Binary file RefereeReport.pdf has changed
--- a/blob to-do Thu Aug 11 12:50:46 2011 -0700
+++ b/blob to-do Thu Aug 11 12:50:50 2011 -0700
@@ -3,8 +3,10 @@
* better discussion of systems of fields from disk-like n-cats
(Is this done by now?)
+
+* ?? say clearly that certain lemmas don't work for TOP; we're only claiming DIFF and PL (requires small changes in many places)
-* need to fix fam-o-homeo argument per discussion with Rob
+* need to fix fam-o-homeo argument per discussion with Rob (or just remove it)
* need to change module axioms to follow changes in n-cat axioms; search for and destroy all the "Homeo_\bd"'s, add a v-cone axiom
--- a/text/a_inf_blob.tex Thu Aug 11 12:50:46 2011 -0700
+++ b/text/a_inf_blob.tex Thu Aug 11 12:50:50 2011 -0700
@@ -418,14 +418,31 @@
\begin{rem}
Lurie has shown in \cite[Theorem 3.8.6]{0911.0018} that the topological chiral homology
of an $n$-manifold $M$ with coefficients in 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.
+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 described in Example \ref{ex:e-n-alg}
that an $E_n$ algebra is roughly equivalent data to an $A_\infty$ $n$-category which
is trivial at levels 0 through $n-1$.
Ricardo Andrade also told us about a similar result.
+
+Specializing still further, Theorem \ref{thm:map-recon} is related to the classical result that for connected spaces $T$
+we have $HH_*(C_*(\Omega T)) \cong H_*(LT)$, that is, the Hochschild homology of based loops in $T$ is isomorphic
+to the homology of the free loop space of $T$ (see \cite{MR793184} and \cite{MR842427}).
+Theorem \ref{thm:map-recon} says that for any space $T$ (connected or not) we have
+$\bc_*(S^1; C_*(\pi^\infty_{\le 1}(T))) \simeq C_*(LT)$.
+Here $C_*(\pi^\infty_{\le 1}(T))$ denotes the singular chain version of the fundamental infinity-groupoid of $T$,
+whose objects are points in $T$ and morphism chain complexes are $C_*(\paths(t_1 \to t_2))$ for $t_1, t_2 \in T$.
+If $T$ is connected then the $A_\infty$ 1-category $C_*(\pi^\infty_{\le 1}(T))$ is Morita equivalent to the
+$A_\infty$ algebra $C_*(\Omega T)$;
+the bimodule for the equivalence is the singular chains of the space of paths which start at the base point of $T$.
+Theorem \ref{thm:hochschild} holds for $A_\infty$ 1-categories (though we do not prove that in this paper),
+which then implies that
+\[
+ Hoch_*(C_*(\Omega T)) \simeq Hoch_*(C_*(\pi^\infty_{\le 1}(T)))
+ \simeq \bc_*(S^1; C_*(\pi^\infty_{\le 1}(T))) \simeq C_*(LT) .
+\]
\end{rem}
-\begin{proof}
+\begin{proof}[Proof of Theorem \ref{thm:map-recon}]
The proof is again similar to that of Theorem \ref{thm:product}.
We begin by constructing a chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$.
--- a/text/deligne.tex Thu Aug 11 12:50:46 2011 -0700
+++ b/text/deligne.tex Thu Aug 11 12:50:50 2011 -0700
@@ -160,7 +160,7 @@
We have now defined a map from the little $n{+}1$-balls operad to the $n$-SC operad,
with contractible fibers.
(The fibers correspond to moving the $D_i$'s in the $x_{n+1}$
-direction without changing their ordering.)
+direction while keeping them disjoint.)
%\nn{issue: we've described this by varying the $R_i$'s, but above we emphasize varying the $f_i$'s.
%does this need more explanation?}
--- a/text/kw_macros.tex Thu Aug 11 12:50:46 2011 -0700
+++ b/text/kw_macros.tex Thu Aug 11 12:50:50 2011 -0700
@@ -64,7 +64,7 @@
% \DeclareMathOperator{\pr}{pr} etc.
\def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}}
-\applytolist{declaremathop}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}{Compat}{Coll}{Cone}{pr};
+\applytolist{declaremathop}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}{Compat}{Coll}{Cone}{pr}{paths};
\DeclareMathOperator*{\colim}{colim}
\DeclareMathOperator*{\hocolim}{hocolim}
--- a/text/ncat.tex Thu Aug 11 12:50:46 2011 -0700
+++ b/text/ncat.tex Thu Aug 11 12:50:50 2011 -0700
@@ -2117,7 +2117,7 @@
associated to $L$ by $\cX$ and $\cC$.
(See \S \ref{ss:ncat_fields} and \S \ref{moddecss}.)
Define $\cl{\cY}(L)$ similarly.
-For $K$ an unmarked 1-ball let $\cl{\cC(K)}$ denote the local homotopy colimit
+For $K$ an unmarked 1-ball let $\cl{\cC}(K)$ denote the local homotopy colimit
construction associated to $K$ by $\cC$.
Then we have an injective gluing map
\[
@@ -2225,7 +2225,7 @@
We only consider those decompositions in which the smaller balls are either
$0$-marked (i.e. intersect the $0$-marking of the large ball in a disc)
or plain (don't intersect the $0$-marking of the large ball).
-We can also take the boundary of a $0$-marked ball, which is $0$-marked sphere.
+We can also take the boundary of a $0$-marked ball, which is a $0$-marked sphere.
Fix $n$-categories $\cA$ and $\cB$.
These will label the two halves of a $0$-marked $k$-ball.
@@ -2618,7 +2618,6 @@
\caption{Moving $B$ from bottom to top}
\label{jun23c}
\end{figure}
-Let $D' = B\cap C$.
It is not hard too show that the above two maps are mutually inverse.
\begin{lem} \label{equator-lemma}