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%!TEX root = ../blob1.tex
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\section{TQFTs via fields}
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\label{sec:fields}
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\label{sec:tqftsviafields}
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In this section we review the construction of TQFTs from ``topological fields".
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For more details see \cite{kw:tqft}.
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We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0
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submanifold of $X$, then $X \setmin Y$ implicitly means the closure
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$\overline{X \setmin Y}$.
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\subsection{Systems of fields}
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Let $\cM_k$ denote the category with objects
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unoriented PL manifolds of dimension
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$k$ and morphisms homeomorphisms.
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(We could equally well work with a different category of manifolds ---
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oriented, topological, smooth, spin, etc. --- but for definiteness we
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will stick with unoriented PL.)
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%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 $\cS = \Kom$.
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A $n$-dimensional {\it system of fields} in $\cS$
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is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$
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together with some additional data and satisfying some additional conditions, all specified below.
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Before finishing the definition of fields, we give two motivating examples
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(actually, families of examples) of systems of fields.
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The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps
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from X to $B$.
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The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be
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the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by
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$j$-morphisms of $C$.
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One can think of such sub-cell-complexes as dual to pasting diagrams for $C$.
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This is described in more detail below.
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Now for the rest of the definition of system of fields.
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\begin{enumerate}
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\item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$,
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and these maps are a natural
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transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$.
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For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of
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$\cC(X)$ which restricts to $c$.
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In this context, we will call $c$ a boundary condition.
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\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 $\cS$.
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\item $\cC_k$ is compatible with the symmetric monoidal
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structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$,
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compatibly with homeomorphisms, restriction to boundary, and orientation reversal.
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We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$
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restriction maps.
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\item Gluing without corners.
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Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds.
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Let $X\sgl$ denote $X$ glued to itself along $\pm Y$.
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Using the boundary restriction, disjoint union, and (in one case) orientation reversal
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maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two
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copies of $Y$ in $\bd X$.
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Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps.
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Then (here's the axiom/definition part) there is an injective ``gluing" map
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\[
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\Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) ,
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\]
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and this gluing map is compatible with all of the above structure (actions
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of homeomorphisms, boundary restrictions, orientation reversal, disjoint union).
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Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity,
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the gluing map is surjective.
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From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the
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gluing surface, we say that fields in the image of the gluing map
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are transverse to $Y$ or splittable along $Y$.
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\item Gluing with corners.
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Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries.
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Let $X\sgl$ denote $X$ glued to itself along $\pm Y$.
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Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself
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(without corners) along two copies of $\bd Y$.
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Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a splittable field on $W\sgl$ and let
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$c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$.
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Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$.
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(This restriction map uses the gluing without corners map above.)
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Using the boundary restriction, gluing without corners, and (in one case) orientation reversal
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maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two
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copies of $Y$ in $\bd X$.
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Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps.
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Then (here's the axiom/definition part) there is an injective ``gluing" map
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\[
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) ,
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\]
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and this gluing map is compatible with all of the above structure (actions
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of homeomorphisms, boundary restrictions, orientation reversal, disjoint union).
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Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity,
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the gluing map is surjective.
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From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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gluing surface, we say that fields in the image of the gluing map
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are transverse to $Y$ or splittable along $Y$.
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\item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted
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$c \mapsto c\times I$.
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These maps comprise a natural transformation of functors, and commute appropriately
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with all the structure maps above (disjoint union, boundary restriction, etc.).
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Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism
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103 |
covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$.
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104 |
\end{enumerate}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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105 |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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106 |
There are two notations we commonly use for gluing.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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107 |
One is
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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108 |
\[
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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109 |
x\sgl \deq \gl(x) \in \cC(X\sgl) ,
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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110 |
\]
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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111 |
for $x\in\cC(X)$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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112 |
The other is
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
113 |
\[
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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114 |
x_1\bullet x_2 \deq \gl(x_1\otimes x_2) \in \cC(X\sgl) ,
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
115 |
\]
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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116 |
in the case that $X = X_1 \du X_2$, with $x_i \in \cC(X_i)$.
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117 |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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118 |
\medskip
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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119 |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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120 |
Using the functoriality and $\cdot\times I$ properties above, together
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121 |
with boundary collar homeomorphisms of manifolds, we can define the notion of
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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122 |
{\it extended isotopy}.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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123 |
Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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124 |
of $\bd M$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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125 |
Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is splittable along $\bd Y$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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126 |
Let $c$ be $x$ restricted to $Y$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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127 |
Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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128 |
Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$.
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|
129 |
Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
130 |
Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
131 |
More generally, we define extended isotopy to be the equivalence relation on fields
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
132 |
on $M$ generated by isotopy plus all instance of the above construction
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
133 |
(for all appropriate $Y$ and $x$).
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
134 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
135 |
\nn{should also say something about pseudo-isotopy}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
136 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
137 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
138 |
\nn{remark that if top dimensional fields are not already linear
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
139 |
then we will soon linearize them(?)}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
140 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
141 |
We now describe in more detail systems of fields coming from sub-cell-complexes labeled
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
142 |
by $n$-category morphisms.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
143 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
144 |
Given an $n$-category $C$ with the right sort of duality
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
145 |
(e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category),
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
146 |
we can construct a system of fields as follows.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
147 |
Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
148 |
with codimension $i$ cells labeled by $i$-morphisms of $C$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
149 |
We'll spell this out for $n=1,2$ and then describe the general case.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
150 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
151 |
If $X$ has boundary, we require that the cell decompositions are in general
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
152 |
position with respect to the boundary --- the boundary intersects each cell
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
153 |
transversely, so cells meeting the boundary are mere half-cells.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
154 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
155 |
Put another way, the cell decompositions we consider are dual to standard cell
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
156 |
decompositions of $X$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
157 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
158 |
We will always assume that our $n$-categories have linear $n$-morphisms.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
159 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
160 |
For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
161 |
an object (0-morphism) of the 1-category $C$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
162 |
A field on a 1-manifold $S$ consists of
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
163 |
\begin{itemize}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
164 |
\item A cell decomposition of $S$ (equivalently, a finite collection
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
165 |
of points in the interior of $S$);
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
166 |
\item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
167 |
by an object (0-morphism) of $C$;
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
168 |
\item a transverse orientation of each 0-cell, thought of as a choice of
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
169 |
``domain" and ``range" for the two adjacent 1-cells; and
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
170 |
\item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
171 |
domain and range determined by the transverse orientation and the labelings of the 1-cells.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
172 |
\end{itemize}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
173 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
174 |
If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
175 |
of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
176 |
interior of $S$, each transversely oriented and each labeled by an element (1-morphism)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
177 |
of the algebra.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
178 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
179 |
\medskip
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
180 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
181 |
For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
182 |
that are common in the literature.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
183 |
We describe these carefully here.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
184 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
185 |
A field on a 0-manifold $P$ is a labeling of each point of $P$ with
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
186 |
an object of the 2-category $C$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
187 |
A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
188 |
A field on a 2-manifold $Y$ consists of
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
189 |
\begin{itemize}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
190 |
\item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
191 |
that each component of the complement is homeomorphic to a disk);
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
192 |
\item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
193 |
by a 0-morphism of $C$;
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
194 |
\item a transverse orientation of each 1-cell, thought of as a choice of
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
195 |
``domain" and ``range" for the two adjacent 2-cells;
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
196 |
\item a labeling of each 1-cell by a 1-morphism of $C$, with
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
197 |
domain and range determined by the transverse orientation of the 1-cell
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
198 |
and the labelings of the 2-cells;
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
199 |
\item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
200 |
of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
201 |
to $\pm 1 \in S^1$; and
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
202 |
\item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
203 |
determined by the labelings of the 1-cells and the parameterizations of the previous
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
204 |
bullet.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
205 |
\end{itemize}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
206 |
\nn{need to say this better; don't try to fit everything into the bulleted list}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
207 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
208 |
For general $n$, a field on a $k$-manifold $X^k$ consists of
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
209 |
\begin{itemize}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
210 |
\item A cell decomposition of $X$;
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
211 |
\item an explicit general position homeomorphism from the link of each $j$-cell
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
212 |
to the boundary of the standard $(k-j)$-dimensional bihedron; and
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
213 |
\item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
214 |
domain and range determined by the labelings of the link of $j$-cell.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
215 |
\end{itemize}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
216 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
217 |
%\nn{next definition might need some work; I think linearity relations should
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
218 |
%be treated differently (segregated) from other local relations, but I'm not sure
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
219 |
%the next definition is the best way to do it}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
220 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
221 |
\medskip
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
222 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
223 |
For top dimensional ($n$-dimensional) manifolds, we're actually interested
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
224 |
in the linearized space of fields.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
225 |
By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
226 |
the vector space of finite
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
227 |
linear combinations of fields on $X$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
228 |
If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
229 |
Thus the restriction (to boundary) maps are well defined because we never
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
230 |
take linear combinations of fields with differing boundary conditions.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
231 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
232 |
In some cases we don't linearize the default way; instead we take the
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
233 |
spaces $\lf(X; a)$ to be part of the data for the system of fields.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
234 |
In particular, for fields based on linear $n$-category pictures we linearize as follows.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
235 |
Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
236 |
obvious relations on 0-cell labels.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
237 |
More specifically, let $L$ be a cell decomposition of $X$
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
238 |
and let $p$ be a 0-cell of $L$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
239 |
Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
240 |
$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
241 |
Then the subspace $K$ is generated by things of the form
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
242 |
$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
243 |
to infer the meaning of $\alpha_{\lambda c + d}$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
244 |
Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
245 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
246 |
\nn{Maybe comment further: if there's a natural basis of morphisms, then no need;
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
247 |
will do something similar below; in general, whenever a label lives in a linear
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
248 |
space we do something like this; ? say something about tensor
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
249 |
product of all the linear label spaces? Yes:}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
250 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
251 |
For top dimensional ($n$-dimensional) manifolds, we linearize as follows.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
252 |
Define an ``almost-field" to be a field without labels on the 0-cells.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
253 |
(Recall that 0-cells are labeled by $n$-morphisms.)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
254 |
To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
255 |
space determined by the labeling of the link of the 0-cell.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
256 |
(If the 0-cell were labeled, the label would live in this space.)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
257 |
We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell).
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
258 |
We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
259 |
above tensor products.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
260 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
261 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
262 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
263 |
\subsection{Local relations}
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\label{sec:local-relations}
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A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$,
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for all $n$-manifolds $B$ which are
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homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$,
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satisfying the following properties.
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\begin{enumerate}
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\item functoriality:
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$f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$
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\item local relations imply extended isotopy:
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if $x, y \in \cC(B; c)$ and $x$ is extended isotopic
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to $y$, then $x-y \in U(B; c)$.
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\item ideal with respect to gluing:
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if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$
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\end{enumerate}
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See \cite{kw:tqft} for details.
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For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$,
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where $a$ and $b$ are maps (fields) which are homotopic rel boundary.
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For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map
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$\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into
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domain and range.
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\nn{maybe examples of local relations before general def?}
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\subsection{Constructing a TQFT}
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In this subsection we briefly review the construction of a TQFT from a system of fields and local relations.
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(For more details, see \cite{kw:tqft}.)
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Let $W$ be an $n{+}1$-manifold.
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We can think of the path integral $Z(W)$ as assigning to each
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boundary condition $x\in \cC(\bd W)$ a complex number $Z(W)(x)$.
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In other words, $Z(W)$ lies in $\c^{\lf(\bd W)}$, the vector space of linear
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maps $\lf(\bd W)\to \c$.
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The locality of the TQFT implies that $Z(W)$ in fact lies in a subspace
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$Z(\bd W) \sub \c^{\lf(\bd W)}$ defined by local projections.
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The linear dual to this subspace, $A(\bd W) = Z(\bd W)^*$,
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can be thought of as finite linear combinations of fields modulo local relations.
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(In other words, $A(\bd W)$ is a sort of generalized skein module.)
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This is the motivation behind the definition of fields and local relations above.
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In more detail, let $X$ be an $n$-manifold.
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%To harmonize notation with the next section,
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%let $\bc_0(X)$ be the vector space of finite linear combinations of fields on $X$, so
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%$\bc_0(X) = \lf(X)$.
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Define $U(X) \sub \lf(X)$ to be the space of local relations in $\lf(X)$;
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$U(X)$ is generated by things of the form $u\bullet r$, where
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$u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$.
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Define
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\[
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A(X) \deq \lf(X) / U(X) .
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\]
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(The blob complex, defined in the next section,
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is in some sense the derived version of $A(X)$.)
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If $X$ has boundary we can similarly define $A(X; c)$ for each
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boundary condition $c\in\cC(\bd X)$.
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The above construction can be extended to higher codimensions, assigning
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a $k$-category $A(Y)$ to an $n{-}k$-manifold $Y$, for $0 \le k \le n$.
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These invariants fit together via actions and gluing formulas.
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We describe only the case $k=1$ below.
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(The construction of the $n{+}1$-dimensional part of the theory (the path integral)
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requires that the starting data (fields and local relations) satisfy additional
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conditions.
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We do not assume these conditions here, so when we say ``TQFT" we mean a decapitated TQFT
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that lacks its $n{+}1$-dimensional part.)
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Let $Y$ be an $n{-}1$-manifold.
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Define a (linear) 1-category $A(Y)$ as follows.
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The objects of $A(Y)$ are $\cC(Y)$.
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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 $Y\times I$.
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Composition is given by gluing of cylinders.
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Let $X$ be an $n$-manifold with boundary and consider the collection of vector spaces
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$A(X; \cdot) \deq \{A(X; c)\}$ where $c$ ranges through $\cC(\bd X)$.
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This collection of vector spaces affords a representation of the category $A(\bd X)$, where
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the action is given by gluing a collar $\bd X\times I$ to $X$.
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Given a splitting $X = X_1 \cup_Y X_2$ of a closed $n$-manifold $X$ along an $n{-}1$-manifold $Y$,
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we have left and right actions of $A(Y)$ on $A(X_1; \cdot)$ and $A(X_2; \cdot)$.
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The gluing theorem for $n$-manifolds states that there is a natural isomorphism
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\[
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A(X) \cong A(X_1; \cdot) \otimes_{A(Y)} A(X_2; \cdot) .
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\]
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