154 which is natural with respect to the actions of homeomorphisms. |
154 which is natural with respect to the actions of homeomorphisms. |
155 \end{axiom} |
155 \end{axiom} |
156 |
156 |
157 \begin{figure}[!ht] |
157 \begin{figure}[!ht] |
158 $$ |
158 $$ |
159 \begin{tikzpicture}[every label/.style={green}] |
159 \begin{tikzpicture}[%every label/.style={green} |
160 \node[fill=black, circle, label=below:$E$](S) at (0,0) {}; |
160 ] |
161 \node[fill=black, circle, label=above:$E$](N) at (0,2) {}; |
161 \node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {}; |
|
162 \node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {}; |
162 \draw (S) arc (-90:90:1); |
163 \draw (S) arc (-90:90:1); |
163 \draw (N) arc (90:270:1); |
164 \draw (N) arc (90:270:1); |
164 \node[left] at (-1,1) {$B_1$}; |
165 \node[left] at (-1,1) {$B_1$}; |
165 \node[right] at (1,1) {$B_2$}; |
166 \node[right] at (1,1) {$B_2$}; |
166 \end{tikzpicture} |
167 \end{tikzpicture} |
167 $$ |
168 $$ |
168 $$\mathfig{.4}{tempkw/blah3}$$ |
169 \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
169 \caption{Combining two balls to get a full boundary |
|
170 \nn{maybe smaller dots for $E$ in pdf fig}}\label{blah3}\end{figure} |
|
171 |
170 |
172 Note that we insist on injectivity above. |
171 Note that we insist on injectivity above. |
173 |
172 |
174 Let $\cC(S)_E$ denote the image of $\gl_E$. |
173 Let $\cC(S)_E$ denote the image of $\gl_E$. |
175 We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
174 We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
213 If $k < n$ we require that $\gl_Y$ is injective. |
212 If $k < n$ we require that $\gl_Y$ is injective. |
214 (For $k=n$, see below.) |
213 (For $k=n$, see below.) |
215 \end{axiom} |
214 \end{axiom} |
216 |
215 |
217 \begin{figure}[!ht] |
216 \begin{figure}[!ht] |
|
217 $$ |
|
218 \begin{tikzpicture}[%every label/.style={green}, |
|
219 x=1.5cm,y=1.5cm] |
|
220 \node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {}; |
|
221 \node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {}; |
|
222 \draw (S) arc (-90:90:1); |
|
223 \draw (N) arc (90:270:1); |
|
224 \draw (N) -- (S); |
|
225 \node[left] at (-1/4,1) {$B_1$}; |
|
226 \node[right] at (1/4,1) {$B_2$}; |
|
227 \node at (1/6,3/2) {$Y$}; |
|
228 \end{tikzpicture} |
|
229 $$ |
218 $$\mathfig{.4}{tempkw/blah5}$$ |
230 $$\mathfig{.4}{tempkw/blah5}$$ |
219 \caption{From two balls to one ball}\label{blah5}\end{figure} |
231 \caption{From two balls to one ball.}\label{blah5}\end{figure} |
220 |
232 |
221 \begin{axiom}[Strict associativity] \label{nca-assoc} |
233 \begin{axiom}[Strict associativity] \label{nca-assoc} |
222 The composition (gluing) maps above are strictly associative. |
234 The composition (gluing) maps above are strictly associative. |
223 \end{axiom} |
235 \end{axiom} |
224 |
236 |
225 \begin{figure}[!ht] |
237 \begin{figure}[!ht] |
226 $$\mathfig{.65}{tempkw/blah6}$$ |
238 $$\mathfig{.65}{tempkw/blah6}$$ |
227 \caption{An example of strict associativity}\label{blah6}\end{figure} |
239 \caption{An example of strict associativity.}\label{blah6}\end{figure} |
228 |
240 |
229 \nn{figure \ref{blah6} (blah6) needs a dotted line in the south split ball} |
241 \nn{figure \ref{blah6} (blah6) needs a dotted line in the south split ball} |
230 |
242 |
231 Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$. |
243 Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$. |
232 In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
244 In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
261 and these various $m$-fold composition maps satisfy an |
273 and these various $m$-fold composition maps satisfy an |
262 operad-type strict associativity condition (Figure \ref{blah7}).} |
274 operad-type strict associativity condition (Figure \ref{blah7}).} |
263 |
275 |
264 \begin{figure}[!ht] |
276 \begin{figure}[!ht] |
265 $$\mathfig{.8}{tempkw/blah7}$$ |
277 $$\mathfig{.8}{tempkw/blah7}$$ |
266 \caption{Operadish composition and associativity}\label{blah7}\end{figure} |
278 \caption{Operad composition and associativity}\label{blah7}\end{figure} |
267 |
279 |
268 The next axiom is related to identity morphisms, though that might not be immediately obvious. |
280 The next axiom is related to identity morphisms, though that might not be immediately obvious. |
269 |
281 |
270 \begin{axiom}[Product (identity) morphisms] |
282 \begin{axiom}[Product (identity) morphisms] |
271 For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. These maps must satisfy the following conditions. |
283 For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. These maps must satisfy the following conditions. |
518 \begin{example}[Traditional $n$-categories] |
530 \begin{example}[Traditional $n$-categories] |
519 \rm |
531 \rm |
520 \label{ex:traditional-n-categories} |
532 \label{ex:traditional-n-categories} |
521 Given a `traditional $n$-category with strong duality' $C$ |
533 Given a `traditional $n$-category with strong duality' $C$ |
522 define $\cC(X)$, for $X$ a $k$-ball or $k$-sphere with $k < n$, |
534 define $\cC(X)$, for $X$ a $k$-ball or $k$-sphere with $k < n$, |
523 to be the set of all $C$-labeled sub cell complexes of $X$. |
535 to be the set of all $C$-labeled sub cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
524 (See Subsection \ref{sec:fields}.) |
|
525 For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear |
536 For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear |
526 combinations of $C$-labeled sub cell complexes of $X$ |
537 combinations of $C$-labeled sub cell complexes of $X$ |
527 modulo the kernel of the evaluation map. |
538 modulo the kernel of the evaluation map. |
528 Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, |
539 Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, |
529 with each cell labelled by the $m$-th iterated identity morphism of the corresponding cell for $a$. |
540 with each cell labelled by the $m$-th iterated identity morphism of the corresponding cell for $a$. |
731 |
742 |
732 |
743 |
733 |
744 |
734 \subsection{Modules} |
745 \subsection{Modules} |
735 |
746 |
736 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
747 Next we define topological and $A_\infty$ $n$-category modules. |
737 a.k.a.\ actions). |
|
738 The definition will be very similar to that of $n$-categories, |
748 The definition will be very similar to that of $n$-categories, |
739 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
749 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
740 \nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
750 \nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
741 %\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} |
751 %\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} |
742 |
752 |
743 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
753 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
744 in the context of an $m{+}1$-dimensional TQFT. |
754 in the context of an $m{+}1$-dimensional TQFT. |
745 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
755 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
746 This will be explained in more detail as we present the axioms. |
756 This will be explained in more detail as we present the axioms. |
747 |
757 |
748 Fix an $n$-category $\cC$. |
758 Throughout, we fix an $n$-category $\cC$. For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. We state the final axiom, on actions of homeomorphisms, differently in the two cases. |
749 |
759 |
750 Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
760 Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
751 (standard $k$-ball, northern hemisphere in boundary of standard $k$-ball). |
761 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
752 We call $B$ the ball and $N$ the marking. |
762 We call $B$ the ball and $N$ the marking. |
753 A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
763 A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
754 restricts to a homeomorphism of markings. |
764 restricts to a homeomorphism of markings. |
755 |
765 |
756 \mmpar{Module axiom 1}{Module morphisms} |
766 \mmpar{Module axiom 1}{Module morphisms} |
829 of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
839 of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
830 (See Figure \ref{zzz3}.) |
840 (See Figure \ref{zzz3}.) |
831 |
841 |
832 \begin{figure}[!ht] |
842 \begin{figure}[!ht] |
833 \begin{equation*} |
843 \begin{equation*} |
834 \mathfig{.63}{tempkw/zz3} |
844 \mathfig{.4}{ncat/zz3} |
835 \end{equation*} |
845 \end{equation*} |
836 \caption{Module composition (top); $n$-category action (bottom)} |
846 \caption{Module composition (top); $n$-category action (bottom).} |
837 \label{zzz3} |
847 \label{zzz3} |
838 \end{figure} |
848 \end{figure} |
839 |
849 |
840 First, we can compose two module morphisms to get another module morphism. |
850 First, we can compose two module morphisms to get another module morphism. |
841 |
851 |
842 \mmpar{Module axiom 6}{Module composition} |
852 \mmpar{Module axiom 6}{Module composition} |
843 {Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls ($0\le k\le n$) |
853 {Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls (with $0\le k\le n$) |
844 and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
854 and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
845 Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
855 Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
846 Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
856 Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
847 We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$. |
857 We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$. |
848 Let $\cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E$ denote the fibered product of these two maps. |
858 Let $\cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E$ denote the fibered product of these two maps. |
1013 where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and |
1023 where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and |
1014 each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, |
1024 each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, |
1015 with $M_{ib}\cap Y_i$ being the marking. |
1025 with $M_{ib}\cap Y_i$ being the marking. |
1016 (See Figure \ref{mblabel}.) |
1026 (See Figure \ref{mblabel}.) |
1017 \begin{figure}[!ht]\begin{equation*} |
1027 \begin{figure}[!ht]\begin{equation*} |
1018 \mathfig{.9}{tempkw/mblabel} |
1028 \mathfig{.6}{ncat/mblabel} |
1019 \end{equation*}\caption{A permissible decomposition of a manifold |
1029 \end{equation*}\caption{A permissible decomposition of a manifold |
1020 whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$.}\label{mblabel}\end{figure} |
1030 whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure} |
1021 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
1031 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
1022 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
1032 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
1023 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
1033 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
1024 (The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
1034 (The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
1025 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
1035 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
1085 \label{ssec:spherecat} |
1095 \label{ssec:spherecat} |
1086 |
1096 |
1087 In this subsection we define an $n{+}1$-category $\cS$ of ``sphere modules" |
1097 In this subsection we define an $n{+}1$-category $\cS$ of ``sphere modules" |
1088 whose objects correspond to $n$-categories. |
1098 whose objects correspond to $n$-categories. |
1089 This is a version of the familiar algebras-bimodules-intertwiners 2-category. |
1099 This is a version of the familiar algebras-bimodules-intertwiners 2-category. |
1090 (Terminology: It is clearly appropriate to call an $S^0$ modules a bimodule, |
1100 (Terminology: It is clearly appropriate to call an $S^0$ module a bimodule, |
1091 since a 0-sphere has an obvious bi-ness. |
1101 but this is much less true for higher dimensional spheres, |
1092 This is much less true for higher dimensional spheres, |
|
1093 so we prefer the term ``sphere module" for the general case.) |
1102 so we prefer the term ``sphere module" for the general case.) |
1094 |
1103 |
1095 The $0$- through $n$-dimensional parts of $\cC$ are various sorts of modules, and we describe |
1104 The $0$- through $n$-dimensional parts of $\cC$ are various sorts of modules, and we describe |
1096 these first. |
1105 these first. |
1097 The $n{+}1$-dimensional part of $\cS$ consist of intertwiners |
1106 The $n{+}1$-dimensional part of $\cS$ consist of intertwiners |
1144 Corresponding to this decomposition we have an action and/or composition map |
1153 Corresponding to this decomposition we have an action and/or composition map |
1145 from the product of these various sets into $\cM(X)$. |
1154 from the product of these various sets into $\cM(X)$. |
1146 |
1155 |
1147 \medskip |
1156 \medskip |
1148 |
1157 |
1149 Part of the structure of an $n$-cat 0-sphere module is captured my saying it is |
1158 Part of the structure of an $n$-category 0-sphere module is captured by saying it is |
1150 a collection $\cD^{ab}$ of $n{-}1$-categories, indexed by pairs $(a, b)$ of objects (0-morphisms) |
1159 a collection $\cD^{ab}$ of $n{-}1$-categories, indexed by pairs $(a, b)$ of objects (0-morphisms) |
1151 of $\cA$ and $\cB$. |
1160 of $\cA$ and $\cB$. |
1152 Let $J$ be some standard 0-marked 1-ball (i.e.\ an interval with a marked point in its interior). |
1161 Let $J$ be some standard 0-marked 1-ball (i.e.\ an interval with a marked point in its interior). |
1153 Given a $j$-ball $X$, $0\le j\le n-1$, we define |
1162 Given a $j$-ball $X$, $0\le j\le n-1$, we define |
1154 \[ |
1163 \[ |