# HG changeset patch # User Scott Morrison # Date 1285105457 25200 # Node ID c9f41c18a96f37ae63b83776549e4899b3481190 # Parent 4d2dad357a495ad9cec93c1439623d3f4f2f5552 deleting nn's diff -r 4d2dad357a49 -r c9f41c18a96f text/a_inf_blob.tex --- a/text/a_inf_blob.tex Tue Sep 21 07:37:41 2010 -0700 +++ b/text/a_inf_blob.tex Tue Sep 21 14:44:17 2010 -0700 @@ -282,7 +282,7 @@ or $M\to Y$, instead of an undecorated colimit with fancier $k$-categories over $Y$. Information about the specific map to $Y$ has been taken out of the categories and put into sphere modules and decorations. -\nn{...} +\nn{just say that one could do something along these lines} %Let $F \to E \to Y$ be a fiber bundle as above. %Choose a decomposition $Y = \cup X_i$ @@ -442,9 +442,4 @@ It is now easy to see that $\psi\circ\phi$ is the identity on the nose. Another acyclic models argument shows that $\phi\circ\psi$ is homotopic to the identity. (See the proof of Theorem \ref{thm:product} for more details.) -\end{proof} - -\nn{maybe should also mention version where we enrich over -spaces rather than chain complexes;} - - +\end{proof} \ No newline at end of file diff -r 4d2dad357a49 -r c9f41c18a96f text/appendixes/famodiff.tex --- a/text/appendixes/famodiff.tex Tue Sep 21 07:37:41 2010 -0700 +++ b/text/appendixes/famodiff.tex Tue Sep 21 14:44:17 2010 -0700 @@ -234,8 +234,6 @@ \medskip -\nn{need to clean up references from the main text to the lemmas of this section} - %%%%%% Lo, \noop{...} \noop{ diff -r 4d2dad357a49 -r c9f41c18a96f text/appendixes/moam.tex --- a/text/appendixes/moam.tex Tue Sep 21 07:37:41 2010 -0700 +++ b/text/appendixes/moam.tex Tue Sep 21 14:44:17 2010 -0700 @@ -32,7 +32,7 @@ \begin{proof} (Sketch) -This is a standard result; see, for example, \nn{need citations}. +This is a standard result; see, for example, \nn{need citations: Spanier}. We will build a chain map $f\in \Compat(D^\bullet_*)$ inductively. Choose $f(x_{0j})\in D^{0j}_0$ for all $j$ diff -r 4d2dad357a49 -r c9f41c18a96f text/blobdef.tex --- a/text/blobdef.tex Tue Sep 21 07:37:41 2010 -0700 +++ b/text/blobdef.tex Tue Sep 21 14:44:17 2010 -0700 @@ -67,13 +67,11 @@ just erasing the blob from the picture (but keeping the blob label $u$). -\nn{it seems rather strange to make this a theorem} -\nn{it's a theorem because it's stated in the introduction, and I wanted everything there to have numbers that pointed into the paper --S} Note that directly from the definition we have -\begin{thm} +\begin{prop} \label{thm:skein-modules} The skein module $A(X)$ is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. -\end{thm} +\end{prop} This also establishes the second half of Property \ref{property:contractibility}. @@ -292,7 +290,6 @@ and $s:C \to \cF(B_i)$ is some fixed section of $e$.) For lack of a better name, -\nn{can we think of a better name?} we'll call elements of $P$ cone-product polyhedra, and say that blob diagrams have the structure of a cone-product set (analogous to simplicial set). \end{remark} diff -r 4d2dad357a49 -r c9f41c18a96f text/comm_alg.tex --- a/text/comm_alg.tex Tue Sep 21 07:37:41 2010 -0700 +++ b/text/comm_alg.tex Tue Sep 21 14:44:17 2010 -0700 @@ -135,7 +135,7 @@ 0, $\z/j \z$ in odd degrees, and 0 in positive even degrees. The point $\Sigma^0(S^1)$ contributes the homology of $BS^1$ which is $\z$ in even degrees and 0 in odd degrees. -This agrees with the calculation in \cite[3.1.7]{MR1600246}. +This agrees with the calculation in \cite[\S 3.1.7]{MR1600246}. \medskip @@ -189,7 +189,5 @@ \begin{itemize} \item compare the topological computation for truncated polynomial algebra with \cite{MR1600246} \item multivariable truncated polynomial algebras (at least mention them) -\item ideally, say something more about higher hochschild homology (maybe sketch idea for proof of equivalence) -\item say something about SMCs as $n$-categories, e.g. Vect and K-theory. \end{itemize} diff -r 4d2dad357a49 -r c9f41c18a96f text/evmap.tex --- a/text/evmap.tex Tue Sep 21 07:37:41 2010 -0700 +++ b/text/evmap.tex Tue Sep 21 14:44:17 2010 -0700 @@ -191,7 +191,7 @@ and with $\supp(x_k) = U$. We can now take $d_j \deq \sum x_k$. It is clear that $\bd d_j = \sum (g_{j-1}(e_k) - g_j(e_k)) = g_{j-1}(s(\bd b)) - g_{j}(s(\bd b))$, as desired. -\nn{should maybe have figure} +\nn{should have figure} We now define \[ @@ -210,8 +210,6 @@ For sufficiently fine $\cV_{l-1}$ this will be possible. Since $C_*$ is finite, the process terminates after finitely many, say $r$, steps. We take $\cV_r = \cU$. - -\nn{should probably be more specific at the end} \end{proof} @@ -222,8 +220,6 @@ We give $\BD_k$ the finest topology such that \begin{itemize} \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. -\item \nn{don't we need something for collaring maps?} -\nn{KW: no, I don't think so. not unless we wanted some extension of $CH_*$ to act} \item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous. \item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous, where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on @@ -418,7 +414,6 @@ We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. (For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general than simplices --- they can be based on any cone-product polyhedron (see Remark \ref{blobsset-remark}).) -\nn{this note about our non-standard should probably go earlier in the paper, maybe intro} \begin{thm} \label{thm:CH} For $n$-manifolds $X$ and $Y$ there is a chain map diff -r 4d2dad357a49 -r c9f41c18a96f text/intro.tex --- a/text/intro.tex Tue Sep 21 07:37:41 2010 -0700 +++ b/text/intro.tex Tue Sep 21 14:44:17 2010 -0700 @@ -8,7 +8,7 @@ \begin{itemize} \item The 0-th homology $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. -(See Theorem \ref{thm:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) +(See Proposition \ref{thm:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. (See Theorem \ref{thm:hochschild} and \S \ref{sec:hochschild}.) @@ -124,7 +124,7 @@ } (FU.100); \draw[->] (C) -- node[above left=3pt] {restrict to \\ standard balls} (tC); \draw[->] (FU.80) -- node[right] {restrict \\ to balls} (C.-80); -\draw[->] (BC) -- node[right] {$H_0$ \\ c.f. Theorem \ref{thm:skein-modules}} (A); +\draw[->] (BC) -- node[right] {$H_0$ \\ c.f. Proposition \ref{thm:skein-modules}} (A); \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); \draw[<->] (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); @@ -286,7 +286,7 @@ The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology. -\newtheorem*{thm:skein-modules}{Theorem \ref{thm:skein-modules}} +\newtheorem*{thm:skein-modules}{Proposition \ref{thm:skein-modules}} \begin{thm:skein-modules}[Skein modules] The $0$-th blob homology of $X$ is the usual @@ -308,7 +308,7 @@ \end{equation*} \end{thm:hochschild} -Theorem \ref{thm:skein-modules} is immediate from the definition, and +Proposition \ref{thm:skein-modules} is immediate from the definition, and Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}. We also note \S \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of certain commutative algebras thought of as $n$-categories. diff -r 4d2dad357a49 -r c9f41c18a96f text/ncat.tex --- a/text/ncat.tex Tue Sep 21 07:37:41 2010 -0700 +++ b/text/ncat.tex Tue Sep 21 14:44:17 2010 -0700 @@ -1034,8 +1034,7 @@ is more involved. We will describe two different (but homotopy equivalent) versions of the homotopy colimit of $\psi_{\cC;W}$. The first is the usual one, which works for any indexing category. -The second construction, we we call the {\it local} homotopy colimit, -\nn{give it a different name?} +The second construction, which we call the {\it local} homotopy colimit, is more closely related to the blob complex construction of \S \ref{sec:blob-definition} and takes advantage of local (gluing) properties of the indexing category $\cell(W)$. @@ -1351,7 +1350,7 @@ plain ball case. Note that a marked pinched product can be decomposed into either two marked pinched products or a plain pinched product and a marked pinched product. -\nn{should give figure} +\nn{should maybe give figure} \begin{module-axiom}[Product (identity) morphisms] For each pinched product $\pi:E\to M$, with $M$ a marked $k$-ball and $E$ a marked @@ -1828,7 +1827,7 @@ where $B^j$ is the standard $j$-ball. A 1-marked $k$-ball can be decomposed in various ways into smaller balls, which are either (a) smaller 1-marked $k$-balls, (b) 0-marked $k$-balls, or (c) plain $k$-balls. -(See Figure \nn{need figure}.) +(See Figure \nn{need figure, and improve caption on other figure}.) We now proceed as in the above module definitions. \begin{figure}[t] \centering @@ -2190,7 +2189,7 @@ \begin{lem} Assume $n\ge 2$ and fix $E$ and $E'$ as above. -The any two sequences of elementary moves connecting $E$ to $E'$ +Then any two sequences of elementary moves connecting $E$ to $E'$ are related by a sequence of the two movie moves defined above. \end{lem} @@ -2211,7 +2210,7 @@ rotating the 0-sphere $E$ around the 1-sphere $\bd X$. But if $n=1$, then we are in the case of ordinary algebroids and bimodules, and this is just the well-known ``Frobenius reciprocity" result for bimodules. -\nn{find citation for this. Evans and Kawahigashi?} +\nn{find citation for this. Evans and Kawahigashi? Bisch!} \medskip @@ -2240,7 +2239,7 @@ \medskip -\nn{Stuff that remains to be done (either below or in an appendix or in a separate section or in -a separate paper): discuss Morita equivalence; functors} +%\nn{Stuff that remains to be done (either below or in an appendix or in a separate section or in +%a separate paper): discuss Morita equivalence; functors}