# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1323366316 28800
# Node ID 6cfc2dc6ec6edc56ecc36d7f1125e1b5a660f378
# Parent  d73a88d7849860fb432836bfac666eb714bce897
beginning final read-through; minor changes to intro

diff -r d73a88d78498 -r 6cfc2dc6ec6e text/intro.tex
--- a/text/intro.tex	Wed Dec 07 23:00:54 2011 -0800
+++ b/text/intro.tex	Thu Dec 08 09:45:16 2011 -0800
@@ -177,7 +177,7 @@
 
 For non-semi-simple TQFTs, this approach is less satisfactory.
 Our main motivating example (though we will not develop it in this paper)
-is the (decapitated) $4{+}1$-dimensional TQFT associated to Khovanov homology.
+is the $(4{+}\varepsilon)$-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 $A_{Kh}(B^4, L)$.
@@ -209,10 +209,10 @@
 
 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
+In other words, we want 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.)
+is that it has just this sort of manifest invariance.
 
 The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient
 \[
@@ -225,7 +225,8 @@
 Here $\bc_0$ is linear combinations of fields on $W$,
 $\bc_1$ is linear combinations of local relations on $W$,
 $\bc_2$ is linear combinations of relations amongst relations on $W$,
-and so on. We now have a short exact sequence of chain complexes relating resolutions of the link $L$ 
+and so on. 
+We now have a long exact sequence of chain complexes relating resolutions of the link $L$ 
 (c.f. Lemma \ref{lem:hochschild-exact} which shows exactness 
 with respect to boundary conditions in the context of Hochschild homology).
 
@@ -425,7 +426,7 @@
 (see \S \ref{ss:product-formula}).
 
 Fix a disk-like $n$-category $\cC$, which we'll omit from the notation.
-Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ 1-category.
+Recall that for any $(n{-}1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ 1-category.
 (See Appendix \ref{sec:comparing-A-infty} for the translation between disk-like $A_\infty$ $1$-categories 
 and the usual algebraic notion of an $A_\infty$ category.)