109 the gluing map is surjective. |
109 the gluing map is surjective. |
110 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 |
111 are transverse to $Y$ or splittable along $Y$. |
111 are transverse to $Y$ or splittable along $Y$. |
112 \item Gluing with corners. |
112 \item Gluing with corners. |
113 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 |
114 $W$ might intersect along their boundaries. |
114 $W$ might intersect along their boundaries. \todo{Really? I thought we wanted the boundaries of the two copies of Y to be disjoint} |
115 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 $Y$ |
116 (Figure xxxx). |
116 (Figure \ref{fig:???}). |
117 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 |
118 (without corners) along two copies of $\bd Y$. |
118 (without corners) along two copies of $\bd Y$. |
119 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 |
120 $c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
120 $c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
121 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 $W$. |
243 as a ``string diagram". |
243 as a ``string diagram". |
244 It can be thought of as (geometrically) dual to a pasting diagram. |
244 It can be thought of as (geometrically) dual to a pasting diagram. |
245 One of the advantages of string diagrams over pasting diagrams is that one has more |
245 One of the advantages of string diagrams over pasting diagrams is that one has more |
246 flexibility in slicing them up in various ways. |
246 flexibility in slicing them up in various ways. |
247 In addition, string diagrams are traditional in quantum topology. |
247 In addition, string diagrams are traditional in quantum topology. |
248 The diagrams predate by many years the terms ``string diagram" and ``quantum topology", e.g. \cite{ |
248 The diagrams predate by many years the terms ``string diagram" and ``quantum topology", e.g. \cite{MR0281657,MR776784} % both penrose |
249 MR0281657,MR776784 % penrose |
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250 } |
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251 |
249 |
252 If $X$ has boundary, we require that the cell decompositions are in general |
250 If $X$ has boundary, we require that the cell decompositions are in general |
253 position with respect to the boundary --- the boundary intersects each cell |
251 position with respect to the boundary --- the boundary intersects each cell |
254 transversely, so cells meeting the boundary are mere half-cells. |
252 transversely, so cells meeting the boundary are mere half-cells. |
255 Put another way, the cell decompositions we consider are dual to standard cell |
253 Put another way, the cell decompositions we consider are dual to standard cell |