32 |
32 |
33 \medskip |
33 \medskip |
34 |
34 |
35 The axioms for an $n$-category are spread throughout this section. |
35 The axioms for an $n$-category are spread throughout this section. |
36 Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, |
36 Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, |
37 \ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:vcones} and |
37 \ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:extended-isotopies} and \ref{axiom:vcones}. |
38 \ref{axiom:extended-isotopies}. |
38 For an enriched $n$-category we add Axiom \ref{axiom:enriched}. |
39 For an enriched $n$-category we add \ref{axiom:enriched}. |
|
40 For an $A_\infty$ $n$-category, we replace |
39 For an $A_\infty$ $n$-category, we replace |
41 Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. |
40 Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. |
42 |
41 |
43 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms |
42 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms |
44 for $k{-}1$-morphisms. |
43 for $k{-}1$-morphisms. |
577 |
576 |
578 \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] |
577 \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] |
579 Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which |
578 Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which |
580 acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. |
579 acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. |
581 (Keep in mind the important special case where $f$ restricted to $\bd X$ is the identity.) |
580 (Keep in mind the important special case where $f$ restricted to $\bd X$ is the identity.) |
582 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which |
581 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which act |
583 trivially on $\bd b$. |
582 trivially on $\bd b$. |
584 Then $f(b) = b$. |
583 Then $f(b) = b$. |
585 In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially on |
584 In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially on |
586 all of $\cC(X)$. |
585 all of $\cC(X)$. |
587 \end{axiom} |
586 \end{axiom} |
652 isotopic (rel boundary) to the identity {\it extended isotopy}. |
651 isotopic (rel boundary) to the identity {\it extended isotopy}. |
653 |
652 |
654 The revised axiom is |
653 The revised axiom is |
655 |
654 |
656 %\addtocounter{axiom}{-1} |
655 %\addtocounter{axiom}{-1} |
657 \begin{axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$] |
656 \begin{axiom}[Extended isotopy invariance in dimension $n$] |
658 \label{axiom:extended-isotopies} |
657 \label{axiom:extended-isotopies} |
659 Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which |
658 Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which |
660 acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. |
659 acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. |
661 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which |
660 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which |
662 act trivially on $\bd b$. |
661 act trivially on $\bd b$. |
874 |
873 |
875 When the enriching category $\cS$ is chain complexes or topological spaces, |
874 When the enriching category $\cS$ is chain complexes or topological spaces, |
876 or more generally an appropriate sort of $\infty$-category, |
875 or more generally an appropriate sort of $\infty$-category, |
877 we can modify the extended isotopy axiom \ref{axiom:extended-isotopies} |
876 we can modify the extended isotopy axiom \ref{axiom:extended-isotopies} |
878 to require that families of homeomorphisms act |
877 to require that families of homeomorphisms act |
879 and obtain an $A_\infty$ $n$-category. |
878 and obtain what we shall call an $A_\infty$ $n$-category. |
880 |
879 |
881 \noop{ |
880 \noop{ |
882 We believe that abstract definitions should be guided by diverse collections |
881 We believe that abstract definitions should be guided by diverse collections |
883 of concrete examples, and a lack of diversity in our present collection of examples of $A_\infty$ $n$-categories |
882 of concrete examples, and a lack of diversity in our present collection of examples of $A_\infty$ $n$-categories |
884 makes us reluctant to commit to an all-encompassing general definition. |
883 makes us reluctant to commit to an all-encompassing general definition. |
890 Note that the morphisms $\Homeo(X,c; X', c')$ from $(X, c)$ to $(X', c')$ form a topological space. |
889 Note that the morphisms $\Homeo(X,c; X', c')$ from $(X, c)$ to $(X', c')$ form a topological space. |
891 Let $\cS$ be an appropriate $\infty$-category (e.g.\ chain complexes) |
890 Let $\cS$ be an appropriate $\infty$-category (e.g.\ chain complexes) |
892 and let $\cJ$ be an $\infty$-functor from topological spaces to $\cS$ |
891 and let $\cJ$ be an $\infty$-functor from topological spaces to $\cS$ |
893 (e.g.\ the singular chain functor $C_*$). |
892 (e.g.\ the singular chain functor $C_*$). |
894 |
893 |
895 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
894 \begin{axiom}[\textup{\textbf{[$A_\infty$ replacement for Axiom \ref{axiom:extended-isotopies}]}} Families of homeomorphisms act in dimension $n$.] |
896 \label{axiom:families} |
895 \label{axiom:families} |
897 For each pair of $n$-balls $X$ and $X'$ and each pair $c\in \cl{\cC}(\bd X)$ and $c'\in \cl{\cC}(\bd X')$ we have an $\cS$-morphism |
896 For each pair of $n$-balls $X$ and $X'$ and each pair $c\in \cl{\cC}(\bd X)$ and $c'\in \cl{\cC}(\bd X')$ we have an $\cS$-morphism |
898 \[ |
897 \[ |
899 \cJ(\Homeo(X,c; X', c')) \ot \cC(X; c) \to \cC(X'; c') . |
898 \cJ(\Homeo(X,c; X', c')) \ot \cC(X; c) \to \cC(X'; c') . |
900 \] |
899 \] |
911 \end{axiom} |
910 \end{axiom} |
912 |
911 |
913 We now describe the topology on $\Coll(X; c)$. |
912 We now describe the topology on $\Coll(X; c)$. |
914 We retain notation from the above definition of collar map. |
913 We retain notation from the above definition of collar map. |
915 Each collaring homeomorphism $X \cup (Y\times J) \to X$ determines a map from points $p$ of $\bd X$ to |
914 Each collaring homeomorphism $X \cup (Y\times J) \to X$ determines a map from points $p$ of $\bd X$ to |
916 (possibly zero-width) embedded intervals in $X$ terminating at $p$. |
915 (possibly length zero) embedded intervals in $X$ terminating at $p$. |
917 If $p \in Y$ this interval is the image of $\{p\}\times J$. |
916 If $p \in Y$ this interval is the image of $\{p\}\times J$. |
918 If $p \notin Y$ then $p$ is assigned the zero-width interval $\{p\}$. |
917 If $p \notin Y$ then $p$ is assigned the length zero interval $\{p\}$. |
919 Such collections of intervals have a natural topology, and $\Coll(X; c)$ inherits its topology from this. |
918 Such collections of intervals have a natural topology, and $\Coll(X; c)$ inherits its topology from this. |
920 Note in particular that parts of the collar are allowed to shrink continuously to zero width. |
919 Note in particular that parts of the collar are allowed to shrink continuously to zero length. |
921 (This is the real content; if nothing shrinks to zero width then the action of families of collar |
920 (This is the real content; if nothing shrinks to zero length then the action of families of collar |
922 maps follows from the action of families of homeomorphisms and compatibility with gluing.) |
921 maps follows from the action of families of homeomorphisms and compatibility with gluing.) |
923 |
922 |
924 The $k=n$ case of Axiom \ref{axiom:morphisms} posits a {\it strictly} associative action of {\it sets} |
923 The $k=n$ case of Axiom \ref{axiom:morphisms} posits a {\it strictly} associative action of {\it sets} |
925 $\Homeo(X,c; X', c') \times \cC(X; c) \to \cC(X'; c')$, and at first it might seem that this would force the above |
924 $\Homeo(X,c; X', c') \times \cC(X; c) \to \cC(X'; c')$, and at first it might seem that this would force the above |
926 action of $\cJ(\Homeo(X,c; X', c'))$ to be strictly associative as well (assuming the two actions are compatible). |
925 action of $\cJ(\Homeo(X,c; X', c'))$ to be strictly associative as well (assuming the two actions are compatible). |
1116 $W \to W'$ which restricts to the identity on the boundary. |
1115 $W \to W'$ which restricts to the identity on the boundary. |
1117 For $n=1$ we have the familiar bordism 1-category of $d$-manifolds. |
1116 For $n=1$ we have the familiar bordism 1-category of $d$-manifolds. |
1118 The case $n=d$ captures the $n$-categorical nature of bordisms. |
1117 The case $n=d$ captures the $n$-categorical nature of bordisms. |
1119 The case $n > 2d$ captures the full symmetric monoidal $n$-category structure. |
1118 The case $n > 2d$ captures the full symmetric monoidal $n$-category structure. |
1120 \end{example} |
1119 \end{example} |
1121 \begin{remark} |
1120 \begin{rem} |
1122 Working with the smooth bordism category would require careful attention to either collars, corners or halos. |
1121 Working with the smooth bordism category would require careful attention to either collars, corners or halos. |
1123 \end{remark} |
1122 \end{rem} |
1124 |
1123 |
1125 %\nn{the next example might be an unnecessary distraction. consider deleting it.} |
1124 %\nn{the next example might be an unnecessary distraction. consider deleting it.} |
1126 |
1125 |
1127 %\begin{example}[Variation on the above examples] |
1126 %\begin{example}[Variation on the above examples] |
1128 %We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
1127 %We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
1342 We also assume, inductively, that we have gluing and restriction maps for colimits of $k{-}1$-manifolds. |
1341 We also assume, inductively, that we have gluing and restriction maps for colimits of $k{-}1$-manifolds. |
1343 Gluing and restriction maps for colimits of $k$-manifolds will be defined later in this subsection. |
1342 Gluing and restriction maps for colimits of $k$-manifolds will be defined later in this subsection. |
1344 |
1343 |
1345 Let $\du_a X_a = M_0\to\cdots\to M_m = W$ be a ball decomposition compatible with $x$. |
1344 Let $\du_a X_a = M_0\to\cdots\to M_m = W$ be a ball decomposition compatible with $x$. |
1346 Let $\bd M_i = N_i \cup Y_i \cup Y'_i$, where $Y_i$ and $Y'_i$ are glued together to produce $M_{i+1}$. |
1345 Let $\bd M_i = N_i \cup Y_i \cup Y'_i$, where $Y_i$ and $Y'_i$ are glued together to produce $M_{i+1}$. |
1347 We will define $\psi_{\cC;W}(x)$ be be the subset of $\prod_a \cC(X_a)$ which satisfies a series of conditions |
1346 We will define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies a series of conditions |
1348 related to the gluings $M_{i-1} \to M_i$, $1\le i \le m$. |
1347 related to the gluings $M_{i-1} \to M_i$, $1\le i \le m$. |
1349 By Axiom \ref{nca-boundary}, we have a map |
1348 By Axiom \ref{nca-boundary}, we have a map |
1350 \[ |
1349 \[ |
1351 \prod_a \cC(X_a) \to \cl\cC(\bd M_0) . |
1350 \prod_a \cC(X_a) \to \cl\cC(\bd M_0) . |
1352 \] |
1351 \] |
1359 $\cl\cC(N_0) \to \cl\cC(\bd M_1)$. |
1358 $\cl\cC(N_0) \to \cl\cC(\bd M_1)$. |
1360 The second condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_1)$ is splittable |
1359 The second condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_1)$ is splittable |
1361 along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\cl\cC(Y_1)$ and $\cl\cC(Y'_1)$ agree |
1360 along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\cl\cC(Y_1)$ and $\cl\cC(Y'_1)$ agree |
1362 (with respect to the identification of $Y_1$ and $Y'_1$ provided by the gluing map). |
1361 (with respect to the identification of $Y_1$ and $Y'_1$ provided by the gluing map). |
1363 The $i$-th condition is defined similarly. |
1362 The $i$-th condition is defined similarly. |
1364 Note that these conditions depend on on the boundaries of elements of $\prod_a \cC(X_a)$. |
1363 Note that these conditions depend on the boundaries of elements of $\prod_a \cC(X_a)$. |
1365 |
1364 |
1366 We define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies the |
1365 We define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies the |
1367 above conditions for all $i$ and also all |
1366 above conditions for all $i$ and also all |
1368 ball decompositions compatible with $x$. |
1367 ball decompositions compatible with $x$. |
1369 (If $x$ is a nice, non-pathological cell decomposition, then it is easy to see that gluing |
1368 (If $x$ is a nice, non-pathological cell decomposition, then it is easy to see that gluing |
1438 However, using the fact that $\bd y_i$ splits along $\bd Y$ and applying Axiom \ref{axiom:vcones}, |
1437 However, using the fact that $\bd y_i$ splits along $\bd Y$ and applying Axiom \ref{axiom:vcones}, |
1439 we can choose the decomposition $\du_{a} X_{ia}$ so that its restriction to $\bd W_i$ is a refinement |
1438 we can choose the decomposition $\du_{a} X_{ia}$ so that its restriction to $\bd W_i$ is a refinement |
1440 of the splitting along $\bd Y$, and this implies that the combined decomposition $\du_{ia} X_{ia}$ |
1439 of the splitting along $\bd Y$, and this implies that the combined decomposition $\du_{ia} X_{ia}$ |
1441 is permissible. |
1440 is permissible. |
1442 We can now define the gluing $y_1\bullet y_2$ in the obvious way, and a further application of Axiom \ref{axiom:vcones} |
1441 We can now define the gluing $y_1\bullet y_2$ in the obvious way, and a further application of Axiom \ref{axiom:vcones} |
1443 shows that this is independebt of the choices of representatives of $y_i$. |
1442 shows that this is independent of the choices of representatives of $y_i$. |
1444 |
1443 |
1445 |
1444 |
1446 \medskip |
1445 \medskip |
1447 |
1446 |
1448 We now give more concrete descriptions of the above colimits. |
1447 We now give more concrete descriptions of the above colimits. |
1452 \[ |
1451 \[ |
1453 \cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) \Bigg/ \sim , |
1452 \cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) \Bigg/ \sim , |
1454 \] |
1453 \] |
1455 where $x$ runs through decompositions of $W$, and $\sim$ is the obvious equivalence relation |
1454 where $x$ runs through decompositions of $W$, and $\sim$ is the obvious equivalence relation |
1456 induced by refinement and gluing. |
1455 induced by refinement and gluing. |
1457 If $\cC$ is enriched over vector spaces and $W$ is an $n$-manifold, |
1456 If $\cC$ is enriched over, for example, vector spaces and $W$ is an $n$-manifold, |
1458 we can take |
1457 we can take |
1459 \begin{equation*} |
1458 \begin{equation*} |
1460 \cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K, |
1459 \cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K, |
1461 \end{equation*} |
1460 \end{equation*} |
1462 where $K$ is the vector space spanned by elements $a - g(a)$, with |
1461 where $K$ is the vector space spanned by elements $a - g(a)$, with |