377 such that $E$ is a $k{+}m$-ball, $X$ is a $k$-ball ($m\ge 1$), and $\pi$ is locally modeled |
377 such that $E$ is a $k{+}m$-ball, $X$ is a $k$-ball ($m\ge 1$), and $\pi$ is locally modeled |
378 on a standard iterated degeneracy map |
378 on a standard iterated degeneracy map |
379 \[ |
379 \[ |
380 d: \Delta^{k+m}\to\Delta^k . |
380 d: \Delta^{k+m}\to\Delta^k . |
381 \] |
381 \] |
382 In other words, \nn{each point has a neighborhood blah blah...} |
|
383 (We thank Kevin Costello for suggesting this approach.) |
382 (We thank Kevin Costello for suggesting this approach.) |
384 |
383 |
385 Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball, |
384 Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball, |
386 and for for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension |
385 and for for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension |
387 $l \le m$, with $l$ depending on $x$. |
386 $l \le m$, with $l$ depending on $x$. |
517 The last axiom (below), concerning actions of |
516 The last axiom (below), concerning actions of |
518 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
517 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
519 |
518 |
520 We start with the plain $n$-category case. |
519 We start with the plain $n$-category case. |
521 |
520 |
522 \begin{axiom}[Isotopy invariance in dimension $n$]{\textup{\textbf{[preliminary]}}} |
521 \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] |
523 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
522 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
524 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
523 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
525 Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. |
524 Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. |
526 \end{axiom} |
525 \end{axiom} |
527 |
526 |
591 isotopic (rel boundary) to the identity {\it extended isotopy}. |
590 isotopic (rel boundary) to the identity {\it extended isotopy}. |
592 |
591 |
593 The revised axiom is |
592 The revised axiom is |
594 |
593 |
595 \addtocounter{axiom}{-1} |
594 \addtocounter{axiom}{-1} |
596 \begin{axiom}{\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$.} |
595 \begin{axiom}[\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$.] |
597 \label{axiom:extended-isotopies} |
596 \label{axiom:extended-isotopies} |
598 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
597 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
599 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
598 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
600 Then $f$ acts trivially on $\cC(X)$. |
599 Then $f$ acts trivially on $\cC(X)$. |
601 In addition, collar maps act trivially on $\cC(X)$. |
600 In addition, collar maps act trivially on $\cC(X)$. |
609 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
608 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
610 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
609 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
611 |
610 |
612 |
611 |
613 \addtocounter{axiom}{-1} |
612 \addtocounter{axiom}{-1} |
614 \begin{axiom}{\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.} |
613 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
615 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
614 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
616 \[ |
615 \[ |
617 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
616 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
618 \] |
617 \] |
619 These action maps are required to be associative up to homotopy |
618 These action maps are required to be associative up to homotopy |
627 To do this we need to explain how collar maps form a topological space. |
626 To do this we need to explain how collar maps form a topological space. |
628 Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
627 Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
629 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
628 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
630 Having chains on the space of collar maps act gives rise to coherence maps involving |
629 Having chains on the space of collar maps act gives rise to coherence maps involving |
631 weak identities. |
630 weak identities. |
632 We will not pursue this in this draft of the paper. |
631 We will not pursue this in detail here. |
633 |
632 |
634 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
633 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
635 into a plain $n$-category (enriched over graded groups). |
634 into a plain $n$-category (enriched over graded groups). |
636 In a different direction, if we enrich over topological spaces instead of chain complexes, |
635 In a different direction, if we enrich over topological spaces instead of chain complexes, |
637 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
636 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
915 We will first define the ``cell-decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
914 We will first define the ``cell-decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
916 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
915 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
917 and we will define $\cl{\cC}(W)$ as a suitable colimit |
916 and we will define $\cl{\cC}(W)$ as a suitable colimit |
918 (or homotopy colimit in the $A_\infty$ case) of this functor. |
917 (or homotopy colimit in the $A_\infty$ case) of this functor. |
919 We'll later give a more explicit description of this colimit. |
918 We'll later give a more explicit description of this colimit. |
920 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data), |
919 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-balls with boundary data), |
921 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex). |
920 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex). |
922 |
921 |
923 Define a {\it permissible decomposition} of $W$ to be a cell decomposition |
922 Define a {\it permissible decomposition} of $W$ to be a cell decomposition |
924 \[ |
923 \[ |
925 W = \bigcup_a X_a , |
924 W = \bigcup_a X_a , |
973 means the subset of $\cC(X_a)$ whose restriction to $\bd X_a$ agress with $\beta$. |
972 means the subset of $\cC(X_a)$ whose restriction to $\bd X_a$ agress with $\beta$. |
974 If we are enriching over $\cS$ and $k=n$, then $\cC(X_a; \beta)$ is an object in |
973 If we are enriching over $\cS$ and $k=n$, then $\cC(X_a; \beta)$ is an object in |
975 $\cS$ and the coproduct and product in Equation \ref{eq:psi-CC} should be replaced by the approriate |
974 $\cS$ and the coproduct and product in Equation \ref{eq:psi-CC} should be replaced by the approriate |
976 operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect). |
975 operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect). |
977 |
976 |
978 Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
977 Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
979 |
978 |
980 \begin{defn}[System of fields functor] |
979 \begin{defn}[System of fields functor] |
981 \label{def:colim-fields} |
980 \label{def:colim-fields} |
982 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
981 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
983 That is, for each decomposition $x$ there is a map |
982 That is, for each decomposition $x$ there is a map |
1038 Then we kill the extra homology we just introduced with mapping |
1037 Then we kill the extra homology we just introduced with mapping |
1039 cylinders between the mapping cylinders (filtration degree 2), and so on. |
1038 cylinders between the mapping cylinders (filtration degree 2), and so on. |
1040 |
1039 |
1041 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1040 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1042 |
1041 |
1043 \todo{This next sentence is circular: these maps are an axiom, not a consequence of anything. -S} It is easy to see that |
1042 It is easy to see that |
1044 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
1043 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
1045 comprise a natural transformation of functors. |
1044 comprise a natural transformation of functors. |
1046 |
1045 |
1047 \begin{lem} |
1046 \begin{lem} |
1048 \label{lem:colim-injective} |
1047 \label{lem:colim-injective} |
1340 We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ |
1339 We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ |
1341 for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs |
1340 for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs |
1342 $(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all |
1341 $(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all |
1343 such maps modulo homotopies fixed on $\bdy B \setminus N$. |
1342 such maps modulo homotopies fixed on $\bdy B \setminus N$. |
1344 This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. |
1343 This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. |
|
1344 \end{example} |
1345 Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and |
1345 Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and |
1346 \ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to |
1346 \ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to |
1347 Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains. |
1347 Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains. |
1348 \end{example} |
|
1349 |
1348 |
1350 \subsection{Modules as boundary labels (colimits for decorated manifolds)} |
1349 \subsection{Modules as boundary labels (colimits for decorated manifolds)} |
1351 \label{moddecss} |
1350 \label{moddecss} |
1352 |
1351 |
1353 Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. |
1352 Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. |