# HG changeset patch # User Scott Morrison # Date 1289777943 28800 # Node ID fd6e53389f2c1f9e56eee385b9010e8de5266e79 # Parent 455106e40a611e7d21f2c047c6702542afe06044 futzing with preambles diff -r 455106e40a61 -r fd6e53389f2c pnas/pnas.tex --- a/pnas/pnas.tex Sat Nov 13 20:58:40 2010 -0800 +++ b/pnas/pnas.tex Sun Nov 14 15:39:03 2010 -0800 @@ -74,7 +74,6 @@ %\def\s{\sigma} \input{preamble} -\input{../text/kw_macros} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Don't type in anything in the following section: @@ -374,7 +373,7 @@ Product morphisms are compatible with gluing. Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ be pinched products with $E = E_1\cup E_2$. -Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$. +Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\subset X$. Then \[ \pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . @@ -401,7 +400,7 @@ \end{axiom} To state the next axiom we need the notion of {\it collar maps} on $k$-morphisms. -Let $X$ be a $k$-ball and $Y\sub\bd X$ be a $(k{-}1)$-ball. +Let $X$ be a $k$-ball and $Y\subset\bd X$ be a $(k{-}1)$-ball. Let $J$ be a 1-ball. Let $Y\times_p J$ denote $Y\times J$ pinched along $(\bd Y)\times J$. A collar map is an instance of the composition @@ -434,7 +433,7 @@ \label{axiom:families} For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes \[ - C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . + C_*(\Homeo_\bd(X))\tensor \cC(X; c) \to \cC(X; c) . \] These action maps are required to be associative up to homotopy, and also compatible with composition (gluing) in the sense that @@ -464,7 +463,6 @@ Define product morphisms via product cell decompositions. -\nn{also do bordism category} \subsection{The blob complex} \subsubsection{Decompositions of manifolds} @@ -497,7 +495,7 @@ a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets (possibly with additional structure if $k=n$). Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, -and there is a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries +and there is a subset $\cC(X)\spl \subset \cC(X)$ of morphisms whose boundaries are splittable along this decomposition. \begin{defn} @@ -505,7 +503,7 @@ For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset \begin{equation*} %\label{eq:psi-C} - \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl + \psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl \end{equation*} where the restrictions to the various pieces of shared boundaries amongst the cells $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). When $k=n$, the `subset' and `product' in the above formula should be interpreted in the appropriate enriching category. @@ -673,10 +671,10 @@ (using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). \begin{equation*} \xymatrix@C+0.3cm{ - \CH{X} \otimes \bc_*(X) - \ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y} & + \CH{X} \tensor \bc_*(X) + \ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \tensor \gl_Y} & \bc_*(X) \ar[d]_{\gl_Y} \\ - \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]_<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) + \CH{X \bigcup_Y \selfarrow} \tensor \bc_*(X \bigcup_Y \selfarrow) \ar[r]_<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) } \end{equation*} \end{enumerate} @@ -782,7 +780,7 @@ The little disks operad $LD$ is homotopy equivalent to the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cohains $Hoch^*(C, C)$. The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map \[ - C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} + C_*(LD_k)\tensor \overbrace{Hoch^*(C, C)\tensor\cdots\tensor Hoch^*(C, C)}^{\text{$k$ copies}} \to Hoch^*(C, C), \] which we now see to be a specialization of Theorem \ref{thm:deligne}. diff -r 455106e40a61 -r fd6e53389f2c pnas/preamble.tex --- a/pnas/preamble.tex Sat Nov 13 20:58:40 2010 -0800 +++ b/pnas/preamble.tex Sun Nov 14 15:39:03 2010 -0800 @@ -39,11 +39,16 @@ \newcommand{\tensor}{\otimes} \newcommand{\Tensor}{\bigotimes} +\newcommand{\Bord}{\operatorname{Bord}} + \newcommand{\selfarrow}{\ensuremath{\smash{\tikz[baseline]{\clip (0,0.36) rectangle (0.39,-0.16); \draw[->] (0,0.2) .. controls (0.5,0.6) and (0.5,-0.4) .. (0,0);}}}} +\newcommand{\bd}{\partial} \newcommand{\bdy}{\partial} +\newcommand{\du}{\sqcup} \newcommand{\compose}{\circ} \newcommand{\eset}{\emptyset} +\def\spl{_\pitchfork} \newcommand{\id}{\boldsymbol{1}} @@ -71,6 +76,42 @@ \def\mathclapinternal#1#2{% \clap{$\mathsurround=0pt#1{#2}$}} +% tricky way to iterate macros over a list +\def\semicolon{;} +\def\applytolist#1{ + \expandafter\def\csname multi#1\endcsname##1{ + \def\multiack{##1}\ifx\multiack\semicolon + \def\next{\relax} + \else + \csname #1\endcsname{##1} + \def\next{\csname multi#1\endcsname} + \fi + \next} + \csname multi#1\endcsname} + +% \def\cA{{\cal A}} for A..Z +\def\calc#1{\expandafter\def\csname c#1\endcsname{{\mathcal #1}}} +\applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM; + +% \def\bbA{{\mathbb A}} for A..Z +\def\bbc#1{\expandafter\def\csname bb#1\endcsname{{\mathbb #1}}} +\applytolist{bbc}QWERTYUIOPLKJHGFDSAZXCVBNM; + +% \def\bA{{\mathbf A}} for A..Z +\def\bc#1{\expandafter\def\csname b#1\endcsname{{\mathbf #1}}} +\applytolist{bc}QWERTYUIOPLKJHGFDSAZXCVBNM; + +% \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}; + +% \todo, \nn and \noop +\newcommand{\todo}[1]{\textbf{\color[rgb]{.8,.2,.5}\small TODO: #1}} +\def\nn#1{{{\color[rgb]{.2,.5,.6} \small [[#1]]}}} +\long\def\noop#1{} + + + % references \newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\tt arXiv:\nolinkurl{#1}}} @@ -80,8 +121,6 @@ \newcommand{\googlebooks}[1]{(preview at \href{http://books.google.com/books?id=#1}{google books})} -\newcommand{\todo}[1]{\textbf{\color[rgb]{.8,.2,.5}\small TODO: #1}} - % figures diff -r 455106e40a61 -r fd6e53389f2c preamble.tex --- a/preamble.tex Sat Nov 13 20:58:40 2010 -0800 +++ b/preamble.tex Sun Nov 14 15:39:03 2010 -0800 @@ -173,6 +173,8 @@ \newcommand{\Kom}{\text{\textbf{Kom}}} \newcommand{\Cat}{\mathcal{C}} +\newcommand{\Bord}{\operatorname{Bord}} + \newcommand{\Inv}[1]{\operatorname{Inv}\left(#1\right)} \newcommand{\Hom}[3]{\operatorname{Hom}_{#1}\left(#2,#3\right)} \newcommand{\End}[1]{\operatorname{End}\left(#1\right)} diff -r 455106e40a61 -r fd6e53389f2c text/ncat.tex --- a/text/ncat.tex Sat Nov 13 20:58:40 2010 -0800 +++ b/text/ncat.tex Sun Nov 14 15:39:03 2010 -0800 @@ -771,7 +771,6 @@ } -\newcommand{\Bord}{\operatorname{Bord}} \begin{example}[The bordism $n$-category, plain version] \label{ex:bord-cat} \rm