# HG changeset patch # User Kevin Walker # Date 1279046878 21600 # Node ID 9576c3d68a3ddaa1545dd63af5e89910389786e6 # Parent c4c1a01a900920983cb93635290b752b1402916b# Parent 84834a1fdd50d6b4e15d4998e970c85973146a60 Automated merge with https://tqft.net/hg/blob/ diff -r c4c1a01a9009 -r 9576c3d68a3d text/evmap.tex --- a/text/evmap.tex Mon Jul 12 21:46:32 2010 -0600 +++ b/text/evmap.tex Tue Jul 13 12:47:58 2010 -0600 @@ -122,7 +122,7 @@ Now for a little more detail. (But we're still just motivating the full, gory details, which will follow.) -Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of balls of radius $\gamma$. +Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of $X$ by balls of radius $\gamma$. By Lemma \ref{extension_lemma} we can restrict our attention to $k$-parameter families $p$ of homeomorphisms such that $\supp(p)$ is contained in the union of $k$ $\gamma$-balls. For fixed blob diagram $b$ and fixed $k$, it's not hard to show that for $\gamma$ small enough @@ -153,7 +153,7 @@ We'll use the notation $|b| = \supp(b)$ and $|p| = \supp(p)$. Choose a metric on $X$. -Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging monotonically to zero +Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero (e.g.\ $\ep_i = 2^{-i}$). Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$ converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$). @@ -177,7 +177,7 @@ is homeomorphic to a disjoint union of balls and \[ N_{i,k}(p\ot b) \subeq V_0 \subeq N_{i,k+1}(p\ot b) - \subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) . + \subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) , \] and further $\bd(p\ot b) \in G_*^{i,m}$. We also require that $b$ is splitable (transverse) along the boundary of each $V_l$. @@ -345,7 +345,8 @@ \begin{proof} There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ . -(Here we are using the fact that the blobs are piecewise-linear and thatthat $\bd c$ is collared.) +(Here we are using the fact that the blobs are +piecewise smooth or piecewise-linear and that $\bd c$ is collared.) We need to consider all such $c$ because all generators appearing in iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.) diff -r c4c1a01a9009 -r 9576c3d68a3d text/ncat.tex --- a/text/ncat.tex Mon Jul 12 21:46:32 2010 -0600 +++ b/text/ncat.tex Tue Jul 13 12:47:58 2010 -0600 @@ -1434,11 +1434,6 @@ let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. -%Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), -%and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary -%component $\bd_i W$ of $W$. -%(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.) - We will define a set $\cC(W, \cN)$ using a colimit construction similar to the one appearing in \S \ref{ss:ncat_fields} above. (If $k = n$ and our $n$-categories are enriched, then @@ -1448,15 +1443,18 @@ \[ W = \left(\bigcup_a X_a\right) \cup \left(\bigcup_{i,b} M_{ib}\right) , \] -where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and -each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, +where each $X_a$ is a plain $k$-ball (disjoint from $\cup Y_i$) and +each $M_{ib}$ is a marked $k$-ball intersecting $Y_i$, with $M_{ib}\cap Y_i$ being the marking. (See Figure \ref{mblabel}.) -\begin{figure}[!ht]\begin{equation*} +\begin{figure}[t] +\begin{equation*} \mathfig{.4}{ncat/mblabel} -\end{equation*}\caption{A permissible decomposition of a manifold +\end{equation*} +\caption{A permissible decomposition of a manifold whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. -Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure} +Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel} +\end{figure} Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. This defines a partial ordering $\cell(W)$, which we will think of as a category. @@ -1472,23 +1470,25 @@ \] such that the restrictions to the various pieces of shared boundaries amongst the $X_a$ and $M_{ib}$ all agree. -(That is, the fibered product over the boundary maps.) +(That is, the fibered product over the boundary restriction maps.) If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. We now define the set $\cC(W, \cN)$ to be the colimit of the functor $\psi_\cN$. -(As usual, if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means -homotopy colimit.) +(As in \S\ref{ss:ncat-coend}, if $k=n$ we take a colimit in whatever +category we are enriching over, and if additionally we are in the $A_\infty$ case, +then we use a homotopy colimit.) + +\medskip If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define $\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold $D\times Y_i \sub \bd(D\times W)$. It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$ -has the structure of an $n{-}k$-category, which we call $\cT(W, \cN)(D)$. +has the structure of an $n{-}k$-category. \medskip - We will use a simple special case of the above construction to define tensor products of modules. @@ -1497,7 +1497,7 @@ a left module and the other a right module.) Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. Define the tensor product $\cM_1 \tensor \cM_2$ to be the -$n{-}1$-category $\cT(J, \{\cM_1, \cM_2\})$. +$n{-}1$-category associated as above to $J$ with its boundary labeled by $\cM_1$ and $\cM_2$. This of course depends (functorially) on the choice of 1-ball $J$.