83 $\cC(X)$ which restricts to $c$. |
83 $\cC(X)$ which restricts to $c$. |
84 In this context, we will call $c$ a boundary condition. |
84 In this context, we will call $c$ a boundary condition. |
85 \item The subset $\cC_n(X;c)$ of top-dimensional fields |
85 \item The subset $\cC_n(X;c)$ of top-dimensional fields |
86 with a given boundary condition is an object in our symmetric monoidal category $\cS$. |
86 with a given boundary condition is an object in our symmetric monoidal category $\cS$. |
87 (This condition is of course trivial when $\cS = \Set$.) |
87 (This condition is of course trivial when $\cS = \Set$.) |
88 If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), |
88 If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$ (chain complexes)), |
89 then this extra structure is considered part of the definition of $\cC_n$. |
89 then this extra structure is considered part of the definition of $\cC_n$. |
90 Any maps mentioned below between fields on $n$-manifolds must be morphisms in $\cS$. |
90 Any maps mentioned below between fields on $n$-manifolds must be morphisms in $\cS$. |
91 \item $\cC_k$ is compatible with the symmetric monoidal |
91 \item $\cC_k$ is compatible with the symmetric monoidal |
92 structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
92 structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
93 compatibly with homeomorphisms and restriction to boundary. |
93 compatibly with homeomorphisms and restriction to boundary. |
297 ``domain" and ``range" for the two adjacent 1-cells; and |
297 ``domain" and ``range" for the two adjacent 1-cells; and |
298 \item a labeling of each 0-cell by a 1-morphism of $C$, with |
298 \item a labeling of each 0-cell by a 1-morphism of $C$, with |
299 domain and range determined by the transverse orientation and the labelings of the 1-cells. |
299 domain and range determined by the transverse orientation and the labelings of the 1-cells. |
300 \end{itemize} |
300 \end{itemize} |
301 |
301 |
302 We want fields on 1-manifolds to be enriched over Vect, so we also allow formal linear combinations |
302 We want fields on 1-manifolds to be enriched over $\Vect$, so we also allow formal linear combinations |
303 of the above fields on a 1-manifold $X$ so long as these fields restrict to the same field on $\bd X$. |
303 of the above fields on a 1-manifold $X$ so long as these fields restrict to the same field on $\bd X$. |
304 |
304 |
305 In addition, we mod out by the relation which replaces |
305 In addition, we mod out by the relation which replaces |
306 a 1-morphism label $a$ of a 0-cell $p$ with $a^*$ and reverse the transverse orientation of $p$. |
306 a 1-morphism label $a$ of a 0-cell $p$ with $a^*$ and reverse the transverse orientation of $p$. |
307 |
307 |
369 |
369 |
370 |
370 |
371 \subsection{Local relations} |
371 \subsection{Local relations} |
372 \label{sec:local-relations} |
372 \label{sec:local-relations} |
373 |
373 |
374 For convenience we assume that fields are enriched over Vect. |
374 For convenience we assume that fields are enriched over $\Vect$. |
375 |
375 |
376 Local relations are subspaces $U(B; c)\sub \cC(B; c)$ of the fields on balls which form an ideal under gluing. |
376 Local relations are subspaces $U(B; c)\sub \cC(B; c)$ of the fields on balls which form an ideal under gluing. |
377 Again, we give the examples first. |
377 Again, we give the examples first. |
378 |
378 |
379 \addtocounter{subsection}{-2} |
379 \addtocounter{subsection}{-2} |
398 homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, |
398 homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, |
399 satisfying the following properties. |
399 satisfying the following properties. |
400 \begin{enumerate} |
400 \begin{enumerate} |
401 \item Functoriality: |
401 \item Functoriality: |
402 $f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$ |
402 $f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$ |
403 \item Local relations imply extended isotopy: |
403 \item Local relations imply extended isotopy invariance: |
404 if $x, y \in \cC(B; c)$ and $x$ is extended isotopic |
404 if $x, y \in \cC(B; c)$ and $x$ is extended isotopic |
405 to $y$, then $x-y \in U(B; c)$. |
405 to $y$, then $x-y \in U(B; c)$. |
406 \item Ideal with respect to gluing: |
406 \item Ideal with respect to gluing: |
407 if $B = B' \cup B''$, $x\in U(B')$, and $r\in \cC(B'')$, then $x\bullet r \in U(B)$ |
407 if $B = B' \cup B''$, $x\in U(B')$, and $r\in \cC(B'')$, then $x\bullet r \in U(B)$ |
408 \end{enumerate} |
408 \end{enumerate} |