author | Scott Morrison <scott@tqft.net> |
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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 notion of a ``system of fields and local relations". |
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For more details see \cite{kw:tqft}. |
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From a system of fields and local relations we can readily construct TQFT invariants of manifolds. |
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This is described in \S \ref{sec:constructing-a-tqft}. |
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A system of fields is very closely related to an $n$-category. |
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In Example \ref{ex:traditional-n-categories(fields)}, which runs throughout this section, |
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we sketch the construction of a system of fields from an $n$-category. |
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We make this more precise for $n=1$ or $2$ in \S \ref{sec:example:traditional-n-categories(fields)}, |
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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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we explain precisely how to go back and forth between a topological $n$-category and a system of fields and local relations. |
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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 symmetric monoidal category $\cS$. |
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While reading the definition, you should just think about the cases $\cS = \Set$ or $\cS = \Vect$. |
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The presentation here requires that the objects of $\cS$ have an underlying set, but this could probably be avoided if desired. |
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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 of systems of fields. |
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\begin{example} |
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\label{ex:maps-to-a-space(fields)} |
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Fix a target space $T$, and let $\cC(X)$ be the set of continuous maps |
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from $X$ to $T$. |
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\end{example} |
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\begin{example} |
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\label{ex:traditional-n-categories(fields)} |
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Fix an $n$-category $C$, and let $\cC(X)$ be |
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the set of embedded cell complexes in $X$ with codimension-$j$ cells labeled by |
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$j$-morphisms of $C$. |
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One can think of such embedded cell complexes as dual to pasting diagrams for $C$. |
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This is described in more detail in \S \ref{sec:example:traditional-n-categories(fields)}. |
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\end{example} |
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Now for the rest of the definition of system of fields. |
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(Readers desiring a more precise definition should refer to Subsection \ref{ss:n-cat-def} |
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and replace $k$-balls with $k$-manifolds.) |
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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 comprise 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 $\cS$. |
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(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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then this extra structure is considered part of the definition of $\cC_n$. |
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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 and restriction to boundary. |
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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 the two copies of $Y$. |
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Using the boundary restriction and disjoint union |
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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, 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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We say that fields on $X\sgl$ 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 the two copies of $Y$ and |
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$W$ might intersect along their boundaries. |
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Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$. |
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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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\[ |
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\Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) , |
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\] |
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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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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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covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
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\end{enumerate} |
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There are two notations we commonly use for gluing. |
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One is |
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\[ |
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x\sgl \deq \gl(x) \in \cC(X\sgl) , |
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\] |
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for $x\in\cC(X)$. |
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The other is |
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\[ |
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x_1\bullet x_2 \deq \gl(x_1\otimes x_2) \in \cC(X\sgl) , |
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\] |
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in the case that $X = X_1 \du X_2$, with $x_i \in \cC(X_i)$. |
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\medskip |
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Using the functoriality and $\cdot\times I$ properties above, together |
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with boundary collar homeomorphisms of manifolds, we can define the notion of |
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{\it extended isotopy}. |
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Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
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of $\bd M$. |
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Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is splittable along $\bd Y$. |
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Let $c$ be $x$ restricted to $Y$. |
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Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. |
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Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$. |
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Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. |
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Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$. |
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More generally, we define extended isotopy to be the equivalence relation on fields |
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on $M$ generated by isotopy plus all instance of the above construction |
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(for all appropriate $Y$ and $x$). |
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\nn{the following discussion of linearizing fields is kind of lame. |
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maybe just assume things are already linearized.} |
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\nn{remark that if top dimensional fields are not already linear |
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then we will soon linearize them(?)} |
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For top dimensional ($n$-dimensional) manifolds, we're actually interested |
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in the linearized space of fields. |
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By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is |
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the vector space of finite |
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linear combinations of fields on $X$. |
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If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$. |
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Thus the restriction (to boundary) maps are well defined because we never |
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take linear combinations of fields with differing boundary conditions. |
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In some cases we don't linearize the default way; instead we take the |
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spaces $\lf(X; a)$ to be part of the data for the system of fields. |
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In particular, for fields based on linear $n$-category pictures we linearize as follows. |
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Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
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obvious relations on 0-cell labels. |
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More specifically, let $L$ be a cell decomposition of $X$ |
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and let $p$ be a 0-cell of $L$. |
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Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
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$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
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Then the subspace $K$ is generated by things of the form |
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$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader |
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to infer the meaning of $\alpha_{\lambda c + d}$. |
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Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. |
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\nn{Maybe comment further: if there's a natural basis of morphisms, then no need; |
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will do something similar below; in general, whenever a label lives in a linear |
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space we do something like this; ? say something about tensor |
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product of all the linear label spaces? Yes:} |
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For top dimensional ($n$-dimensional) manifolds, we linearize as follows. |
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Define an ``almost-field" to be a field without labels on the 0-cells. |
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(Recall that 0-cells are labeled by $n$-morphisms.) |
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To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
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space determined by the labeling of the link of the 0-cell. |
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(If the 0-cell were labeled, the label would live in this space.) |
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We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
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We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the |
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above tensor products. |
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\subsection{Systems of fields from $n$-categories} |
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\label{sec:example:traditional-n-categories(fields)} |
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We now describe in more detail Example \ref{ex:traditional-n-categories(fields)}, |
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systems of fields coming from embedded cell complexes labeled |
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by $n$-category morphisms. |
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Given an $n$-category $C$ with the right sort of duality |
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(e.g. a pivotal 2-category, *-1-category), |
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we can construct a system of fields as follows. |
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Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
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with codimension $i$ cells labeled by $i$-morphisms of $C$. |
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We'll spell this out for $n=1,2$ and then describe the general case. |
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If $X$ has boundary, we require that the cell decompositions are in general |
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position with respect to the boundary --- the boundary intersects each cell |
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transversely, so cells meeting the boundary are mere half-cells. |
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Put another way, the cell decompositions we consider are dual to standard cell |
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decompositions of $X$. |
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We will always assume that our $n$-categories have linear $n$-morphisms. |
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For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
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an object (0-morphism) of the 1-category $C$. |
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A field on a 1-manifold $S$ consists of |
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\begin{itemize} |
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\item a cell decomposition of $S$ (equivalently, a finite collection |
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of points in the interior of $S$); |
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\item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) |
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by an object (0-morphism) of $C$; |
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\item a transverse orientation of each 0-cell, thought of as a choice of |
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``domain" and ``range" for the two adjacent 1-cells; and |
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\item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with |
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domain and range determined by the transverse orientation and the labelings of the 1-cells. |
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\end{itemize} |
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If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels |
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of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the |
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interior of $S$, each transversely oriented and each labeled by an element (1-morphism) |
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of the algebra. |
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\medskip |
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For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories) |
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that are common in the literature. |
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We describe these carefully here. |
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A field on a 0-manifold $P$ is a labeling of each point of $P$ with |
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an object of the 2-category $C$. |
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A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$. |
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A field on a 2-manifold $Y$ consists of |
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\begin{itemize} |
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\item a cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such |
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that each component of the complement is homeomorphic to a disk); |
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\item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$) |
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by a 0-morphism of $C$; |
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\item a transverse orientation of each 1-cell, thought of as a choice of |
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``domain" and ``range" for the two adjacent 2-cells; |
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\item a labeling of each 1-cell by a 1-morphism of $C$, with |
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domain and range determined by the transverse orientation of the 1-cell |
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and the labelings of the 2-cells; |
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\item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood |
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of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped |
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to $\pm 1 \in S^1$; and |
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\item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range |
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determined by the labelings of the 1-cells and the parameterizations of the previous |
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bullet. |
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\end{itemize} |
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\nn{need to say this better; don't try to fit everything into the bulleted list} |
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For general $n$, a field on a $k$-manifold $X^k$ consists of |
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\begin{itemize} |
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\item A cell decomposition of $X$; |
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\item an explicit general position homeomorphism from the link of each $j$-cell |
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to the boundary of the standard $(k-j)$-dimensional bihedron; and |
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\item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
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domain and range determined by the labelings of the link of $j$-cell. |
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\end{itemize} |
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%\nn{next definition might need some work; I think linearity relations should |
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%be treated differently (segregated) from other local relations, but I'm not sure |
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%the next definition is the best way to do it} |
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\medskip |
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\subsection{Local relations} |
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\label{sec:local-relations} |
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Local relations are certain subspaces of the fields on balls, which form an ideal under gluing. |
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Again, we give the examples first. |
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\addtocounter{prop}{-2} |
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\begin{example}[contd.] |
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For maps into spaces, $U(B; c)$ is generated by fields 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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\end{example} |
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\begin{example}[contd.] |
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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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\end{example} |
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These motivate the following definition. |
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\begin{defn} |
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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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\end{defn} |
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See \cite{kw:tqft} for further details. |
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\subsection{Constructing a TQFT} |
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\label{sec: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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\begin{defn} |
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\label{defn:TQFT-invariant} |
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The TQFT invariant of $X$ associated to a system of fields $\cF$ and local relations $\cU$ is |
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$$A(X) \deq \lf(X) / U(X),$$ |
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where $\cU(X) \sub \lf(X)$ to be the space of local relations in $\lf(X)$; |
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$\cU(X)$ is generated by things of the form $u\bullet r$, where |
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$u\in \cU(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$. |
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\end{defn} |
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(The blob complex, defined in the next section, |
355 |
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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359 |
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 $A(Y\times I; a, b)$, |
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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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381 |
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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