
diff -r fbd394dc95fa -r f38801a419f7 text/intro.tex
--- a/text/intro.tex	Sun Nov 01 02:03:20 2009 +0000
+++ b/text/intro.tex	Sun Nov 01 16:28:24 2009 +0000
@@ -42,7 +42,7 @@
 \nn{some more things to cover in the intro}
 \begin{itemize}
 \item related: we are being unsophisticated from a homotopy theory point of
-view and using chain complexes in many places where we could be by with spaces
+view and using chain complexes in many places where we could get by with spaces
 \item ? one of the points we make (far) below is that there is not really much
 difference between (a) systems of fields and local relations and (b) $n$-cats;
 thus we tend to switch between talking in terms of one or the other
@@ -53,19 +53,24 @@
 \subsection{Motivations}
 \label{sec:motivations}
 
-[Old outline for intro]
-\begin{itemize}
-\item Starting point: TQFTs via fields and local relations.
+We will briefly sketch our original motivation for defining the blob complex.
+\nn{this is adapted from an old draft of the intro; it needs further modification
+in order to better integrate it into the current intro.}
+
+As a starting point, consider TQFTs constructed via fields and local relations.
+(See Section \ref{sec:tqftsviafields} or \cite{kwtqft}.)
 This gives a satisfactory treatment for semisimple TQFTs
 (i.e.\ TQFTs for which the cylinder 1-category associated to an
 $n{-}1$-manifold $Y$ is semisimple for all $Y$).
-\item For non-semiemple TQFTs, this approach is less satisfactory.
+
+For non-semiemple TQFTs, this approach is less satisfactory.
 Our main motivating example (though we will not develop it in this paper)
-is the $4{+}1$-dimensional TQFT associated to Khovanov homology.
+is the (decapitated) $4{+}1$-dimensional TQFT associated to Khovanov homology.
 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together
 with a link $L \subset \bd W$.
 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$.
-\item How would we go about computing $A_{Kh}(W^4, L)$?
+
+How would we go about computing $A_{Kh}(W^4, L)$?
 For $A_{Kh}(B^4, L)$, the main tool is the exact triangle (long exact sequence)
 \nn{... $L_1, L_2, L_3$}.
 Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt
@@ -74,7 +79,9 @@
 corresponds to taking a coend (self tensor product) over the cylinder category
 associated to $B^3$ (with appropriate boundary conditions).
 The coend is not an exact functor, so the exactness of the triangle breaks.
-\item The obvious solution to this problem is to replace the coend with its derived counterpart.
+
+
+The obvious solution to this problem is to replace the coend with its derived counterpart.
 This presumably works fine for $S^1\times B^3$ (the answer being the Hochschild homology
 of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired.
 If we build our manifold up via a handle decomposition, the computation
@@ -84,13 +91,15 @@
 To show that our definition in terms of derived coends is well-defined, we
 would need to show that the above two sequences of derived coends yield the same answer.
 This is probably not easy to do.
-\item Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$
+
+Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$
 which is manifestly invariant.
 (That is, a definition that does not
 involve choosing a decomposition of $W$.
 After all, one of the virtues of our starting point --- TQFTs via field and local relations ---
 is that it has just this sort of manifest invariance.)
-\item The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient
+
+The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient
 \[
  \text{linear combinations of fields} \;\big/\; \text{local relations} ,
 \]
@@ -102,10 +111,12 @@
 $\bc_1$ is linear combinations of local relations on $W$,
 $\bc_2$ is linear combinations of relations amongst relations on $W$,
 and so on.
-\item None of the above ideas depend on the details of the Khovanov homology example,
+
+None of the above ideas depend on the details of the Khovanov homology example,
 so we develop the general theory in the paper and postpone specific applications
 to later papers.
-\end{itemize}
+
+
 
 \subsection{Formal properties}
 \label{sec:properties}




