# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1279752994 21600
# Node ID ec8e9c920a34592ca0cf83fdbd70073b5d8dd74b
# Parent  a5d5a4c33bc43d8a7be100cce18e267875ae8255
 start on comm alg appendix

diff -r a5d5a4c33bc4 -r ec8e9c920a34 text/comm_alg.tex
--- a/text/comm_alg.tex	Wed Jul 21 10:29:23 2010 -0600
+++ b/text/comm_alg.tex	Wed Jul 21 16:56:34 2010 -0600
@@ -3,14 +3,10 @@
 \section{Commutative algebras as $n$-categories}
 \label{sec:comm_alg}
 
-\nn{should consider leaving this out; for now, make it an appendix.}
-
-\nn{also, this section needs a little updating to be compatible with the rest of the paper.}
-
 If $C$ is a commutative algebra it
 can also be thought of as an $n$-category whose $j$-morphisms are trivial for
 $j<n$ and whose $n$-morphisms are $C$. 
-The goal of this \nn{subsection?} is to compute
+The goal of this appendix is to compute
 $\bc_*(M^n, C)$ for various commutative algebras $C$.
 
 Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative 
@@ -35,24 +31,24 @@
 \end{prop}
 
 \begin{proof}
-To define the chain maps between the two complexes we will use the following lemma:
-
-\begin{lemma}
-Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with
-a basis (e.g.\ blob diagrams or singular simplices).
-For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$
-such that $R(c')_* \sub R(c)_*$ whenever $c'$ is a basis element which is part of $\bd c$.
-Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that
-$f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty).
-\end{lemma}
-
-\begin{proof}
-\nn{easy, but should probably write the details eventually}
-\nn{this is just the standard ``method of acyclic models" set up, so we should just give a reference for that}
-\end{proof}
-
+%To define the chain maps between the two complexes we will use the following lemma:
+%
+%\begin{lemma}
+%Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with
+%a basis (e.g.\ blob diagrams or singular simplices).
+%For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$
+%such that $R(c')_* \sub R(c)_*$ whenever $c'$ is a basis element which is part of $\bd c$.
+%Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that
+%$f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty).
+%\end{lemma}
+%
+%\begin{proof}
+%\nn{easy, but should probably write the details eventually}
+%\nn{this is just the standard ``method of acyclic models" set up, so we should just give a reference for that}
+%\end{proof}
+We will use acyclic models \nn{need ref}.
 Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$
-satisfying the conditions of the above lemma.
+satisfying the conditions of \nn{need ref}.
 If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a 
 finite unordered collection of points of $M$ with multiplicities, which is
 a point in $\Sigma^\infty(M)$.
@@ -66,7 +62,8 @@
 and using this point we can embed $X$ in $\Sigma^\infty(M)$.
 Define $R(B, u, r)_*$ to be the singular chain complex of $X$, thought of as a 
 subspace of $\Sigma^\infty(M)$.
-It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from the above lemma.
+It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from 
+\nn{need ref, or state condition}.
 Thus we have defined (up to homotopy) a map from 
 $\bc_*(M^n, k[t])$ to $C_*(\Sigma^\infty(M))$.
 
@@ -84,7 +81,7 @@
 Choose a neighborhood $D$ of $T$ which is a disjoint union of balls of small diameter.
 \nn{need to say more precisely how small}
 Define $R(c)_*$ to be $\bc_*(D, k[t]) \sub \bc_*(M^n, k[t])$.
-This is contractible by \ref{bcontract}.
+This is contractible by Proposition \ref{bcontract}.
 We can arrange that the boundary/inclusion condition is satisfied if we start with
 low-dimensional simplices and work our way up.
 \nn{need to be more precise}