# HG changeset patch # User Scott Morrison # Date 1277304882 14400 # Node ID facac77e9a72e9a802ea059f88e9fa01672c43c0 # Parent f0518720227ae1daf95d5ca0e13de7a88e57a1f9# Parent eec4b1f9cfc2343af389d05252375f57217a7e61 Automated merge with https://tqft.net/hg/blob/ diff -r eec4b1f9cfc2 -r facac77e9a72 text/appendixes/smallblobs.tex --- a/text/appendixes/smallblobs.tex Wed Jun 23 10:54:29 2010 -0400 +++ b/text/appendixes/smallblobs.tex Wed Jun 23 10:54:42 2010 -0400 @@ -15,10 +15,14 @@ We can't quite do the same with all $\cV_k$ just equal to $\cU$, but we can get by if we give ourselves arbitrarily little room to maneuver, by making the blobs we act on slightly smaller. \end{rem} \begin{proof} +This follows from the remark \nn{number it and cite it?} following the proof of +Proposition \ref{CHprop}. +\end{proof} +\noop{ We choose yet another open cover, $\cW$, which so fine that the union (disjoint or not) of any one open set $V \in \cV$ with $k$ open sets $W_i \in \cW$ is contained in a disjoint union of open sets of $\cU$. Now, in the proof of Proposition \ref{CHprop} -\todo{I think I need to understand better that proof before I can write this!} -\end{proof} +[...] +} \begin{proof}[Proof of Theorem \ref{thm:small-blobs}] diff -r eec4b1f9cfc2 -r facac77e9a72 text/deligne.tex --- a/text/deligne.tex Wed Jun 23 10:54:29 2010 -0400 +++ b/text/deligne.tex Wed Jun 23 10:54:42 2010 -0400 @@ -11,7 +11,7 @@ (Proposition \ref{prop:deligne} below). Then we sketch the proof. -\nn{Does this generalisation encompass Kontsevich's proposed generalisation from \cite[\S2.5]{MR1718044}, +\nn{Does this generalization encompass Kontsevich's proposed generalization from \cite[\S2.5]{MR1718044}, that (I think...) the Hochschild homology of an $E_n$ algebra is an $E_{n+1}$ algebra? -S} %from http://www.ams.org/mathscinet-getitem?mr=1805894 @@ -195,7 +195,7 @@ \stackrel{f_k}{\to} \bc_*(N_0) \] (Recall that the maps $\id\ot\alpha_i$ were defined in \nn{need ref}.) -\nn{issue: haven't we only defined $\id\ot\alpha_i$ when $\alpha_i$ is closed?} +\nn{need to double check case where $\alpha_i$'s are not closed.} It is easy to check that the above definition is compatible with the equivalence relations and also the operad structure. We can reinterpret the above as a chain map diff -r eec4b1f9cfc2 -r facac77e9a72 text/evmap.tex --- a/text/evmap.tex Wed Jun 23 10:54:29 2010 -0400 +++ b/text/evmap.tex Wed Jun 23 10:54:42 2010 -0400 @@ -618,7 +618,6 @@ \end{proof} -\noop{ \nn{this should perhaps be a numbered remark, so we can cite it more easily} @@ -626,11 +625,13 @@ For the proof of xxxx below we will need the following observation on the action constructed above. Let $b$ be a blob diagram and $p:P\times X\to X$ be a family of homeomorphisms. Then we may choose $e$ such that $e(p\ot b)$ is a sum of generators, each -of which has support arbitrarily close to $p(t,|b|)$ for some $t\in P$. -This follows from the fact that the -\nn{not correct, since there could also be small balls far from $|b|$} +of which has support close to $p(t,|b|)$ for some $t\in P$. +More precisely, the support of the generators is contained in a small neighborhood +of $p(t,|b|)$ union some small balls. +(Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.) \end{rem} -} + + \begin{prop} The $CH_*(X, Y)$ actions defined above are associative. diff -r eec4b1f9cfc2 -r facac77e9a72 text/ncat.tex --- a/text/ncat.tex Wed Jun 23 10:54:29 2010 -0400 +++ b/text/ncat.tex Wed Jun 23 10:54:42 2010 -0400 @@ -64,7 +64,7 @@ They could be topological or PL or smooth. %\nn{need to check whether this makes much difference} (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need -to be fussier about corners.) +to be fussier about corners and boundaries.) For each flavor of manifold there is a corresponding flavor of $n$-category. We will concentrate on the case of PL unoriented manifolds. @@ -1521,36 +1521,37 @@ gluing subintervals together and/or omitting some of the rightmost subintervals. (See Figure \ref{fig:lmar}.) \begin{figure}[t]$$ -\begin{tikzpicture} +\definecolor{arcolor}{rgb}{.75,.4,.1} +\begin{tikzpicture}[line width=1pt] \fill (0,0) circle (.1); \draw (0,0) -- (2,0); \draw (1,0.1) -- (1,-0.1); -\draw [->,red] (1,0.25) -- (1,0.75); +\draw [->, arcolor] (1,0.25) -- (1,0.75); \fill (0,1) circle (.1); \draw (0,1) -- (2,1); \end{tikzpicture} \qquad -\begin{tikzpicture} +\begin{tikzpicture}[line width=1pt] \fill (0,0) circle (.1); \draw (0,0) -- (2,0); \draw (1,0.1) -- (1,-0.1); -\draw [->,red] (1,0.25) -- (1,0.75); +\draw [->, arcolor] (1,0.25) -- (1,0.75); \fill (0,1) circle (.1); \draw (0,1) -- (1,1); \end{tikzpicture} \qquad -\begin{tikzpicture} +\begin{tikzpicture}[line width=1pt] \fill (0,0) circle (.1); \draw (0,0) -- (3,0); \foreach \x in {0.5, 1.0, 1.25, 1.5, 2.0, 2.5} { \draw (\x,0.1) -- (\x,-0.1); } -\draw [->,red] (1,0.25) -- (1,0.75); +\draw [->, arcolor] (1,0.25) -- (1,0.75); \fill (0,1) circle (.1); \draw (0,1) -- (2,1); @@ -1586,8 +1587,7 @@ where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and $\cbar''$ corresponds to the subintervals which are dropped off the right side. -(Either $\cbar'$ or $\cbar''$ might be empty.) -\nn{surely $\cbar'$ can't be empy: we don't allow $D_1$ to be empty.} +(If no such subintervals are dropped, then $\cbar''$ is empty.) Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ appearing in Equation \eqref{eq:tensor-product-boundary}, we have \begin{eqnarray*} @@ -1645,6 +1645,11 @@ \[ g\ot\id : \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ . \] +\nn{...} +More generally, we have a chain map +\[ + \hom_\cC(\cX_\cC \to \cY_\cC) \ot \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ . +\] \nn{not sure whether to do low degree examples or try to state the general case; ideally both, but maybe just low degrees for now.} @@ -1677,10 +1682,12 @@ whose objects are $n$-categories. When $n=2$ this is a version of the familiar algebras-bimodules-intertwiners $2$-category. -While it is clearly appropriate to call an $S^0$ module a bimodule, +It is clearly appropriate to call an $S^0$ module a bimodule, but this is much less true for higher dimensional spheres, so we prefer the term ``sphere module" for the general case. +For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces. + The $0$- through $n$-dimensional parts of $\cC$ are various sorts of modules, and we describe these first. The $n{+}1$-dimensional part of $\cS$ consists of intertwiners @@ -1706,7 +1713,7 @@ Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. \begin{figure}[!ht] -$$\tikz[baseline,line width=2pt]{\draw[blue] (-2,0)--(2,0); \fill[red] (0,0) circle (0.1);} \qquad \qquad \tikz[baseline,line width=2pt]{\draw[blue] (0,0) circle (2 and 1); \draw[red] (0,1)--(0,-1);}$$ +$$\tikz[baseline,line width=2pt]{\draw[blue] (-2,0)--(2,0); \fill[red] (0,0) circle (0.1);} \qquad \qquad \tikz[baseline,line width=2pt]{\draw[blue][fill=blue!30!white] (0,0) circle (2 and 1); \draw[red] (0,1)--(0,-1);}$$ \caption{0-marked 1-ball and 0-marked 2-ball} \label{feb21a} \end{figure} @@ -1731,7 +1738,7 @@ or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side) or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball). Corresponding to this decomposition we have an action and/or composition map -from the product of these various sets into $\cM(X)$. +from the product of these various sets into $\cM_k(X)$. \medskip @@ -1756,7 +1763,7 @@ \draw (2,0) -- (4,0) node[below] {$J$}; \fill[red] (3,0) circle (0.1); -\draw (6,0) node(a) {} arc (135:90:4) node(top) {} arc (90:45:4) node(b) {} arc (-45:-90:4) node(bottom) {} arc(-90:-135:4); +\draw[fill=blue!30!white] (6,0) node(a) {} arc (135:90:4) node(top) {} arc (90:45:4) node(b) {} arc (-45:-90:4) node(bottom) {} arc(-90:-135:4); \draw[red] (top.center) -- (bottom.center); \fill (a) circle (0.1) node[left] {\color{green!50!brown} $a$}; \fill (b) circle (0.1) node[right] {\color{green!50!brown} $b$}; @@ -1831,7 +1838,7 @@ \begin{figure}[!ht] $$ \begin{tikzpicture}[baseline,line width = 2pt] -\draw[blue] (0,0) circle (2); +\draw[blue][fill=blue!15!white] (0,0) circle (2); \fill[red] (0,0) circle (0.1); \foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} { \draw[red] (0,0) -- (\qm:2); @@ -1871,7 +1878,7 @@ \medskip -We can now define the $n$- or less dimensional part of our $n{+}1$-category $\cS$. +We can now define the $n$-or-less-dimensional part of our $n{+}1$-category $\cS$. Choose some collection of $n$-categories, then choose some collections of bimodules for these $n$-categories, then choose some collection of 1-sphere modules for the various possible marked 1-spheres labeled by the $n$-categories and bimodules, and so on. @@ -1892,7 +1899,7 @@ Thus the $k$-morphisms of $\cS$ (for $k\le n$) can be thought of as $n$-category $k{-}1$-sphere modules (generalizations of bimodules). -On the other hand, we can equally think of the $k$-morphisms as decorations on $k$-balls, +On the other hand, we can equally well think of the $k$-morphisms as decorations on $k$-balls, and from this (official) point of view it is clear that they satisfy all of the axioms of an $n{+}1$-category. (All of the axioms for the less-than-$n{+}1$-dimensional part of an $n{+}1$-category, that is.) @@ -1900,12 +1907,40 @@ \medskip Next we define the $n{+}1$-morphisms of $\cS$. +The construction of the 0- through $n$-morphisms was easy and tautological, but the +$n{+}1$-morphisms will require a bit of combinatorial topology effort, as well as addition +duality assumptions on the lower morphisms. - +Let $X$ be an $n{+}1$-ball, and let $c$ be a decoration of its boundary +by a cell complex labeled by 0- through $n$-morphisms, as above. +Choose an $n{-}1$-sphere $E\sub \bd X$ which divides +$\bd X$ into ``incoming" and ``outgoing" boundary $\bd_-X$ and $\bd_+X$. +Let $E_c$ denote $E$ decorated by the restriction of $c$ to $E$. +Recall from above the associated 1-category $\cS(E_c)$. +We can also have $\cS(E_c)$ modules $\cS(\bd_-X_c)$ and $\cS(\bd_+X_c)$. +Define +\[ + \cS(X; c; E) \deq \hom_{\cS(E_c)}(\cS(\bd_-X_c), \cS(\bd_+X_c)) . +\] - +We will show that if the sphere modules are equipped with a compatible family of +non-degenerate inner products, then there is a coherent family of isomorphisms +$\cS(X; c; E) \cong \cS(X; c; E')$ for all pairs of choices $E$ and $E'$. +This will allow us to define $\cS(X; e)$ independently of the choice of $E$. - +Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image. +(We assume we are working in the unoriented category.) +Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$ +along their common boundary. +An {\it inner product} on $\cS(Y)$ is a dual vector +\[ + z_Y : \cS(Y\cup\ol{Y}) \to \c. +\] +We will also use the notation +\[ + \langle a, b\rangle \deq z_Y(a\bullet \ol{b}) \in \c . +\] +An inner product is {\it non-degenerate} if \nn{...} @@ -1924,10 +1959,7 @@ Stuff that remains to be done (either below or in an appendix or in a separate section or in a separate paper): \begin{itemize} -\item spell out what difference (if any) Top vs PL vs Smooth makes \item discuss Morita equivalence -\item morphisms of modules; show that it's adjoint to tensor product -(need to define dual module for this) \item functors \end{itemize}