
diff -r 58707c93f5e7 -r 1df2e5b38eb2 blob1.tex
--- a/blob1.tex	Tue May 05 17:27:21 2009 +0000
+++ b/blob1.tex	Sun May 24 20:30:45 2009 +0000
@@ -112,6 +112,7 @@
 \begin{itemize}
 \item Derive Hochschild standard results from blob point of view?
 \item Kh
+\item Mention somewhere \cite{MR1624157} ``Skein homology''; it's not directly related, but has similar motivations.
 \end{itemize}
 \end{itemize}
 
@@ -293,11 +294,10 @@
 unoriented, topological, smooth, spin, etc. --- but for definiteness we
 will stick with oriented PL.)
 
-Fix a top dimension $n$.
+Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$.
 
-A {\it system of fields}
-is a collection of functors $\cC_k$, for $k \le n$, from $\cM_k$ to the
-category of sets,
+A $n$-dimensional {\it system of fields} in $\cS$
+is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$
 together with some additional data and satisfying some additional conditions, all specified below.
 
 \nn{refer somewhere to my TQFT notes \cite{kw:tqft}, and possibly also to paper with Chris}
@@ -322,11 +322,12 @@
 For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of 
 $\cC(X)$ which restricts to $c$.
 In this context, we will call $c$ a boundary condition.
+\item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields must be morphisms in $\cS$.
 \item There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps
 again comprise a natural transformation of functors.
 In addition, the orientation reversal maps are compatible with the boundary restriction maps.
 \item $\cC_k$ is compatible with the symmetric monoidal
-structures on $\cM_k$ and sets: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$,
+structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$,
 compatibly with homeomorphisms, restriction to boundary, and orientation reversal.
 We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$
 restriction maps.
@@ -400,16 +401,16 @@
 
 \nn{should also say something about pseudo-isotopy}
 
-\bigskip
-\hrule
-\bigskip
-
-\input{text/fields.tex}
-
-
-\bigskip
-\hrule
-\bigskip
+%\bigskip
+%\hrule
+%\bigskip
+%
+%\input{text/fields.tex}
+%
+%
+%\bigskip
+%\hrule
+%\bigskip
 
 \nn{note: probably will suppress from notation the distinction
 between fields and their (orientation-reversal) duals}
@@ -726,7 +727,7 @@
 \]
 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above.
 $\overline{c}$ runs over all boundary conditions, again as described above.
-$j$ runs over all indices of twig blobs.
+$j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are cuttable along all of the blobs in $\overline{B}$.
 
 The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows.
 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram.




