2091 |
2091 |
2092 There are two alternatives for the next axiom, according whether we are defining |
2092 There are two alternatives for the next axiom, according whether we are defining |
2093 modules for ordinary $n$-categories or $A_\infty$ $n$-categories. |
2093 modules for ordinary $n$-categories or $A_\infty$ $n$-categories. |
2094 In the ordinary case we require |
2094 In the ordinary case we require |
2095 |
2095 |
2096 \begin{module-axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$] |
2096 \begin{module-axiom}[Extended isotopy invariance in dimension $n$] |
2097 {Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
2097 Let $M$ be a marked $n$-ball, $b \in \cM(M)$, and $f: M\to M$ be a homeomorphism which |
2098 to the identity on $\bd M$ and is isotopic (rel boundary) to the identity. |
2098 acts trivially on the restriction $\bd b$ of $b$ to $\bd M$. |
2099 Then $f$ acts trivially on $\cM(M)$.} |
2099 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which |
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2100 act trivially on $\bd b$. |
|
2101 Then $f(b) = b$. |
2100 In addition, collar maps act trivially on $\cM(M)$. |
2102 In addition, collar maps act trivially on $\cM(M)$. |
2101 \end{module-axiom} |
2103 \end{module-axiom} |
2102 |
2104 |
2103 We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
2105 We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
2104 In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
2106 In other words, if $M = (B, N)$ then we require only that isotopies are fixed |