--- a/text/blobdef.tex Wed Jul 28 11:16:36 2010 -0700
+++ b/text/blobdef.tex Wed Jul 28 11:20:28 2010 -0700
@@ -4,16 +4,16 @@
\label{sec:blob-definition}
Let $X$ be an $n$-manifold.
-Let $\cC$ be a fixed system of fields and local relations.
+Let $(\cF,U)$ be a fixed system of fields and local relations.
We'll assume it is enriched over \textbf{Vect};
if it is not we can make it so by allowing finite
-linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$.
+linear combinations of elements of $\cF(X; c)$, for fixed $c\in \cF(\bd X)$.
-%In this section we will usually suppress boundary conditions on $X$ from the notation, e.g. by writing $\lf(X)$ instead of $\lf(X; c)$.
+%In this section we will usually suppress boundary conditions on $X$ from the notation, e.g. by writing $\cF(X)$ instead of $\cF(X; c)$.
We want to replace the quotient
\[
- A(X) \deq \lf(X) / U(X)
+ A(X) \deq \cF(X) / U(X)
\]
of Definition \ref{defn:TQFT-invariant} with a resolution
\[
@@ -32,8 +32,8 @@
into which we cut manifolds below are not themselves manifolds, and it requires special attention
to define fields on these pieces.
-We of course define $\bc_0(X) = \lf(X)$.
-(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$ for each $c \in \lf(\bdy X)$.
+We of course define $\bc_0(X) = \cF(X)$.
+(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cF(X; c)$ for each $c \in \cF(\bdy X)$.
We'll omit such boundary conditions from the notation in the rest of this section.)
In other words, $\bc_0(X)$ is just the vector space of all fields on $X$.
@@ -42,8 +42,8 @@
Thus we say a $1$-blob diagram consists of:
\begin{itemize}
\item An closed ball in $X$ (``blob") $B \sub X$.
-\item A boundary condition $c \in \cC(\bdy B) = \cC(\bd(X \setmin B))$.
-\item A field $r \in \cC(X \setmin B; c)$.
+\item A boundary condition $c \in \cF(\bdy B) = \cF(\bd(X \setmin B))$.
+\item A field $r \in \cF(X \setmin B; c)$.
\item A local relation field $u \in U(B; c)$.
\end{itemize}
(See Figure \ref{blob1diagram}.) Since $c$ is implicitly determined by $u$ or $r$, we usually omit it from the notation.
@@ -52,10 +52,10 @@
\end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure}
In order to get the linear structure correct, we define
\[
- \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) .
+ \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \cF(X \setmin B; c) .
\]
The first direct sum is indexed by all blobs $B\subset X$, and the second
-by all boundary conditions $c \in \cC(\bd B)$.
+by all boundary conditions $c \in \cF(\bd B)$.
Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$.
Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by
@@ -83,8 +83,8 @@
A disjoint 2-blob diagram consists of
\begin{itemize}
\item A pair of closed balls (blobs) $B_1, B_2 \sub X$ with disjoint interiors.
-\item A field $r \in \cC(X \setmin (B_1 \cup B_2); c_1, c_2)$
-(where $c_i \in \cC(\bd B_i)$).
+\item A field $r \in \cF(X \setmin (B_1 \cup B_2); c_1, c_2)$
+(where $c_i \in \cF(\bd B_i)$).
\item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$.
\end{itemize}
(See Figure \ref{blob2ddiagram}.)
@@ -103,9 +103,9 @@
A nested 2-blob diagram consists of
\begin{itemize}
\item A pair of nested balls (blobs) $B_1 \subseteq B_2 \subseteq X$.
-\item A field $r' \in \cC(B_2 \setminus B_1; c_1, c_2)$
-(for some $c_1 \in \cC(\bdy B_1)$ and $c_2 \in \cC(\bdy B_2)$).
-\item A field $r \in \cC(X \setminus B_2; c_2)$.
+\item A field $r' \in \cF(B_2 \setminus B_1; c_1, c_2)$
+(for some $c_1 \in \cF(\bdy B_1)$ and $c_2 \in \cF(\bdy B_2)$).
+\item A field $r \in \cF(X \setminus B_2; c_2)$.
\item A local relation field $u \in U(B_1; c_1)$.
\end{itemize}
(See Figure \ref{blob2ndiagram}.)
@@ -124,11 +124,11 @@
\bc_2(X) & \deq &
\left(
\bigoplus_{B_1, B_2\; \text{disjoint}} \bigoplus_{c_1, c_2}
- U(B_1; c_1) \otimes U(B_2; c_2) \otimes \lf(X\setmin (B_1\cup B_2); c_1, c_2)
+ U(B_1; c_1) \otimes U(B_2; c_2) \otimes \cF(X\setmin (B_1\cup B_2); c_1, c_2)
\right) \bigoplus \\
&& \quad\quad \left(
\bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2}
- U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1, c_2) \tensor \cC(X \setminus B_2; c_2)
+ U(B_1; c_1) \otimes \cF(B_2 \setmin B_1; c_1, c_2) \tensor \cF(X \setminus B_2; c_2)
\right) .
\end{eqnarray*}
% __ (already said this above)
@@ -197,10 +197,10 @@
A $k$-blob diagram on $X$ consists of
\begin{itemize}
\item a configuration $\{B_1, \ldots B_k\}$ of $k$ blobs in $X$,
-\item and a field $r \in \cC(X)$ which is splittable along some gluing decomposition compatible with that configuration,
+\item and a field $r \in \cF(X)$ which is splittable along some gluing decomposition compatible with that configuration,
\end{itemize}
such that
-the restriction $u_i$ of $r$ to each twig blob $B_i$ lies in the subspace $U(B_i) \subset \cC(B_i)$. (See Figure \ref{blobkdiagram}.) More precisely, each twig blob $B_i$ is the image of some ball $M_r'$ as above, and it is really the restriction to $M_r'$ that must lie in the subspace $U(M_r')$.
+the restriction $u_i$ of $r$ to each twig blob $B_i$ lies in the subspace $U(B_i) \subset \cF(B_i)$. (See Figure \ref{blobkdiagram}.) More precisely, each twig blob $B_i$ is the image of some ball $M_r'$ as above, and it is really the restriction to $M_r'$ that must lie in the subspace $U(M_r')$.
\end{defn}
\begin{figure}[t]\begin{equation*}
\mathfig{.7}{definition/k-blobs}
@@ -241,7 +241,7 @@
For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while
a diagram of $k$ disjoint blobs corresponds to a $k$-cube.
(This correspondence works best if we think of each twig label $u_i$ as having the form
-$x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cC(B_i) \to C$ is the evaluation map,
-and $s:C \to \cC(B_i)$ is some fixed section of $e$. \todo{This parenthetical remark mysteriously specialises to the category case})
+$x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cF(B_i) \to C$ is the evaluation map,
+and $s:C \to \cF(B_i)$ is some fixed section of $e$. \todo{This parenthetical remark mysteriously specialises to the category case})