2094 %In the ordinary case we require |
2094 %In the ordinary case we require |
2095 |
2095 |
2096 The remaining module axioms are very similar to their counterparts in \S\ref{ss:n-cat-def}. |
2096 The remaining module axioms are very similar to their counterparts in \S\ref{ss:n-cat-def}. |
2097 |
2097 |
2098 \begin{module-axiom}[Extended isotopy invariance in dimension $n$] |
2098 \begin{module-axiom}[Extended isotopy invariance in dimension $n$] |
|
2099 \label{ei-module-axiom} |
2099 Let $M$ be a marked $n$-ball, $b \in \cM(M)$, and $f: M\to M$ be a homeomorphism which |
2100 Let $M$ be a marked $n$-ball, $b \in \cM(M)$, and $f: M\to M$ be a homeomorphism which |
2100 acts trivially on the restriction $\bd b$ of $b$ to $\bd M$. |
2101 acts trivially on the restriction $\bd b$ of $b$ to $\bd M$. |
2101 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which |
2102 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which |
2102 act trivially on $\bd b$. |
2103 act trivially on $\bd b$. |
2103 Then $f(b) = b$. |
2104 Then $f(b) = b$. |
2104 In addition, collar maps act trivially on $\cM(M)$. |
2105 In addition, collar maps act trivially on $\cM(M)$. |
2105 \end{module-axiom} |
2106 \end{module-axiom} |
2106 |
2107 |
2107 We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
2108 We emphasize that the $\bd M$ above (and below) means boundary in the marked $k$-ball sense. |
2108 In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
2109 In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
2109 on $\bd B \setmin N$. |
2110 on $\bd B \setmin N$. |
2110 |
2111 |
2111 \begin{module-axiom}[Splittings] |
2112 \begin{module-axiom}[Splittings] |
2112 Let $c\in \cM_k(M)$, with $1\le k < n$. |
2113 Let $c\in \cM_k(M)$, with $1\le k < n$. |
2126 category $\mbc$ of {\it marked $n$-balls with boundary conditions} as follows. |
2127 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 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 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 homeomorphisms $f:M\to M'$ such that $f|_{\bd M}(c) = c'$. |
2130 |
2131 |
2131 |
2132 Let $\cS$ be a distributive symmetric monoidal category, and assume that $\cC$ is enriched in $\cS$. |
2132 |
2133 A $\cC$-module enriched in $\cS$ is defined analogously to \ref{axiom:enriched}. |
2133 \nn{resume revising here} |
2134 The top-dimensional part of the module $\cM_n$ is required to be a functor from $\mbc$ to $\cS$. |
2134 |
2135 The top-dimensional gluing maps (module composition and $n$-category action) are $\cS$-maps whose |
2135 For $A_\infty$ modules we require |
2136 domain is a direct sub of tensor products, as in \ref{axiom:enriched}. |
2136 |
2137 |
2137 %\addtocounter{module-axiom}{-1} |
2138 If $\cC$ is an $A_\infty$ $n$-category (see \ref{axiom:families}), we replace module axiom \ref{ei-module-axiom} |
2138 \begin{module-axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act] |
2139 with the following axiom. |
2139 For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
2140 Retain notation from \ref{axiom:families}. |
2140 \[ |
2141 |
2141 C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
2142 \begin{module-axiom}[Families of homeomorphisms act in dimension $n$.] |
2142 \] |
2143 For each pair of marked $n$-balls $M$ and $M'$ and each pair $c\in \cl{\cM}(\bd M)$ and $c'\in \cl{\cM}(\bd M')$ |
2143 Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |
2144 we have an $\cS$-morphism |
2144 which fix $\bd M$. |
2145 \[ |
2145 These action maps are required to be associative up to homotopy, as in Theorem \ref{thm:CH-associativity}, |
2146 \cJ(\Homeo(M;c \to M'; c')) \ot \cM(M; c) \to \cM(M'; c') . |
|
2147 \] |
|
2148 Similarly, we have an $\cS$-morphism |
|
2149 \[ |
|
2150 \cJ(\Coll(M,c)) \ot \cM(M; c) \to \cM(M; c), |
|
2151 \] |
|
2152 where $\Coll(M,c)$ denotes the space of collar maps. |
|
2153 These action maps are required to be associative up to coherent homotopy, |
2146 and also compatible with composition (gluing) in the sense that |
2154 and also compatible with composition (gluing) in the sense that |
2147 a diagram like the one in Theorem \ref{thm:CH} commutes. |
2155 a diagram like the one in Theorem \ref{thm:CH} commutes. |
2148 \end{module-axiom} |
2156 \end{module-axiom} |
2149 |
2157 |
2150 As with the $n$-category version of the above axiom, we should also have families of collar maps act. |
|
2151 |
2158 |
2152 \medskip |
2159 \medskip |
2153 |
2160 |
2154 Note that the above axioms imply that an $n$-category module has the structure |
2161 Note that the above axioms imply that an $n$-category module has the structure |
2155 of an $n{-}1$-category. |
2162 of an $n{-}1$-category. |