# HG changeset patch # User Scott Morrison # Date 1280341228 25200 # Node ID e9ef2270eb6106e1ace53d25e5e7cf24a83a3af3 # Parent bdbd890086eba03610f9fa11d6ff23bfe66a6e75 changing all the \cC's to \cF's in blobdef diff -r bdbd890086eb -r e9ef2270eb61 text/blobdef.tex --- 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})