376 such that $E$ is a $k{+}m$-ball, $X$ is a $k$-ball ($m\ge 1$), and $\pi$ is locally modeled |
376 such that $E$ is a $k{+}m$-ball, $X$ is a $k$-ball ($m\ge 1$), and $\pi$ is locally modeled |
377 on a standard iterated degeneracy map |
377 on a standard iterated degeneracy map |
378 \[ |
378 \[ |
379 d: \Delta^{k+m}\to\Delta^k . |
379 d: \Delta^{k+m}\to\Delta^k . |
380 \] |
380 \] |
381 In other words, \nn{each point has a neighborhood blah blah...} |
|
382 (We thank Kevin Costello for suggesting this approach.) |
381 (We thank Kevin Costello for suggesting this approach.) |
383 |
382 |
384 Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball, |
383 Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball, |
385 and for for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension |
384 and for for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension |
386 $l \le m$, with $l$ depending on $x$. |
385 $l \le m$, with $l$ depending on $x$. |
516 The last axiom (below), concerning actions of |
515 The last axiom (below), concerning actions of |
517 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
516 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
518 |
517 |
519 We start with the plain $n$-category case. |
518 We start with the plain $n$-category case. |
520 |
519 |
521 \begin{axiom}[Isotopy invariance in dimension $n$]{\textup{\textbf{[preliminary]}}} |
520 \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] |
522 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
521 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
523 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
522 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
524 Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. |
523 Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. |
525 \end{axiom} |
524 \end{axiom} |
526 |
525 |
590 isotopic (rel boundary) to the identity {\it extended isotopy}. |
589 isotopic (rel boundary) to the identity {\it extended isotopy}. |
591 |
590 |
592 The revised axiom is |
591 The revised axiom is |
593 |
592 |
594 \addtocounter{axiom}{-1} |
593 \addtocounter{axiom}{-1} |
595 \begin{axiom}{\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$.} |
594 \begin{axiom}[\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$.] |
596 \label{axiom:extended-isotopies} |
595 \label{axiom:extended-isotopies} |
597 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
596 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
598 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
597 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
599 Then $f$ acts trivially on $\cC(X)$. |
598 Then $f$ acts trivially on $\cC(X)$. |
600 In addition, collar maps act trivially on $\cC(X)$. |
599 In addition, collar maps act trivially on $\cC(X)$. |
608 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
607 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
609 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
608 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
610 |
609 |
611 |
610 |
612 \addtocounter{axiom}{-1} |
611 \addtocounter{axiom}{-1} |
613 \begin{axiom}{\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.} |
612 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
614 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
613 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
615 \[ |
614 \[ |
616 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
615 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
617 \] |
616 \] |
618 These action maps are required to be associative up to homotopy |
617 These action maps are required to be associative up to homotopy |
626 To do this we need to explain how collar maps form a topological space. |
625 To do this we need to explain how collar maps form a topological space. |
627 Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
626 Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
628 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
627 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
629 Having chains on the space of collar maps act gives rise to coherence maps involving |
628 Having chains on the space of collar maps act gives rise to coherence maps involving |
630 weak identities. |
629 weak identities. |
631 We will not pursue this in this draft of the paper. |
630 We will not pursue this in detail here. |
632 |
631 |
633 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
632 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
634 into a plain $n$-category (enriched over graded groups). |
633 into a plain $n$-category (enriched over graded groups). |
635 In a different direction, if we enrich over topological spaces instead of chain complexes, |
634 In a different direction, if we enrich over topological spaces instead of chain complexes, |
636 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
635 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
914 We will first define the ``cell-decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
913 We will first define the ``cell-decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
915 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
914 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
916 and we will define $\cC(W)$ as a suitable colimit |
915 and we will define $\cC(W)$ as a suitable colimit |
917 (or homotopy colimit in the $A_\infty$ case) of this functor. |
916 (or homotopy colimit in the $A_\infty$ case) of this functor. |
918 We'll later give a more explicit description of this colimit. |
917 We'll later give a more explicit description of this colimit. |
919 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data), |
918 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-balls with boundary data), |
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). |
919 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). |
921 |
920 |
922 \begin{defn} |
921 \begin{defn} |
923 Say that a ``permissible decomposition" of $W$ is a cell decomposition |
922 Say that a ``permissible decomposition" of $W$ is a cell decomposition |
924 \[ |
923 \[ |
969 (Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.) |
968 (Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.) |
970 Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we |
969 Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we |
971 fix a field on $\bd W$ |
970 fix a field on $\bd W$ |
972 (i.e. fix an element of the colimit associated to $\bd W$). |
971 (i.e. fix an element of the colimit associated to $\bd W$). |
973 |
972 |
974 Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
973 Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
975 |
974 |
976 \begin{defn}[System of fields functor] |
975 \begin{defn}[System of fields functor] |
977 \label{def:colim-fields} |
976 \label{def:colim-fields} |
978 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
977 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
979 That is, for each decomposition $x$ there is a map |
978 That is, for each decomposition $x$ there is a map |
1034 Then we kill the extra homology we just introduced with mapping |
1033 Then we kill the extra homology we just introduced with mapping |
1035 cylinders between the mapping cylinders (filtration degree 2), and so on. |
1034 cylinders between the mapping cylinders (filtration degree 2), and so on. |
1036 |
1035 |
1037 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1036 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1038 |
1037 |
1039 \todo{This next sentence is circular: these maps are an axiom, not a consequence of anything. -S} It is easy to see that |
1038 It is easy to see that |
1040 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
1039 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
1041 comprise a natural transformation of functors. |
1040 comprise a natural transformation of functors. |
1042 |
1041 |
1043 \begin{lem} |
1042 \begin{lem} |
1044 \label{lem:colim-injective} |
1043 \label{lem:colim-injective} |
1336 We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ |
1335 We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ |
1337 for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs |
1336 for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs |
1338 $(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all |
1337 $(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all |
1339 such maps modulo homotopies fixed on $\bdy B \setminus N$. |
1338 such maps modulo homotopies fixed on $\bdy B \setminus N$. |
1340 This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. |
1339 This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. |
|
1340 \end{example} |
1341 Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and |
1341 Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and |
1342 \ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to |
1342 \ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to |
1343 Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains. |
1343 Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains. |
1344 \end{example} |
|
1345 |
1344 |
1346 \subsection{Modules as boundary labels (colimits for decorated manifolds)} |
1345 \subsection{Modules as boundary labels (colimits for decorated manifolds)} |
1347 \label{moddecss} |
1346 \label{moddecss} |
1348 |
1347 |
1349 Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. |
1348 Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. |