# HG changeset patch # User Kevin Walker # Date 1306343296 21600 # Node ID 91d32d0cb2efd715233e8286b34834c155e8eddb # Parent 36cffad93a4a838cd29cb5f3f2a3077599bcadf2 corrected statement of module to category restrictions; note that this affects the numbering of items in subsection 6.4 diff -r 36cffad93a4a -r 91d32d0cb2ef blob to-do --- a/blob to-do Wed May 25 09:48:01 2011 -0600 +++ b/blob to-do Wed May 25 11:08:16 2011 -0600 @@ -47,11 +47,6 @@ modules: -* Lemma 6.4.5 needs to actually construct this map! Needs more input! Do -we actually need this as written? - - KW will look at it; probably needs to be weakened - - * SCOTT: typo in delfig3a -- upper g should be g^{-1} * SCOTT: make sure acknowledge list doesn't omit anyone from blob seminar who should be included (I think I have all the speakers; does anyone other than the speakers rate a mention?) diff -r 36cffad93a4a -r 91d32d0cb2ef blob_changes_v3 --- a/blob_changes_v3 Wed May 25 09:48:01 2011 -0600 +++ b/blob_changes_v3 Wed May 25 11:08:16 2011 -0600 @@ -20,9 +20,10 @@ - added remarks about categories of defects - clarified that the "cell complexes" in string diagrams are actually a bit more general - added remark to insure that the poset of decompositions is a small category -- rewrote definition of colimit (in "From Balls to Manifolds" subsection) to allow for more general decompositions; also added more details -- +- corrected statement of module to category restrictions +INCOMPLETE: +- rewrote definition of colimit (in "From Balls to Manifolds" subsection) to allow for more general decompositions; also added more details diff -r 36cffad93a4a -r 91d32d0cb2ef text/ncat.tex --- a/text/ncat.tex Wed May 25 09:48:01 2011 -0600 +++ b/text/ncat.tex Wed May 25 11:08:16 2011 -0600 @@ -1365,13 +1365,23 @@ Let $\cl\cM(H)\trans E$ denote the image of $\gl_E$. We will refer to elements of $\cl\cM(H)\trans E$ as ``splittable along $E$" or ``transverse to $E$". +\noop{ %%%%%%% \begin{lem}[Module to category restrictions] {For each marked $k$-hemisphere $H$ there is a restriction map -$\cl\cM(H)\to \cC(H)$. +$\cl\cM(H)\to \cC(H)$. ($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) These maps comprise a natural transformation of functors.} \end{lem} +} %%%%%%% end \noop +It follows from the definition of the colimit $\cl\cM(H)$ that +given any (unmarked) $k{-}1$-ball $Y$ in the interior of $H$ there is a restriction map +from a subset $\cl\cM(H)_{\trans{\bdy Y}}$ of $\cl\cM(H)$ to $\cC(Y)$. +Combining this with the boundary map $\cM(B,N) \to \cl\cM(\bd(B,N))$, we also have a restriction +map from a subset $\cM(B,N)_{\trans{\bdy Y}}$ of $\cM(B,N)$ to $\cC(Y)$ whenever $Y$ is in the interior of $\bd B \setmin N$. +This fact will be used below. + +\noop{ %%%% Note that combining the various boundary and restriction maps above (for both modules and $n$-categories) we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ @@ -1379,6 +1389,7 @@ This subset $\cM(B,N)\trans{\bdy Y}$ is the subset of morphisms which are appropriately splittable (transverse to the cutting submanifolds). This fact will be used below. +} %%%%% end \noop In our example, the various restriction and gluing maps above come from restricting and gluing maps into $T$.