9 \subsection{Definition of $n$-categories} |
9 \subsection{Definition of $n$-categories} |
10 \label{ss:n-cat-def} |
10 \label{ss:n-cat-def} |
11 |
11 |
12 Before proceeding, we need more appropriate definitions of $n$-categories, |
12 Before proceeding, we need more appropriate definitions of $n$-categories, |
13 $A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
13 $A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
14 (As is the case throughout this paper, by ``$n$-category" we implicitly intend some notion of |
14 (As is the case throughout this paper, by ``$n$-category" we mean some notion of |
15 a `weak' $n$-category with `strong duality'.) |
15 a ``weak" $n$-category with ``strong duality".) |
16 |
16 |
17 The definitions presented below tie the categories more closely to the topology |
17 The definitions presented below tie the categories more closely to the topology |
18 and avoid combinatorial questions about, for example, the minimal sufficient |
18 and avoid combinatorial questions about, for example, the minimal sufficient |
19 collections of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. |
19 collections of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. |
20 For examples of topological origin, it is typically easy to show that they |
20 For examples of topological origin |
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21 (e.g.\ categories whose morphisms are maps into spaces or decorated balls), |
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22 it is easy to show that they |
21 satisfy our axioms. |
23 satisfy our axioms. |
22 For examples of a more purely algebraic origin, one would typically need the combinatorial |
24 For examples of a more purely algebraic origin, one would typically need the combinatorial |
23 results that we have avoided here. |
25 results that we have avoided here. |
24 |
26 |
25 \medskip |
27 \medskip |
34 (This allows for strict associativity.) |
36 (This allows for strict associativity.) |
35 Still other definitions (see, for example, \cite{MR2094071}) |
37 Still other definitions (see, for example, \cite{MR2094071}) |
36 model the $k$-morphisms on more complicated combinatorial polyhedra. |
38 model the $k$-morphisms on more complicated combinatorial polyhedra. |
37 |
39 |
38 For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball. |
40 For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball. |
39 Thus we expect to associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |
41 Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |
40 to the standard $k$-ball. |
42 to the standard $k$-ball. |
41 By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
43 By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
42 standard $k$-ball. |
44 standard $k$-ball. |
43 We {\it do not} assume that it is equipped with a |
45 We {\it do not} assume that it is equipped with a |
44 preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
46 preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
56 \end{axiom} |
58 \end{axiom} |
57 |
59 |
58 |
60 |
59 (Note: We usually omit the subscript $k$.) |
61 (Note: We usually omit the subscript $k$.) |
60 |
62 |
61 We are so far being deliberately vague about what flavor of $k$-balls |
63 We are being deliberately vague about what flavor of $k$-balls |
62 we are considering. |
64 we are considering. |
63 They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
65 They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
64 They could be topological or PL or smooth. |
66 They could be topological or PL or smooth. |
65 %\nn{need to check whether this makes much difference} |
67 %\nn{need to check whether this makes much difference} |
66 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
68 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
67 to be fussier about corners and boundaries.) |
69 to be fussier about corners and boundaries.) |
68 For each flavor of manifold there is a corresponding flavor of $n$-category. |
70 For each flavor of manifold there is a corresponding flavor of $n$-category. |
69 We will concentrate on the case of PL unoriented manifolds. |
71 For simplicity, we will concentrate on the case of PL unoriented manifolds. |
70 |
72 |
71 (The ambitious reader may want to keep in mind two other classes of balls. |
73 (The ambitious reader may want to keep in mind two other classes of balls. |
72 The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). |
74 The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). |
73 This will be used below to describe the blob complex of a fiber bundle with |
75 This will be used below to describe the blob complex of a fiber bundle with |
74 base space $Y$. |
76 base space $Y$. |
75 The second is balls equipped with a section of the the tangent bundle, or the frame |
77 The second is balls equipped with a section of the tangent bundle, or the frame |
76 bundle (i.e.\ framed balls), or more generally some flag bundle associated to the tangent bundle. |
78 bundle (i.e.\ framed balls), or more generally some flag bundle associated to the tangent bundle. |
77 These can be used to define categories with less than the ``strong" duality we assume here, |
79 These can be used to define categories with less than the ``strong" duality we assume here, |
78 though we will not develop that idea fully in this paper.) |
80 though we will not develop that idea fully in this paper.) |
79 |
81 |
80 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
82 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
92 boundary of a morphism. |
94 boundary of a morphism. |
93 Morphisms are modeled on balls, so their boundaries are modeled on spheres. |
95 Morphisms are modeled on balls, so their boundaries are modeled on spheres. |
94 In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for |
96 In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for |
95 $1\le k \le n$. |
97 $1\le k \le n$. |
96 At first it might seem that we need another axiom for this, but in fact once we have |
98 At first it might seem that we need another axiom for this, but in fact once we have |
97 all the axioms in the subsection for $0$ through $k-1$ we can use a colimit |
99 all the axioms in this subsection for $0$ through $k-1$ we can use a colimit |
98 construction, as described in Subsection \ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ |
100 construction, as described in Subsection \ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ |
99 to spheres (and any other manifolds): |
101 to spheres (and any other manifolds): |
100 |
102 |
101 \begin{lem} |
103 \begin{lem} |
102 \label{lem:spheres} |
104 \label{lem:spheres} |
105 homeomorphisms to the category of sets and bijections. |
107 homeomorphisms to the category of sets and bijections. |
106 \end{lem} |
108 \end{lem} |
107 |
109 |
108 We postpone the proof of this result until after we've actually given all the axioms. |
110 We postpone the proof of this result until after we've actually given all the axioms. |
109 Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, |
111 Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, |
110 along with the data described in the other Axioms at lower levels. |
112 along with the data described in the other axioms at lower levels. |
111 |
113 |
112 %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
114 %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
113 |
115 |
114 \begin{axiom}[Boundaries]\label{nca-boundary} |
116 \begin{axiom}[Boundaries]\label{nca-boundary} |
115 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
117 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
129 category structure. |
131 category structure. |
130 Note that this auxiliary structure is only in dimension $n$; |
132 Note that this auxiliary structure is only in dimension $n$; |
131 $\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. |
133 $\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. |
132 |
134 |
133 \medskip |
135 \medskip |
134 \nn{ |
136 |
135 %At the moment I'm a little confused about orientations, and more specifically |
137 (In order to simplify the exposition we have concentrated on the case of |
136 %about the role of orientation-reversing maps of boundaries when gluing oriented manifolds. |
138 unoriented PL manifolds and avoided the question of what exactly we mean by |
137 Maybe need a discussion about what the boundary of a manifold with a |
139 the boundary a manifold with extra structure, such as an oriented manifold. |
138 structure (e.g. orientation) means. |
140 In general, all manifolds of dimension less than $n$ should be equipped with the germ |
139 Tentatively, I think we need to redefine the oriented boundary of an oriented $n$-manifold. |
141 of a thickening to dimension $n$, and this germ should carry whatever structure we have |
140 Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal |
142 on $n$-manifolds. |
141 first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold |
143 In addition, lower dimensional manifolds should be equipped with a framing |
142 equipped with an orientation of its once-stabilized tangent bundle. |
144 of their normal bundle in the thickening; the framing keeps track of which |
143 Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of |
145 side (iterated) bounded manifolds lie on. |
144 their $k$ times stabilized tangent bundles. |
146 For example, the boundary of an oriented $n$-ball |
145 (cf. \cite{MR2079378}.) |
147 should be an $n{-}1$-sphere equipped with an orientation of its once stabilized tangent |
146 Probably should also have a framing of the stabilized dimensions in order to indicate which |
148 bundle and a choice of direction in this bundle indicating |
147 side the bounded manifold is on. |
149 which side the $n$-ball lies on.) |
148 For the moment just stick with unoriented manifolds.} |
150 |
149 \medskip |
151 \medskip |
150 |
152 |
151 We have just argued that the boundary of a morphism has no preferred splitting into |
153 We have just argued that the boundary of a morphism has no preferred splitting into |
152 domain and range, but the converse meets with our approval. |
154 domain and range, but the converse meets with our approval. |
153 That is, given compatible domain and range, we should be able to combine them into |
155 That is, given compatible domain and range, we should be able to combine them into |
172 |
174 |
173 \begin{figure}[!ht] |
175 \begin{figure}[!ht] |
174 $$ |
176 $$ |
175 \begin{tikzpicture}[%every label/.style={green} |
177 \begin{tikzpicture}[%every label/.style={green} |
176 ] |
178 ] |
177 \node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {}; |
179 \node[fill=black, circle, label=below:$E$, inner sep=1.5pt](S) at (0,0) {}; |
178 \node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {}; |
180 \node[fill=black, circle, label=above:$E$, inner sep=1.5pt](N) at (0,2) {}; |
179 \draw (S) arc (-90:90:1); |
181 \draw (S) arc (-90:90:1); |
180 \draw (N) arc (90:270:1); |
182 \draw (N) arc (90:270:1); |
181 \node[left] at (-1,1) {$B_1$}; |
183 \node[left] at (-1,1) {$B_1$}; |
182 \node[right] at (1,1) {$B_2$}; |
184 \node[right] at (1,1) {$B_2$}; |
183 \end{tikzpicture} |
185 \end{tikzpicture} |
184 $$ |
186 $$ |
185 \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
187 \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
186 |
188 |
187 Note that we insist on injectivity above. \todo{Make sure we prove this, as a consequence of the next axiom, later.} |
189 Note that we insist on injectivity above. |
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190 The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. |
188 |
191 |
189 Let $\cl{\cC}(S)_E$ denote the image of $\gl_E$. |
192 Let $\cl{\cC}(S)_E$ denote the image of $\gl_E$. |
190 We will refer to elements of $\cl{\cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
193 We will refer to elements of $\cl{\cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
191 |
194 |
192 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
195 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
259 %Compositions of boundary and restriction maps will also be called restriction maps. |
262 %Compositions of boundary and restriction maps will also be called restriction maps. |
260 %For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
263 %For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
261 %restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
264 %restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
262 |
265 |
263 We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
266 We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
264 We will call elements of $\cC(B)_Y$ morphisms which are `splittable along $Y$' or `transverse to $Y$'. |
267 We will call elements of $\cC(B)_Y$ morphisms which are |
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268 ``splittable along $Y$'' or ``transverse to $Y$''. |
265 We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
269 We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
266 |
270 |
267 More generally, let $\alpha$ be a subdivision of a ball $X$ into smaller balls. |
271 More generally, let $\alpha$ be a subdivision of a ball $X$ into smaller balls. |
268 Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
272 Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
269 the smaller balls to $X$. |
273 the smaller balls to $X$. |
296 For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, |
300 For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, |
297 usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
301 usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
298 These maps must satisfy the following conditions. |
302 These maps must satisfy the following conditions. |
299 \begin{enumerate} |
303 \begin{enumerate} |
300 \item |
304 \item |
301 If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram |
305 If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are homeomorphisms such that the diagram |
302 \[ \xymatrix{ |
306 \[ \xymatrix{ |
303 X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
307 X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
304 X \ar[r]^{f} & X' |
308 X \ar[r]^{f} & X' |
305 } \] |
309 } \] |
306 commutes, then we have |
310 commutes, then we have |
361 \draw[->,red,line width=2pt] (-2.83,-1.5) -- (-2.83,-2.5); |
365 \draw[->,red,line width=2pt] (-2.83,-1.5) -- (-2.83,-2.5); |
362 \end{tikzpicture} |
366 \end{tikzpicture} |
363 $$ |
367 $$ |
364 \caption{Examples of pinched products}\label{pinched_prods} |
368 \caption{Examples of pinched products}\label{pinched_prods} |
365 \end{figure} |
369 \end{figure} |
366 (The need for a strengthened version will become apparent in appendix \ref{sec:comparing-defs} |
370 (The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs} |
367 where we construct a traditional category from a topological category.) |
371 where we construct a traditional category from a topological category.) |
368 Define a {\it pinched product} to be a map |
372 Define a {\it pinched product} to be a map |
369 \[ |
373 \[ |
370 \pi: E\to X |
374 \pi: E\to X |
371 \] |
375 \] |
523 This axiom needs to be strengthened to force product morphisms to act as the identity. |
527 This axiom needs to be strengthened to force product morphisms to act as the identity. |
524 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
528 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
525 Let $J$ be a 1-ball (interval). |
529 Let $J$ be a 1-ball (interval). |
526 We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
530 We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
527 (Here we use the ``pinched" version of $Y\times J$. |
531 (Here we use the ``pinched" version of $Y\times J$. |
528 \nn{need notation for this}) |
532 \nn{do we need notation for this?}) |
529 We define a map |
533 We define a map |
530 \begin{eqnarray*} |
534 \begin{eqnarray*} |
531 \psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
535 \psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
532 a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
536 a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
533 \end{eqnarray*} |
537 \end{eqnarray*} |
575 \begin{equation*} |
579 \begin{equation*} |
576 \xymatrix@C+2cm{\cC(X) \ar[r]^(0.45){\text{glue}} & \cC(X \cup \text{collar}) \ar[r]^(0.55){\text{homeo}} & \cC(X)} |
580 \xymatrix@C+2cm{\cC(X) \ar[r]^(0.45){\text{glue}} & \cC(X \cup \text{collar}) \ar[r]^(0.55){\text{homeo}} & \cC(X)} |
577 \end{equation*} |
581 \end{equation*} |
578 |
582 |
579 \caption{Extended homeomorphism.}\label{glue-collar}\end{figure} |
583 \caption{Extended homeomorphism.}\label{glue-collar}\end{figure} |
580 We say that $\psi_{Y,J}$ is {\it extended isotopic} to the identity map. |
584 We call a map of this form a {\it collar map}. |
581 \nn{bad terminology; fix it later} |
|
582 \nn{also need to make clear that plain old isotopic to the identity implies |
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583 extended isotopic} |
|
584 \nn{maybe remark that in some examples (e.g.\ ones based on sub cell complexes) |
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585 extended isotopies are also plain isotopies, so |
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586 no extension necessary} |
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587 It can be thought of as the action of the inverse of |
585 It can be thought of as the action of the inverse of |
588 a map which projects a collar neighborhood of $Y$ onto $Y$. |
586 a map which projects a collar neighborhood of $Y$ onto $Y$, |
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587 or as the limit of homeomorphisms $X\to X$ which expand a very thin collar of $Y$ |
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588 to a larger collar. |
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589 We call the equivalence relation generated by collar maps and homeomorphisms |
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590 isotopic (rel boundary) to the identity {\it extended isotopy}. |
589 |
591 |
590 The revised axiom is |
592 The revised axiom is |
591 |
593 |
592 \addtocounter{axiom}{-1} |
594 \addtocounter{axiom}{-1} |
593 \begin{axiom}{\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$} |
595 \begin{axiom}{\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$.} |
594 \label{axiom:extended-isotopies} |
596 \label{axiom:extended-isotopies} |
595 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 |
596 to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity. |
598 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
597 Then $f$ acts trivially on $\cC(X)$. |
599 Then $f$ acts trivially on $\cC(X)$. |
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600 In addition, collar maps act trivially on $\cC(X)$. |
598 \end{axiom} |
601 \end{axiom} |
599 |
|
600 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
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601 |
602 |
602 \smallskip |
603 \smallskip |
603 |
604 |
604 For $A_\infty$ $n$-categories, we replace |
605 For $A_\infty$ $n$-categories, we replace |
605 isotopy invariance with the requirement that families of homeomorphisms act. |
606 isotopy invariance with the requirement that families of homeomorphisms act. |
959 (i.e. fix an element of the colimit associated to $\bd W$). |
960 (i.e. fix an element of the colimit associated to $\bd W$). |
960 |
961 |
961 Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
962 Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
962 |
963 |
963 \begin{defn}[System of fields functor] |
964 \begin{defn}[System of fields functor] |
|
965 \label{def:colim-fields} |
964 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
966 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
965 That is, for each decomposition $x$ there is a map |
967 That is, for each decomposition $x$ there is a map |
966 $\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps |
968 $\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps |
967 above, and $\cl{\cC}(W)$ is universal with respect to these properties. |
969 above, and $\cl{\cC}(W)$ is universal with respect to these properties. |
968 \end{defn} |
970 \end{defn} |
1024 |
1026 |
1025 \todo{This next sentence is circular: these maps are an axiom, not a consequence of anything. -S} It is easy to see that |
1027 \todo{This next sentence is circular: these maps are an axiom, not a consequence of anything. -S} It is easy to see that |
1026 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
1028 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
1027 comprise a natural transformation of functors. |
1029 comprise a natural transformation of functors. |
1028 |
1030 |
1029 \todo{Explicitly say somewhere: `this proves Lemma \ref{lem:domain-and-range}'} |
1031 \begin{lem} |
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1032 \label{lem:colim-injective} |
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1033 Let $W$ be a manifold of dimension less than $n$. Then for each |
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1034 decomposition $x$ of $W$ the natural map $\psi_{\cC;W}(x)\to \cl{\cC}(W)$ is injective. |
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1035 \end{lem} |
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1036 \begin{proof} |
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1037 \nn{...} |
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1038 \end{proof} |
1030 |
1039 |
1031 \nn{need to finish explaining why we have a system of fields; |
1040 \nn{need to finish explaining why we have a system of fields; |
1032 need to say more about ``homological" fields? |
1041 need to say more about ``homological" fields? |
1033 (actions of homeomorphisms); |
1042 (actions of homeomorphisms); |
1034 define $k$-cat $\cC(\cdot\times W)$} |
1043 define $k$-cat $\cC(\cdot\times W)$} |