236 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
236 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
237 to the intersection of the boundaries of $B$ and $B_i$. |
237 to the intersection of the boundaries of $B$ and $B_i$. |
238 If $k < n$, |
238 If $k < n$, |
239 or if $k=n$ and we are in the $A_\infty$ case, |
239 or if $k=n$ and we are in the $A_\infty$ case, |
240 we require that $\gl_Y$ is injective. |
240 we require that $\gl_Y$ is injective. |
241 (For $k=n$ in the plain (non-$A_\infty$) case, see below.) |
241 (For $k=n$ in the ordinary (non-$A_\infty$) case, see below.) |
242 \end{axiom} |
242 \end{axiom} |
243 |
243 |
244 \begin{figure}[!ht] \centering |
244 \begin{figure}[!ht] \centering |
245 \begin{tikzpicture}[%every label/.style={green}, |
245 \begin{tikzpicture}[%every label/.style={green}, |
246 x=1.5cm,y=1.5cm] |
246 x=1.5cm,y=1.5cm] |
522 |
522 |
523 All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
523 All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
524 The last axiom (below), concerning actions of |
524 The last axiom (below), concerning actions of |
525 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
525 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
526 |
526 |
527 We start with the plain $n$-category case. |
527 We start with the ordinary $n$-category case. |
528 |
528 |
529 \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] |
529 \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] |
530 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
530 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
531 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
531 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
532 Then $f$ acts trivially on $\cC(X)$; that is $f(a) = a$ for all $a\in \cC(X)$. |
532 Then $f$ acts trivially on $\cC(X)$; that is $f(a) = a$ for all $a\in \cC(X)$. |
597 isotopic (rel boundary) to the identity {\it extended isotopy}. |
597 isotopic (rel boundary) to the identity {\it extended isotopy}. |
598 |
598 |
599 The revised axiom is |
599 The revised axiom is |
600 |
600 |
601 %\addtocounter{axiom}{-1} |
601 %\addtocounter{axiom}{-1} |
602 \begin{axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$.] |
602 \begin{axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$.] |
603 \label{axiom:extended-isotopies} |
603 \label{axiom:extended-isotopies} |
604 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
604 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
605 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
605 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
606 Then $f$ acts trivially on $\cC(X)$. |
606 Then $f$ acts trivially on $\cC(X)$. |
607 In addition, collar maps act trivially on $\cC(X)$. |
607 In addition, collar maps act trivially on $\cC(X)$. |
642 We will not pursue this in detail here. |
642 We will not pursue this in detail here. |
643 |
643 |
644 A variant on the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. |
644 A variant on the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. |
645 |
645 |
646 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
646 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
647 into a plain $n$-category (enriched over graded groups). |
647 into a ordinary $n$-category (enriched over graded groups). |
648 In a different direction, if we enrich over topological spaces instead of chain complexes, |
648 In a different direction, if we enrich over topological spaces instead of chain complexes, |
649 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
649 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
650 instead of $C_*(\Homeo_\bd(X))$. |
650 instead of $C_*(\Homeo_\bd(X))$. |
651 Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex |
651 Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex |
652 type $A_\infty$ $n$-category. |
652 type $A_\infty$ $n$-category. |
653 |
653 |
654 \medskip |
654 \medskip |
655 |
655 |
656 The alert reader will have already noticed that our definition of a (plain) $n$-category |
656 The alert reader will have already noticed that our definition of a (ordinary) $n$-category |
657 is extremely similar to our definition of a system of fields. |
657 is extremely similar to our definition of a system of fields. |
658 There are two differences. |
658 There are two differences. |
659 First, for the $n$-category definition we restrict our attention to balls |
659 First, for the $n$-category definition we restrict our attention to balls |
660 (and their boundaries), while for fields we consider all manifolds. |
660 (and their boundaries), while for fields we consider all manifolds. |
661 Second, in category definition we directly impose isotopy |
661 Second, in category definition we directly impose isotopy |
773 but (string diagrams)/(relations) is isomorphic to |
773 but (string diagrams)/(relations) is isomorphic to |
774 (pasting diagrams composed of smaller string diagrams)/(relations)} |
774 (pasting diagrams composed of smaller string diagrams)/(relations)} |
775 } |
775 } |
776 |
776 |
777 |
777 |
778 \begin{example}[The bordism $n$-category, plain version] |
778 \begin{example}[The bordism $n$-category, ordinary version] |
779 \label{ex:bord-cat} |
779 \label{ex:bord-cat} |
780 \rm |
780 \rm |
781 \label{ex:bordism-category} |
781 \label{ex:bordism-category} |
782 For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
782 For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
783 submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
783 submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
913 |
913 |
914 |
914 |
915 \subsection{From balls to manifolds} |
915 \subsection{From balls to manifolds} |
916 \label{ss:ncat_fields} \label{ss:ncat-coend} |
916 \label{ss:ncat_fields} \label{ss:ncat-coend} |
917 In this section we show how to extend an $n$-category $\cC$ as described above |
917 In this section we show how to extend an $n$-category $\cC$ as described above |
918 (of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
918 (of either the ordinary or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
919 This extension is a certain colimit, and the arrow in the notation is intended as a reminder of this. |
919 This extension is a certain colimit, and the arrow in the notation is intended as a reminder of this. |
920 |
920 |
921 In the case of plain $n$-categories, this construction factors into a construction of a |
921 In the case of ordinary $n$-categories, this construction factors into a construction of a |
922 system of fields and local relations, followed by the usual TQFT definition of a |
922 system of fields and local relations, followed by the usual TQFT definition of a |
923 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
923 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
924 For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
924 For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
925 Recall that we can take a plain $n$-category $\cC$ and pass to the ``free resolution", |
925 Recall that we can take a ordinary $n$-category $\cC$ and pass to the ``free resolution", |
926 an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls |
926 an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls |
927 (recall Example \ref{ex:blob-complexes-of-balls} above). |
927 (recall Example \ref{ex:blob-complexes-of-balls} above). |
928 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
928 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
929 for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the |
929 for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the |
930 same as the original blob complex for $M$ with coefficients in $\cC$. |
930 same as the original blob complex for $M$ with coefficients in $\cC$. |
1158 %\nn{need to finish explaining why we have a system of fields; |
1158 %\nn{need to finish explaining why we have a system of fields; |
1159 %define $k$-cat $\cC(\cdot\times W)$} |
1159 %define $k$-cat $\cC(\cdot\times W)$} |
1160 |
1160 |
1161 \subsection{Modules} |
1161 \subsection{Modules} |
1162 |
1162 |
1163 Next we define plain and $A_\infty$ $n$-category modules. |
1163 Next we define ordinary and $A_\infty$ $n$-category modules. |
1164 The definition will be very similar to that of $n$-categories, |
1164 The definition will be very similar to that of $n$-categories, |
1165 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
1165 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
1166 |
1166 |
1167 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
1167 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
1168 in the context of an $m{+}1$-dimensional TQFT. |
1168 in the context of an $m{+}1$-dimensional TQFT. |
1288 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
1288 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
1289 to the intersection of the boundaries of $M$ and $M_i$. |
1289 to the intersection of the boundaries of $M$ and $M_i$. |
1290 If $k < n$, |
1290 If $k < n$, |
1291 or if $k=n$ and we are in the $A_\infty$ case, |
1291 or if $k=n$ and we are in the $A_\infty$ case, |
1292 we require that $\gl_Y$ is injective. |
1292 we require that $\gl_Y$ is injective. |
1293 (For $k=n$ in the plain (non-$A_\infty$) case, see below.)} |
1293 (For $k=n$ in the ordinary (non-$A_\infty$) case, see below.)} |
1294 \end{module-axiom} |
1294 \end{module-axiom} |
1295 |
1295 |
1296 |
1296 |
1297 Second, we can compose an $n$-category morphism with a module morphism to get another |
1297 Second, we can compose an $n$-category morphism with a module morphism to get another |
1298 module morphism. |
1298 module morphism. |
1312 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
1312 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
1313 to the intersection of the boundaries of $X$ and $M'$. |
1313 to the intersection of the boundaries of $X$ and $M'$. |
1314 If $k < n$, |
1314 If $k < n$, |
1315 or if $k=n$ and we are in the $A_\infty$ case, |
1315 or if $k=n$ and we are in the $A_\infty$ case, |
1316 we require that $\gl_Y$ is injective. |
1316 we require that $\gl_Y$ is injective. |
1317 (For $k=n$ in the plain (non-$A_\infty$) case, see below.)} |
1317 (For $k=n$ in the ordinary (non-$A_\infty$) case, see below.)} |
1318 \end{module-axiom} |
1318 \end{module-axiom} |
1319 |
1319 |
1320 \begin{module-axiom}[Strict associativity] |
1320 \begin{module-axiom}[Strict associativity] |
1321 The composition and action maps above are strictly associative. |
1321 The composition and action maps above are strictly associative. |
1322 Given any decomposition of a large marked ball into smaller marked and unmarked balls |
1322 Given any decomposition of a large marked ball into smaller marked and unmarked balls |
1430 $a$ along a map associated to $\pi$. |
1430 $a$ along a map associated to $\pi$. |
1431 |
1431 |
1432 \medskip |
1432 \medskip |
1433 |
1433 |
1434 There are two alternatives for the next axiom, according whether we are defining |
1434 There are two alternatives for the next axiom, according whether we are defining |
1435 modules for plain $n$-categories or $A_\infty$ $n$-categories. |
1435 modules for ordinary $n$-categories or $A_\infty$ $n$-categories. |
1436 In the plain case we require |
1436 In the ordinary case we require |
1437 |
1437 |
1438 \begin{module-axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$] |
1438 \begin{module-axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$] |
1439 {Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
1439 {Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
1440 to the identity on $\bd M$ and is isotopic (rel boundary) to the identity. |
1440 to the identity on $\bd M$ and is isotopic (rel boundary) to the identity. |
1441 Then $f$ acts trivially on $\cM(M)$.} |
1441 Then $f$ acts trivially on $\cM(M)$.} |
1442 In addition, collar maps act trivially on $\cM(M)$. |
1442 In addition, collar maps act trivially on $\cM(M)$. |
1443 \end{module-axiom} |
1443 \end{module-axiom} |