
diff -r 76ad188dbe68 -r 27cfae8f4330 text/tqftreview.tex
--- a/text/tqftreview.tex	Wed Mar 09 06:48:39 2011 -0700
+++ b/text/tqftreview.tex	Tue Mar 15 07:25:13 2011 -0700
@@ -213,6 +213,7 @@
 
 \medskip
 
+
 Using the functoriality and product field properties above, together
 with boundary collar homeomorphisms of manifolds, we can define 
 {\it collar maps} $\cC(M)\to \cC(M)$.
@@ -231,54 +232,6 @@
 collar neighborhood.
 
 
-% all this linearizing stuff is unnecessary, I think
-\noop{
-
-\nn{the following discussion of linearizing fields is kind of lame.
-maybe just assume things are already linearized.}
-
-\nn{remark that if top dimensional fields are not already linear
-then we will soon linearize them(?)}
-
-For top dimensional ($n$-dimensional) manifolds, we're actually interested
-in the linearized space of fields.
-By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is
-the vector space of finite
-linear combinations of fields on $X$.
-If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$.
-Thus the restriction (to boundary) maps are well defined because we never
-take linear combinations of fields with differing boundary conditions.
-
-In some cases we don't linearize the default way; instead we take the
-spaces $\lf(X; a)$ to be part of the data for the system of fields.
-In particular, for fields based on linear $n$-category pictures we linearize as follows.
-Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by
-obvious relations on 0-cell labels.
-More specifically, let $L$ be a cell decomposition of $X$
-and let $p$ be a 0-cell of $L$.
-Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that
-$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$.
-Then the subspace $K$ is generated by things of the form
-$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader
-to infer the meaning of $\alpha_{\lambda c + d}$.
-Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms.
-
-\nn{Maybe comment further: if there's a natural basis of morphisms, then no need;
-will do something similar below; in general, whenever a label lives in a linear
-space we do something like this; ? say something about tensor
-product of all the linear label spaces?  Yes:}
-
-For top dimensional ($n$-dimensional) manifolds, we linearize as follows.
-Define an ``almost-field" to be a field without labels on the 0-cells.
-(Recall that 0-cells are labeled by $n$-morphisms.)
-To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism
-space determined by the labeling of the link of the 0-cell.
-(If the 0-cell were labeled, the label would live in this space.)
-We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell).
-We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the
-above tensor products.
-
-} % end \noop
 
 
 \subsection{Systems of fields from \texorpdfstring{$n$}{n}-categories}




