2087 In our example, elements $a$ of $\cM(M)$ maps to $T$, and $\pi^*(a)$ is the pullback of |
2087 In our example, elements $a$ of $\cM(M)$ maps to $T$, and $\pi^*(a)$ is the pullback of |
2088 $a$ along a map associated to $\pi$. |
2088 $a$ along a map associated to $\pi$. |
2089 |
2089 |
2090 \medskip |
2090 \medskip |
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 to 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 |
|
2096 The remaining module axioms are very similar to their counterparts in \S\ref{ss:n-cat-def}. |
2095 |
2097 |
2096 \begin{module-axiom}[Extended isotopy invariance in dimension $n$] |
2098 \begin{module-axiom}[Extended isotopy invariance in dimension $n$] |
2097 Let $M$ be a marked $n$-ball, $b \in \cM(M)$, and $f: M\to M$ be a homeomorphism which |
2099 Let $M$ be a marked $n$-ball, $b \in \cM(M)$, and $f: M\to M$ be a homeomorphism which |
2098 acts trivially on the restriction $\bd b$ of $b$ to $\bd M$. |
2100 acts trivially on the restriction $\bd b$ of $b$ to $\bd M$. |
2099 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which |
2101 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which |
2103 \end{module-axiom} |
2105 \end{module-axiom} |
2104 |
2106 |
2105 We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
2107 We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
2106 In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
2108 In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
2107 on $\bd B \setmin N$. |
2109 on $\bd B \setmin N$. |
|
2110 |
|
2111 \begin{module-axiom}[Splittings] |
|
2112 Let $c\in \cM_k(M)$, with $1\le k < n$. |
|
2113 Let $s = \{X_i\}$ be a splitting of M (so $M = \cup_i X_i$, and each $X_i$ is either a marked ball or a plain ball). |
|
2114 Let $\Homeo_\bd(M)$ denote homeomorphisms of $M$ which restrict to the identity on $\bd M$. |
|
2115 \begin{itemize} |
|
2116 \item (Alternative 1) Consider the set of homeomorphisms $g:M\to M$ such that $c$ splits along $g(s)$. |
|
2117 Then this subset of $\Homeo(M)$ is open and dense. |
|
2118 Furthermore, if $s$ restricts to a splitting $\bd s$ of $\bd M$, and if $\bd c$ splits along $\bd s$, then the |
|
2119 intersection of the set of such homeomorphisms $g$ with $\Homeo_\bd(M)$ is open and dense in $\Homeo_\bd(M)$. |
|
2120 \item (Alternative 2) Then there exists an embedded cell complex $S_c \sub M$, called the string locus of $c$, |
|
2121 such that if the splitting $s$ is transverse to $S_c$ then $c$ splits along $s$. |
|
2122 \end{itemize} |
|
2123 \end{module-axiom} |
|
2124 |
|
2125 We define the |
|
2126 category $\mbc$ of {\it marked $n$-balls with boundary conditions} as follows. |
|
2127 Its objects are pairs $(M, c)$, where $M$ is a marked $n$-ball and $c \in \cl\cM(\bd M)$ is the ``boundary condition". |
|
2128 The morphisms from $(M, c)$ to $(M', c')$, denoted $\Homeo(M; c \to M'; c')$, are |
|
2129 homeomorphisms $f:M\to M'$ such that $f|_{\bd M}(c) = c'$. |
|
2130 |
|
2131 |
|
2132 |
|
2133 \nn{resume revising here} |
2108 |
2134 |
2109 For $A_\infty$ modules we require |
2135 For $A_\infty$ modules we require |
2110 |
2136 |
2111 %\addtocounter{module-axiom}{-1} |
2137 %\addtocounter{module-axiom}{-1} |
2112 \begin{module-axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act] |
2138 \begin{module-axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act] |