# HG changeset patch
# User Scott Morrison <scott@tqft.net>
# Date 1275200000 25200
# Node ID 2252c53bd4490fc36e96f0fe3c857b15d390220c
# Parent  52309e058a9537a2789d0e3d43cc540d04b9a73b
minor changes in a few places

diff -r 52309e058a95 -r 2252c53bd449 text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Sat May 29 23:13:03 2010 -0700
+++ b/text/a_inf_blob.tex	Sat May 29 23:13:20 2010 -0700
@@ -279,8 +279,14 @@
 To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$.
 
 \begin{thm} \label{thm:map-recon}
-$\cB^\cT(M) \simeq C_*(\Maps(M\to T))$.
+The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$ is quasi-isomorphic to singular chains on maps from $M$ to $T$.
+$$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$
 \end{thm}
+\begin{rem}
+\nn{This just isn't true, Lurie doesn't do this! I just heard this from Ricardo...}
+Lurie has shown in \cite{0911.0018} that the topological chiral homology of an $n$-manifold $M$ with coefficients in \nn{a certain $E_n$ algebra constructed from $T$} recovers  the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected. This extra hypothesis is not surprising, in view of the idea that an $E_n$ algebra is roughly equivalent data as an $A_\infty$ $n$-category which is trivial at all but the topmost level.
+\end{rem}
+
 \begin{proof}
 We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$.
 We then use \ref{extension_lemma_b} to show that $g$ induces isomorphisms on homology.
diff -r 52309e058a95 -r 2252c53bd449 text/evmap.tex
--- a/text/evmap.tex	Sat May 29 23:13:03 2010 -0700
+++ b/text/evmap.tex	Sat May 29 23:13:20 2010 -0700
@@ -41,7 +41,8 @@
 I lean toward the latter.}
 \medskip
 
-The proof will occupy the the next several pages.
+Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, and then give an outline of the method of proof.
+
 Without loss of generality, we will assume $X = Y$.
 
 \medskip
@@ -108,7 +109,7 @@
 where $r(b_W)$ denotes the obvious action of homeomorphisms on blob diagrams (in
 this case a 0-blob diagram).
 Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$ 
-(by \ref{disjunion} and \ref{bcontract}).
+(by Properties \ref{property:disjoint-union} and \ref{property:contractibility}).
 Assuming inductively that we have already defined $e_{VV'}(\bd(q\otimes b_V))$,
 there is, up to (iterated) homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$
 such that 
@@ -153,8 +154,7 @@
 
 \medskip
 
-Now for the details.
-
+\begin{proof}[Proof of Proposition \ref{CHprop}.]
 Notation: Let $|b| = \supp(b)$, $|p| = \supp(p)$.
 
 Choose a metric on $X$.
@@ -313,7 +313,7 @@
 $G_*^{i,m}$.
 Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero.
 Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$.
-Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is spanned by families of homeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{xxxxx}.
+Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is spanned by families of homeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{extension_lemma}.
 Recall that $h_j$ and also the homotopy connecting it to the identity do not increase
 supports.
 Define
@@ -610,26 +610,10 @@
 \end{itemize}
 
 
-\nn{to be continued....}
-
-\noop{
-
-\begin{lemma}
-
-\end{lemma}
-
-\begin{proof}
-
 \end{proof}
 
-}
+\nn{to be continued....}
 
 
 
 
-%\nn{say something about associativity here}
-
-
-
-
-
diff -r 52309e058a95 -r 2252c53bd449 text/ncat.tex
--- a/text/ncat.tex	Sat May 29 23:13:03 2010 -0700
+++ b/text/ncat.tex	Sat May 29 23:13:20 2010 -0700
@@ -86,6 +86,7 @@
 Morphisms are modeled on balls, so their boundaries are modeled on spheres:
 
 \begin{axiom}[Boundaries (spheres)]
+\label{axiom:spheres}
 For each $0 \le k \le n-1$, we have a functor $\cC_k$ from 
 the category of $k$-spheres and 
 homeomorphisms to the category of sets and bijections.
@@ -735,7 +736,7 @@
 (actions of homeomorphisms);
 define $k$-cat $\cC(\cdot\times W)$}
 
-Recall that Axiom \ref{} for an $n$-category provided functors $\cC$ from $k$-spheres to sets for $0 \leq k < n$. We claim now that these functors automatically agree with the colimits we have associated to spheres in this section. \todo{} \todo{In fact, we probably should do this for balls as well!} For the remainder of this section we will write $\underrightarrow{\cC}(W)$ for the colimit associated to an arbitary manifold $W$, to distinguish it, in the case that $W$ is a ball or a sphere, from $\cC(W)$, which is part of the definition of the $n$-category. After the next three lemmas, there will be no further need for this notational distinction.
+Recall that Axiom \ref{axiom:spheres} for an $n$-category provided functors $\cC$ from $k$-spheres to sets for $0 \leq k < n$. We claim now that these functors automatically agree with the colimits we have associated to spheres in this section. \todo{} \todo{In fact, we probably should do this for balls as well!} For the remainder of this section we will write $\underrightarrow{\cC}(W)$ for the colimit associated to an arbitary manifold $W$, to distinguish it, in the case that $W$ is a ball or a sphere, from $\cC(W)$, which is part of the definition of the $n$-category. After the next three lemmas, there will be no further need for this notational distinction.
 
 \begin{lem}
 For a $k$-ball or $k$-sphere $W$, with $0\leq k < n$, $$\underrightarrow{\cC}(W) = \cC(W).$$