2 |
2 |
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 construction of TQFTs from ``topological fields". |
7 In this section we review the notion of a ``system of fields and local relations". |
8 For more details see \cite{kw:tqft}. |
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. |
9 |
9 |
10 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
10 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 |
11 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
12 $\overline{X \setmin Y}$. |
12 $\overline{X \setmin Y}$. |
13 |
13 |
19 $k$ and morphisms homeomorphisms. |
19 $k$ and morphisms homeomorphisms. |
20 (We could equally well work with a different category of manifolds --- |
20 (We could equally well work with a different category of manifolds --- |
21 oriented, topological, smooth, spin, etc. --- but for definiteness we |
21 oriented, topological, smooth, spin, etc. --- but for definiteness we |
22 will stick with unoriented PL.) |
22 will stick with unoriented PL.) |
23 |
23 |
24 %Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$. |
24 Fix a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the cases $\cS = \Set$ or $\cS = \Vect$. |
25 |
25 |
26 A $n$-dimensional {\it system of fields} in $\cS$ |
26 A $n$-dimensional {\it system of fields} in $\cS$ |
27 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
27 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. |
28 together with some additional data and satisfying some additional conditions, all specified below. |
29 |
29 |
30 Before finishing the definition of fields, we give two motivating examples |
30 Before finishing the definition of fields, we give two motivating examples of systems of fields. |
31 (actually, families of examples) of systems of fields. |
|
32 |
31 |
33 \begin{example} |
32 \begin{example} |
34 \label{ex:maps-to-a-space(fields)} |
33 \label{ex:maps-to-a-space(fields)} |
35 Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps |
34 Fix a target space $T$, and let $\cC(X)$ be the set of continuous maps |
36 from X to $B$. |
35 from X to $B$. |
37 \end{example} |
36 \end{example} |
38 |
37 |
39 \begin{example} |
38 \begin{example} |
40 \label{ex:traditional-n-categories(fields)} |
39 \label{ex:traditional-n-categories(fields)} |
41 Fix an $n$-category $C$, and let $\cC(X)$ be |
40 Fix an $n$-category $C$, and let $\cC(X)$ be |
42 the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by |
41 the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by |
43 $j$-morphisms of $C$. |
42 $j$-morphisms of $C$. |
44 One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. |
43 One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. |
45 This is described in more detail below. |
44 This is described in more detail in \S \ref{sec:example:traditional-n-categories(fields)}. |
46 \end{example} |
45 \end{example} |
47 |
46 |
48 Now for the rest of the definition of system of fields. |
47 Now for the rest of the definition of system of fields. |
49 \begin{enumerate} |
48 \begin{enumerate} |
50 \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
49 \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
142 |
141 |
143 |
142 |
144 \nn{remark that if top dimensional fields are not already linear |
143 \nn{remark that if top dimensional fields are not already linear |
145 then we will soon linearize them(?)} |
144 then we will soon linearize them(?)} |
146 |
145 |
|
146 For top dimensional ($n$-dimensional) manifolds, we're actually interested |
|
147 in the linearized space of fields. |
|
148 By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is |
|
149 the vector space of finite |
|
150 linear combinations of fields on $X$. |
|
151 If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$. |
|
152 Thus the restriction (to boundary) maps are well defined because we never |
|
153 take linear combinations of fields with differing boundary conditions. |
|
154 |
|
155 In some cases we don't linearize the default way; instead we take the |
|
156 spaces $\lf(X; a)$ to be part of the data for the system of fields. |
|
157 In particular, for fields based on linear $n$-category pictures we linearize as follows. |
|
158 Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
|
159 obvious relations on 0-cell labels. |
|
160 More specifically, let $L$ be a cell decomposition of $X$ |
|
161 and let $p$ be a 0-cell of $L$. |
|
162 Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
|
163 $\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
|
164 Then the subspace $K$ is generated by things of the form |
|
165 $\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader |
|
166 to infer the meaning of $\alpha_{\lambda c + d}$. |
|
167 Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. |
|
168 |
|
169 \nn{Maybe comment further: if there's a natural basis of morphisms, then no need; |
|
170 will do something similar below; in general, whenever a label lives in a linear |
|
171 space we do something like this; ? say something about tensor |
|
172 product of all the linear label spaces? Yes:} |
|
173 |
|
174 For top dimensional ($n$-dimensional) manifolds, we linearize as follows. |
|
175 Define an ``almost-field" to be a field without labels on the 0-cells. |
|
176 (Recall that 0-cells are labeled by $n$-morphisms.) |
|
177 To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
|
178 space determined by the labeling of the link of the 0-cell. |
|
179 (If the 0-cell were labeled, the label would live in this space.) |
|
180 We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
|
181 We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the |
|
182 above tensor products. |
|
183 |
|
184 |
|
185 \subsection{Systems of fields from $n$-categories} |
|
186 \label{sec:example:traditional-n-categories(fields)} |
147 We now describe in more detail systems of fields coming from sub-cell-complexes labeled |
187 We now describe in more detail systems of fields coming from sub-cell-complexes labeled |
148 by $n$-category morphisms. |
188 by $n$-category morphisms. |
149 |
189 |
150 Given an $n$-category $C$ with the right sort of duality |
190 Given an $n$-category $C$ with the right sort of duality |
151 (e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
191 (e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
224 %be treated differently (segregated) from other local relations, but I'm not sure |
264 %be treated differently (segregated) from other local relations, but I'm not sure |
225 %the next definition is the best way to do it} |
265 %the next definition is the best way to do it} |
226 |
266 |
227 \medskip |
267 \medskip |
228 |
268 |
229 For top dimensional ($n$-dimensional) manifolds, we're actually interested |
|
230 in the linearized space of fields. |
|
231 By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is |
|
232 the vector space of finite |
|
233 linear combinations of fields on $X$. |
|
234 If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$. |
|
235 Thus the restriction (to boundary) maps are well defined because we never |
|
236 take linear combinations of fields with differing boundary conditions. |
|
237 |
|
238 In some cases we don't linearize the default way; instead we take the |
|
239 spaces $\lf(X; a)$ to be part of the data for the system of fields. |
|
240 In particular, for fields based on linear $n$-category pictures we linearize as follows. |
|
241 Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
|
242 obvious relations on 0-cell labels. |
|
243 More specifically, let $L$ be a cell decomposition of $X$ |
|
244 and let $p$ be a 0-cell of $L$. |
|
245 Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
|
246 $\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
|
247 Then the subspace $K$ is generated by things of the form |
|
248 $\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader |
|
249 to infer the meaning of $\alpha_{\lambda c + d}$. |
|
250 Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. |
|
251 |
|
252 \nn{Maybe comment further: if there's a natural basis of morphisms, then no need; |
|
253 will do something similar below; in general, whenever a label lives in a linear |
|
254 space we do something like this; ? say something about tensor |
|
255 product of all the linear label spaces? Yes:} |
|
256 |
|
257 For top dimensional ($n$-dimensional) manifolds, we linearize as follows. |
|
258 Define an ``almost-field" to be a field without labels on the 0-cells. |
|
259 (Recall that 0-cells are labeled by $n$-morphisms.) |
|
260 To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
|
261 space determined by the labeling of the link of the 0-cell. |
|
262 (If the 0-cell were labeled, the label would live in this space.) |
|
263 We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
|
264 We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the |
|
265 above tensor products. |
|
266 |
269 |
267 |
270 |
268 |
271 |
269 \subsection{Local relations} |
272 \subsection{Local relations} |
270 \label{sec:local-relations} |
273 \label{sec:local-relations} |