equal
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replaced
211 We will show that this 1-morphism is a weak identity. |
211 We will show that this 1-morphism is a weak identity. |
212 This would be easier if our 2-morphisms were shaped like rectangles rather than bigons. |
212 This would be easier if our 2-morphisms were shaped like rectangles rather than bigons. |
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 This means we are free to add or delete product regions from 2-morphisms. |
216 Extended isotopy invariance implies that adding a product collar to a 2-morphism of $\cC$ has no effect, |
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217 and by cutting and regluing we can insert (or delete) product regions in the interior of 2-morphisms as well. |
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218 Figure \nn{triangle.pdf 2.a through 2.d} shows some examples. |
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219 |
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220 |
217 |
221 |
218 Let $a: y\to x$ be a 1-morphism. |
222 Let $a: y\to x$ be a 1-morphism. |
219 Define 2-morphsims $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ |
223 Define 2-morphsims $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ |
220 as shown in Figure \ref{fzo2}. |
224 as shown in Figure \ref{fzo2}. |
221 \begin{figure}[t] |
225 \begin{figure}[t] |
291 \end{tikzpicture} |
295 \end{tikzpicture} |
292 \caption{Producing weak identities from half pinched products} |
296 \caption{Producing weak identities from half pinched products} |
293 \label{fzo2} |
297 \label{fzo2} |
294 \end{figure} |
298 \end{figure} |
295 As suggested by the figure, these are two different reparameterizations |
299 As suggested by the figure, these are two different reparameterizations |
296 of a half-pinched version of $a\times I$. |
300 of a half-pinched version of $a\times I$ |
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301 (i.e.\ two different homeomorphisms from the half-pinched $I\times I$ to the standard bigon). |
297 We must show that the two compositions of these two maps give the identity 2-morphisms |
302 We must show that the two compositions of these two maps give the identity 2-morphisms |
298 on $a$ and $a\bullet \id_x$, as defined above. |
303 on $a$ and $a\bullet \id_x$, as defined above. |
299 Figure \ref{fzo3} shows one case. |
304 Figure \ref{fzo3} shows one case. |
300 \begin{figure}[t] |
305 \begin{figure}[t] |
301 \centering |
306 \centering |
516 |
521 |
517 \end{tikzpicture} |
522 \end{tikzpicture} |
518 \caption{Composition of weak identities, 2} |
523 \caption{Composition of weak identities, 2} |
519 \label{fzo4} |
524 \label{fzo4} |
520 \end{figure} |
525 \end{figure} |
521 We identify a product region and remove it. |
526 We notice that a certain subset of the disk is a product region and remove it. |
522 |
527 |
523 We define horizontal composition $f *_h g$ of 2-morphisms $f$ and $g$ as shown in Figure \ref{fzo5}. |
528 Given 2-morphisms $f$ and $g$, we define the horizontal composition $f *_h g$ to be any of the four |
524 It is not hard to show that this is independent of the arbitrary (left/right) |
529 equal 2-morphisms in Figure \ref{fzo5}. |
525 choice made in the definition, and that it is associative. |
530 \nn{add three remaining cases of triangle.pdf 3.b to fzo5} |
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531 Figure \nn{triangle 3.c, but not necessarily crooked} illustrates part of the proof that these four 2-morphisms are equal. |
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532 Similar arguments show that horizontal composition is associative. |
526 \begin{figure}[t] |
533 \begin{figure}[t] |
527 \begin{equation*} |
534 \begin{equation*} |
528 \raisebox{-.9cm}{ |
535 \raisebox{-.9cm}{ |
529 \begin{tikzpicture} |
536 \begin{tikzpicture} |
530 \draw (0,0) .. controls +(1,.8) and +(-1,.8) .. node[above] {$b$} (2.9,0) |
537 \draw (0,0) .. controls +(1,.8) and +(-1,.8) .. node[above] {$b$} (2.9,0) |
566 \end{tikzpicture}} |
573 \end{tikzpicture}} |
567 \end{equation*} |
574 \end{equation*} |
568 \caption{Horizontal composition of 2-morphisms} |
575 \caption{Horizontal composition of 2-morphisms} |
569 \label{fzo5} |
576 \label{fzo5} |
570 \end{figure} |
577 \end{figure} |
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578 |
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579 Given 1-morphisms $a$, $b$ and $c$ of $D$, we define the associator from $(a\bullet b)\bullet c$ to $a\bullet(b\bullet c)$ |
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580 as in Figure \nn{like triangle 4.a, but more general; use three colors as in that fig}. |
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581 This is just a reparameterization of the pinched product $(a\bullet b\bullet c)\times I$ of $\cC$. |
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582 |
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583 |
571 |
584 |
572 %\nn{need to find a list of axioms for pivotal 2-cats to check} |
585 %\nn{need to find a list of axioms for pivotal 2-cats to check} |
573 |
586 |
574 |
587 |
575 \subsection{\texorpdfstring{$A_\infty$}{A-infinity} 1-categories} |
588 \subsection{\texorpdfstring{$A_\infty$}{A-infinity} 1-categories} |