3 \section{TQFTs via fields} |
3 \section{TQFTs via fields} |
4 \label{sec:fields} |
4 \label{sec:fields} |
5 \label{sec:tqftsviafields} |
5 \label{sec:tqftsviafields} |
6 |
6 |
7 In this section we review the notion of a ``system of fields and local relations". |
7 In this section we review the notion of a ``system of fields and local relations". |
8 For more details see \cite{kw:tqft}. From a system of fields and local relations we can readily construct TQFT invariants of manifolds. This is described in \S \ref{sec:constructing-a-tqft}. A system of fields is very closely related to an $n$-category. In Example \ref{ex:traditional-n-categories(fields)}, which runs throughout this section, we sketch the construction of a system of fields from an $n$-category. We make this more precise for $n=1$ or $2$ in \S \ref{sec:example:traditional-n-categories(fields)}, and much later, after we've have given our own definition of a `topological $n$-category' in \S \ref{sec:ncats}, we explain precisely how to go back and forth between a topological $n$-category and a system of fields and local relations. |
8 For more details see \cite{kw:tqft}. |
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9 From a system of fields and local relations we can readily construct TQFT invariants of manifolds. |
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10 This is described in \S \ref{sec:constructing-a-tqft}. |
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11 A system of fields is very closely related to an $n$-category. |
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12 In Example \ref{ex:traditional-n-categories(fields)}, which runs throughout this section, |
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13 we sketch the construction of a system of fields from an $n$-category. |
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14 We make this more precise for $n=1$ or $2$ in \S \ref{sec:example:traditional-n-categories(fields)}, |
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15 and much later, after we've have given our own definition of a `topological $n$-category' in \S \ref{sec:ncats}, |
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16 we explain precisely how to go back and forth between a topological $n$-category and a system of fields and local relations. |
9 |
17 |
10 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
18 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
11 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
19 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
12 $\overline{X \setmin Y}$. |
20 $\overline{X \setmin Y}$. |
13 |
21 |
19 $k$ and morphisms homeomorphisms. |
27 $k$ and morphisms homeomorphisms. |
20 (We could equally well work with a different category of manifolds --- |
28 (We could equally well work with a different category of manifolds --- |
21 oriented, topological, smooth, spin, etc. --- but for definiteness we |
29 oriented, topological, smooth, spin, etc. --- but for definiteness we |
22 will stick with unoriented PL.) |
30 will stick with unoriented PL.) |
23 |
31 |
24 Fix a symmetric monoidal category $\cS$. While reading the definition, you should just think about the cases $\cS = \Set$ or $\cS = \Vect$. The presentation here requires that the objects of $\cS$ have an underlying set, but this could probably be avoided if desired. |
32 Fix a symmetric monoidal category $\cS$. |
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33 While reading the definition, you should just think about the cases $\cS = \Set$ or $\cS = \Vect$. |
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34 The presentation here requires that the objects of $\cS$ have an underlying set, but this could probably be avoided if desired. |
25 |
35 |
26 A $n$-dimensional {\it system of fields} in $\cS$ |
36 A $n$-dimensional {\it system of fields} in $\cS$ |
27 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
37 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
28 together with some additional data and satisfying some additional conditions, all specified below. |
38 together with some additional data and satisfying some additional conditions, all specified below. |
29 |
39 |
52 and these maps comprise a natural |
62 and these maps comprise a natural |
53 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
63 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
54 For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
64 For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
55 $\cC(X)$ which restricts to $c$. |
65 $\cC(X)$ which restricts to $c$. |
56 In this context, we will call $c$ a boundary condition. |
66 In this context, we will call $c$ a boundary condition. |
57 \item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields must be morphisms in $\cS$. |
67 \item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. |
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68 (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), |
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69 then this extra structure is considered part of the definition of $\cC_n$. |
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70 Any maps mentioned below between top level fields must be morphisms in $\cS$. |
58 \item $\cC_k$ is compatible with the symmetric monoidal |
71 \item $\cC_k$ is compatible with the symmetric monoidal |
59 structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
72 structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
60 compatibly with homeomorphisms and restriction to boundary. |
73 compatibly with homeomorphisms and restriction to boundary. |
61 We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
74 We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
62 restriction maps. |
75 restriction maps. |
183 above tensor products. |
196 above tensor products. |
184 |
197 |
185 |
198 |
186 \subsection{Systems of fields from $n$-categories} |
199 \subsection{Systems of fields from $n$-categories} |
187 \label{sec:example:traditional-n-categories(fields)} |
200 \label{sec:example:traditional-n-categories(fields)} |
188 We now describe in more detail Example \ref{ex:traditional-n-categories(fields)}, systems of fields coming from sub-cell-complexes labeled |
201 We now describe in more detail Example \ref{ex:traditional-n-categories(fields)}, |
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202 systems of fields coming from sub-cell-complexes labeled |
189 by $n$-category morphisms. |
203 by $n$-category morphisms. |
190 |
204 |
191 Given an $n$-category $C$ with the right sort of duality |
205 Given an $n$-category $C$ with the right sort of duality |
192 (e.g. a pivotal 2-category, 1-category with duals, star 1-category), |
206 (e.g. a pivotal 2-category, *-1-category), |
193 we can construct a system of fields as follows. |
207 we can construct a system of fields as follows. |
194 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
208 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
195 with codimension $i$ cells labeled by $i$-morphisms of $C$. |
209 with codimension $i$ cells labeled by $i$-morphisms of $C$. |
196 We'll spell this out for $n=1,2$ and then describe the general case. |
210 We'll spell this out for $n=1,2$ and then describe the general case. |
197 |
211 |
198 If $X$ has boundary, we require that the cell decompositions are in general |
212 If $X$ has boundary, we require that the cell decompositions are in general |
199 position with respect to the boundary --- the boundary intersects each cell |
213 position with respect to the boundary --- the boundary intersects each cell |
200 transversely, so cells meeting the boundary are mere half-cells. Put another way, the cell decompositions we consider are dual to standard cell |
214 transversely, so cells meeting the boundary are mere half-cells. |
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215 Put another way, the cell decompositions we consider are dual to standard cell |
201 decompositions of $X$. |
216 decompositions of $X$. |
202 |
217 |
203 We will always assume that our $n$-categories have linear $n$-morphisms. |
218 We will always assume that our $n$-categories have linear $n$-morphisms. |
204 |
219 |
205 For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
220 For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
268 |
283 |
269 |
284 |
270 |
285 |
271 \subsection{Local relations} |
286 \subsection{Local relations} |
272 \label{sec:local-relations} |
287 \label{sec:local-relations} |
273 Local relations are certain subspaces of the fields on balls, which form an ideal under gluing. Again, we give the examples first. |
288 Local relations are certain subspaces of the fields on balls, which form an ideal under gluing. |
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289 Again, we give the examples first. |
274 |
290 |
275 \addtocounter{prop}{-2} |
291 \addtocounter{prop}{-2} |
276 \begin{example}[contd.] |
292 \begin{example}[contd.] |
277 For maps into spaces, $U(B; c)$ is generated by fields of the form $a-b \in \lf(B; c)$, |
293 For maps into spaces, $U(B; c)$ is generated by fields of the form $a-b \in \lf(B; c)$, |
278 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
294 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
351 that lacks its $n{+}1$-dimensional part.) |
367 that lacks its $n{+}1$-dimensional part.) |
352 |
368 |
353 Let $Y$ be an $n{-}1$-manifold. |
369 Let $Y$ be an $n{-}1$-manifold. |
354 Define a (linear) 1-category $A(Y)$ as follows. |
370 Define a (linear) 1-category $A(Y)$ as follows. |
355 The objects of $A(Y)$ are $\cC(Y)$. |
371 The objects of $A(Y)$ are $\cC(Y)$. |
356 The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$, where $a$ and $b$ label the two boundary components of the cylinder $Y\times I$. |
372 The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$, |
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373 where $a$ and $b$ label the two boundary components of the cylinder $Y\times I$. |
357 Composition is given by gluing of cylinders. |
374 Composition is given by gluing of cylinders. |
358 |
375 |
359 Let $X$ be an $n$-manifold with boundary and consider the collection of vector spaces |
376 Let $X$ be an $n$-manifold with boundary and consider the collection of vector spaces |
360 $A(X; \cdot) \deq \{A(X; c)\}$ where $c$ ranges through $\cC(\bd X)$. |
377 $A(X; \cdot) \deq \{A(X; c)\}$ where $c$ ranges through $\cC(\bd X)$. |
361 This collection of vector spaces affords a representation of the category $A(\bd X)$, where |
378 This collection of vector spaces affords a representation of the category $A(\bd X)$, where |