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 notion of a ``system of fields and local relations". |
7 In this section we review the construction of TQFTs from fields and local relations. |
8 For more details see \cite{kw:tqft}. |
8 For more details see \cite{kw:tqft}. |
9 From a system of fields and local relations we can readily construct TQFT invariants of manifolds. |
9 For our purposes, a TQFT is {\it defined} to be something which arises |
10 This is described in \S \ref{sec:constructing-a-tqft}. |
10 from this construction. |
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11 This is an alternative to the more common definition of a TQFT |
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12 as a functor on cobordism categories satisfying various conditions. |
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13 A fully local (``down to points") version of the cobordism-functor TQFT definition |
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14 should be equivalent to the fields-and-local-relations definition. |
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15 |
11 A system of fields is very closely related to an $n$-category. |
16 A system of fields is very closely related to an $n$-category. |
12 In Example \ref{ex:traditional-n-categories(fields)}, which runs throughout this section, |
17 In one direction, Example \ref{ex:traditional-n-categories(fields)} |
13 we sketch the construction of a system of fields from an $n$-category. |
18 shows how to construct a system of fields from a (traditional) $n$-category. |
14 We make this more precise for $n=1$ or $2$ in \S \ref{sec:example:traditional-n-categories(fields)}, |
19 We do this in detail for $n=1,2$ (Subsection \ref{sec:example:traditional-n-categories(fields)}) |
15 and much later, after we've have given our own definition of a `topological $n$-category' in \S \ref{sec:ncats}, |
20 and more informally for general $n$. |
16 we explain precisely how to go back and forth between a topological $n$-category and a system of fields and local relations. |
21 In the other direction, |
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22 our preferred definition of an $n$-category in Section \ref{sec:ncats} is essentially |
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23 just a system of fields restricted to balls of dimensions 0 through $n$; |
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24 one could call this the ``local" part of a system of fields. |
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25 |
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26 Since this section is intended primarily to motivate |
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27 the blob complex construction of Section \ref{sec:blob-definition}, |
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28 we suppress some technical details. |
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29 In Section \ref{sec:ncats} the analogous details are treated more carefully. |
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30 |
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31 \medskip |
17 |
32 |
18 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
33 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
19 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
34 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
20 $\overline{X \setmin Y}$. |
35 $\overline{X \setmin Y}$. |
21 |
36 |
24 |
39 |
25 Let $\cM_k$ denote the category with objects |
40 Let $\cM_k$ denote the category with objects |
26 unoriented PL manifolds of dimension |
41 unoriented PL manifolds of dimension |
27 $k$ and morphisms homeomorphisms. |
42 $k$ and morphisms homeomorphisms. |
28 (We could equally well work with a different category of manifolds --- |
43 (We could equally well work with a different category of manifolds --- |
29 oriented, topological, smooth, spin, etc. --- but for definiteness we |
44 oriented, topological, smooth, spin, etc. --- but for simplicity we |
30 will stick with unoriented PL.) |
45 will stick with unoriented PL.) |
31 |
46 |
32 Fix a symmetric monoidal category $\cS$. |
47 Fix a symmetric monoidal category $\cS$. |
33 While reading the definition, you should just think about the cases $\cS = \Set$ or $\cS = \Vect$. |
48 Fields on $n$-manifolds will be enriched over $\cS$. |
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49 Good examples to keep in mind are $\cS = \Set$ or $\cS = \Vect$. |
34 The presentation here requires that the objects of $\cS$ have an underlying set, but this could probably be avoided if desired. |
50 The presentation here requires that the objects of $\cS$ have an underlying set, but this could probably be avoided if desired. |
35 |
51 |
36 A $n$-dimensional {\it system of fields} in $\cS$ |
52 A $n$-dimensional {\it system of fields} in $\cS$ |
37 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
53 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
38 together with some additional data and satisfying some additional conditions, all specified below. |
54 together with some additional data and satisfying some additional conditions, all specified below. |
62 and these maps comprise a natural |
78 and these maps comprise a natural |
63 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
79 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
64 For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
80 For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
65 $\cC(X)$ which restricts to $c$. |
81 $\cC(X)$ which restricts to $c$. |
66 In this context, we will call $c$ a boundary condition. |
82 In this context, we will call $c$ a boundary condition. |
67 \item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. |
83 \item The subset $\cC_n(X;c)$ of top-dimensional fields |
68 (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), |
84 with a given boundary condition is an object in our symmetric monoidal category $\cS$. |
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85 (This condition is of course trivial when $\cS = \Set$.) |
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86 If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), |
69 then this extra structure is considered part of the definition of $\cC_n$. |
87 then this extra structure is considered part of the definition of $\cC_n$. |
70 Any maps mentioned below between top level fields must be morphisms in $\cS$. |
88 Any maps mentioned below between fields on $n$-manifolds must be morphisms in $\cS$. |
71 \item $\cC_k$ is compatible with the symmetric monoidal |
89 \item $\cC_k$ is compatible with the symmetric monoidal |
72 structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
90 structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
73 compatibly with homeomorphisms and restriction to boundary. |
91 compatibly with homeomorphisms and restriction to boundary. |
74 We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
92 We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
75 restriction maps. |
93 restriction maps. |
84 \[ |
102 \[ |
85 \Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) , |
103 \Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) , |
86 \] |
104 \] |
87 and this gluing map is compatible with all of the above structure (actions |
105 and this gluing map is compatible with all of the above structure (actions |
88 of homeomorphisms, boundary restrictions, disjoint union). |
106 of homeomorphisms, boundary restrictions, disjoint union). |
89 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
107 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity |
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108 and collaring maps, |
90 the gluing map is surjective. |
109 the gluing map is surjective. |
91 We say that fields on $X\sgl$ in the image of the gluing map |
110 We say that fields on $X\sgl$ in the image of the gluing map |
92 are transverse to $Y$ or splittable along $Y$. |
111 are transverse to $Y$ or splittable along $Y$. |
93 \item Gluing with corners. |
112 \item Gluing with corners. |
94 Let $\bd X = Y \cup Y \cup W$, where the two copies of $Y$ and |
113 Let $\bd X = Y \cup Y \cup W$, where the two copies of $Y$ and |
95 $W$ might intersect along their boundaries. |
114 $W$ might intersect along their boundaries. |
96 Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$. |
115 Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$ |
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116 (Figure xxxx). |
97 Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself |
117 Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself |
98 (without corners) along two copies of $\bd Y$. |
118 (without corners) along two copies of $\bd Y$. |
99 Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a splittable field on $W\sgl$ and let |
119 Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a splittable field on $W\sgl$ and let |
100 $c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
120 $c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
101 Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. |
121 Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. |
102 (This restriction map uses the gluing without corners map above.) |
122 (This restriction map uses the gluing without corners map above.) |
103 Using the boundary restriction, gluing without corners, and (in one case) orientation reversal |
123 Using the boundary restriction and gluing without corners maps, |
104 maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two |
124 we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two |
105 copies of $Y$ in $\bd X$. |
125 copies of $Y$ in $\bd X$. |
106 Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps. |
126 Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps. |
107 Then (here's the axiom/definition part) there is an injective ``gluing" map |
127 Then (here's the axiom/definition part) there is an injective ``gluing" map |
108 \[ |
128 \[ |
109 \Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) , |
129 \Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) , |
110 \] |
130 \] |
111 and this gluing map is compatible with all of the above structure (actions |
131 and this gluing map is compatible with all of the above structure (actions |
112 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
132 of homeomorphisms, boundary restrictions, disjoint union). |
113 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
133 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity |
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134 and collaring maps, |
114 the gluing map is surjective. |
135 the gluing map is surjective. |
115 We say that fields in the image of the gluing map |
136 We say that fields in the image of the gluing map |
116 are transverse to $Y$ or splittable along $Y$. |
137 are transverse to $Y$ or splittable along $Y$. |
117 \item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
138 \item Product fields. |
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139 There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
118 $c \mapsto c\times I$. |
140 $c \mapsto c\times I$. |
119 These maps comprise a natural transformation of functors, and commute appropriately |
141 These maps comprise a natural transformation of functors, and commute appropriately |
120 with all the structure maps above (disjoint union, boundary restriction, etc.). |
142 with all the structure maps above (disjoint union, boundary restriction, etc.). |
121 Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism |
143 Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism |
122 covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
144 covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
134 \] |
156 \] |
135 in the case that $X = X_1 \du X_2$, with $x_i \in \cC(X_i)$. |
157 in the case that $X = X_1 \du X_2$, with $x_i \in \cC(X_i)$. |
136 |
158 |
137 \medskip |
159 \medskip |
138 |
160 |
139 Using the functoriality and $\cdot\times I$ properties above, together |
161 Using the functoriality and product field properties above, together |
140 with boundary collar homeomorphisms of manifolds, we can define the notion of |
162 with boundary collar homeomorphisms of manifolds, we can define |
141 {\it extended isotopy}. |
163 {\it collar maps} $\cC(M)\to \cC(M)$. |
142 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
164 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
143 of $\bd M$. |
165 of $\bd M$. |
144 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is splittable along $\bd Y$. |
166 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is splittable along $\bd Y$. |
145 Let $c$ be $x$ restricted to $Y$. |
167 Let $c$ be $x$ restricted to $Y$. |
146 Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. |
168 Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. |
147 Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$. |
169 Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$. |
148 Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. |
170 Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. |
149 Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$. |
171 Then we call the map $x \mapsto f(x \bullet (c\times I))$ a {\it collar map}. |
150 More generally, we define extended isotopy to be the equivalence relation on fields |
172 We call the equivalence relation generated by collar maps and |
151 on $M$ generated by isotopy plus all instance of the above construction |
173 homeomorphisms isotopic to the identity {\it extended isotopy}, since the collar maps |
152 (for all appropriate $Y$ and $x$). |
174 can be thought of (informally) as the limit of homeomorphisms |
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175 which expand an infinitesimally thin collar neighborhood of $Y$ to a thicker |
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176 collar neighborhood. |
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177 |
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178 |
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179 % all this linearizing stuff is unnecessary, I think |
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180 \noop{ |
153 |
181 |
154 \nn{the following discussion of linearizing fields is kind of lame. |
182 \nn{the following discussion of linearizing fields is kind of lame. |
155 maybe just assume things are already linearized.} |
183 maybe just assume things are already linearized.} |
156 |
184 |
157 \nn{remark that if top dimensional fields are not already linear |
185 \nn{remark that if top dimensional fields are not already linear |
192 space determined by the labeling of the link of the 0-cell. |
220 space determined by the labeling of the link of the 0-cell. |
193 (If the 0-cell were labeled, the label would live in this space.) |
221 (If the 0-cell were labeled, the label would live in this space.) |
194 We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
222 We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
195 We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the |
223 We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the |
196 above tensor products. |
224 above tensor products. |
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225 |
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226 } % end \noop |
197 |
227 |
198 |
228 |
199 \subsection{Systems of fields from $n$-categories} |
229 \subsection{Systems of fields from $n$-categories} |
200 \label{sec:example:traditional-n-categories(fields)} |
230 \label{sec:example:traditional-n-categories(fields)} |
201 We now describe in more detail Example \ref{ex:traditional-n-categories(fields)}, |
231 We now describe in more detail Example \ref{ex:traditional-n-categories(fields)}, |