsims $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ |
224 as shown in Figure \ref{fzo2}. |
301 as shown in Figure \ref{fzo2}. |
656 \end{figure} |
733 \end{figure} |
657 \begin{figure}[t] |
734 \begin{figure}[t] |
658 $$ |
735 $$ |
659 \mathfig{0.6}{triangle/triangle3c} |
736 \mathfig{0.6}{triangle/triangle3c} |
660 $$ |
737 $$ |
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738 $$ |
|
739 \begin{tikzpicture} |
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740 \node (fg1) at (0,0) { |
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741 \begin{tikzpicture}[baseline=-0.6cm] |
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742 \path (0,0) coordinate (f1); |
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743 \path (3,0) coordinate (f2); |
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744 \path (3,-0.5) coordinate (g1); |
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745 \path (6,-0.5) coordinate (g2); |
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746 \node at (1.5,0.125) {$f$}; |
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747 \node at (4.5,-0.625) {$g$}; |
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748 \draw (f1) .. controls +(1,.8) and +(-1,.8) .. (f2); |
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749 \draw[dashed] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2); |
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750 \draw (f1) .. controls +(1,-1) and +(-1,-.4) .. (g1); |
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751 \draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2); |
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752 \draw[dashed] (g1) .. controls +(1,.4) and +(-1,.4) .. (g2); |
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753 \draw (f2) .. controls +(1,.4) and +(-1,1) .. (g2); |
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754 % |
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755 \draw[blue,->] (-0.8,-1.2) node[below] {$(a \circ d) \times I$} -- (1,-0.5) ; |
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756 \path[clip] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2) |
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757 .. controls +(1,.4) and +(-1,1) .. (g2) |
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758 .. controls +(-1,.4) and +(1,.4) .. (g1) |
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759 .. controls +(-1,-.4) and +(1,-1) .. (f1); |
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760 \foreach \x in {0,0.1, ..., 6} { |
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761 \draw[green!50!brown] (\x,-2) -- + (0,4); |
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762 } |
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763 \end{tikzpicture} |
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764 }; |
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765 \node (fg2) at (4,-4) { |
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766 \begin{tikzpicture}[baseline=-0.1cm] |
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767 \path (0,0) coordinate (f1); |
|
768 \path (3,0) coordinate (f2); |
|
769 \path (3,-0.5) coordinate (g1); |
|
770 \path (6,-0.5) coordinate (g2); |
|
771 \node at (1.5,0.125) {$f$}; |
|
772 \node at (4.5,-0.625) {$g$}; |
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773 \draw[dashed] (f1) .. controls +(1,.8) and +(-1,.8) .. (f2); |
|
774 \draw[dashed] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2); |
|
775 \draw (f1) .. controls +(1,-1) and +(-1,-.4) .. (g1); |
|
776 \draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2); |
|
777 \draw[dashed] (g1) .. controls +(1,.4) and +(-1,.4) .. (g2); |
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778 \draw[dashed] (f2) .. controls +(1,.4) and +(-1,1) .. (g2); |
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779 \draw (f1) .. controls +(1,1.5) and +(-1,2)..(g2); |
|
780 % |
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781 \begin{scope} |
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782 \path[clip] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2) |
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783 .. controls +(1,.4) and +(-1,1) .. (g2) |
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784 .. controls +(-1,.4) and +(1,.4) .. (g1) |
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785 .. controls +(-1,-.4) and +(1,-1) .. (f1); |
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786 \foreach \x in {0,0.1, ..., 6} { |
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787 \draw[green!50!brown] (\x,-2) -- + (0,4); |
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788 } |
|
789 \end{scope} |
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790 \begin{scope} |
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791 \path[clip] (f1) .. controls +(1,1.5) and +(-1,2).. (g2) |
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792 .. controls +(-1,1) and +(1,.4) .. (f2) |
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793 .. controls +(-1,.8) and + (1,.8) .. (f1); |
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794 \foreach \x in {0,0.1, ..., 6} { |
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795 \draw[green!50!brown] (\x,-2) -- + (0,4); |
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796 } |
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797 \end{scope} |
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798 \end{tikzpicture} |
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799 }; |
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800 \node (fg3) at (8,0) { |
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801 \begin{tikzpicture}[baseline=-2.45cm] |
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802 \path (0,0) coordinate (f1); |
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803 \path (3,0) coordinate (f2); |
|
804 \path (3,0) coordinate (g1); |
|
805 \path (6,0) coordinate (g2); |
|
806 \node at (1.5,0) {$f$}; |
|
807 \node at (4.5,0) {$g$}; |
|
808 \draw[dashed] (f1) .. controls +(1,.8) and +(-1,.8) .. (f2); |
|
809 \draw (f1) .. controls +(1,-.8) and +(-1,-.8) .. (f2); |
|
810 \draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2); |
|
811 \draw[dashed] (g1) .. controls +(1,.8) and +(-1,.8) .. (g2); |
|
812 \draw (f1) .. controls +(1,1.5) and +(-1,1.5)..(g2); |
|
813 % |
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814 \draw[blue,->] (4,1.75) node[above] {$(b \circ d) \times I$}-- + (0,-1); |
|
815 \begin{scope} |
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816 \path[clip] (f1) .. controls +(1,1.5) and +(-1,1.5).. (g2) |
|
817 .. controls +(-1,.8) and +(1,.8) .. (f2) |
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818 .. controls +(-1,.8) and + (1,.8) .. (f1); |
|
819 \foreach \x in {0,0.1, ..., 6} { |
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820 \draw[green!50!brown] (\x,-2) -- + (0,4); |
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821 } |
|
822 \end{scope} |
|
823 \end{tikzpicture} |
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824 }; |
|
825 \draw[->] ($(fg1.south)+(0,0.5)$) -- node[left=0.5cm] {add $(b \circ d) \times I$} (fg2); |
|
826 \draw[->] (fg2) -- node[right=0.5cm] {remove $(a \circ d) \times I$} ($(fg3.south)+(0,1.75)$); |
|
827 \path (fg1) -- node {$=$} (fg3); |
|
828 \end{tikzpicture} |
|
829 $$ |
661 \caption{Part of the proof that the four different horizontal compositions of 2-morphisms are equal.} |
830 \caption{Part of the proof that the four different horizontal compositions of 2-morphisms are equal.} |
662 \label{fig:horizontal-compositions-equal} |
831 \label{fig:horizontal-compositions-equal} |
663 \end{figure} |
832 \end{figure} |
664 |
833 |
665 Given 1-morphisms $a$, $b$ and $c$ of $D$, we define the associator from $(a\bullet b)\bullet c$ to $a\bullet(b\bullet c)$ |
834 Given 1-morphisms $a$, $b$ and $c$ of $D$, we define the associator from $(a\bullet b)\bullet c$ to $a\bullet(b\bullet c)$ |