diff -r a3631a999462 -r 16efb5711c6f text/ncat.tex --- a/text/ncat.tex Sun Dec 13 01:32:28 2009 +0000 +++ b/text/ncat.tex Wed Dec 16 19:30:13 2009 +0000 @@ -5,15 +5,6 @@ \section{$n$-categories} \label{sec:ncats} -%In order to make further progress establishing properties of the blob complex, -%we need a definition of $A_\infty$ $n$-category that is adapted to our needs. -%(Even in the case $n=1$, we need the new definition given below.) -%It turns out that the $A_\infty$ $n$-category definition and the plain $n$-category -%definition are mostly the same, so we give a new definition of plain -%$n$-categories too. -%We also define modules and tensor products for both plain and $A_\infty$ $n$-categories. - - \subsection{Definition of $n$-categories} Before proceeding, we need more appropriate definitions of $n$-categories, @@ -329,7 +320,46 @@ (See Figure \ref{glue-collar}.) \begin{figure}[!ht] \begin{equation*} -\mathfig{.9}{tempkw/blah10} +\begin{tikzpicture} +\def\rad{1} +\def\srad{0.75} +\def\gap{4.5} +\foreach \i in {0, 1, 2} { + \node(\i) at ($\i*(\gap,0)$) [draw, circle through = {($\i*(\gap,0)+(\rad,0)$)}] {}; + \node(\i-small) at (\i.east) [circle through={($(\i.east)+(\srad,0)$)}] {}; + \foreach \n in {1,2} { + \fill (intersection \n of \i-small and \i) node(\i-intersection-\n) {} circle (2pt); + } +} + +\begin{scope}[decoration={brace,amplitude=10,aspect=0.5}] + \draw[decorate] (0-intersection-1.east) -- (0-intersection-2.east); +\end{scope} +\node[right=1mm] at (0.east) {$a$}; +\draw[->] ($(0.east)+(0.75,0)$) -- ($(1.west)+(-0.2,0)$); + +\draw (1-small) circle (\srad); +\foreach \theta in {90, 72, ..., -90} { + \draw[blue] (1) -- ($(1)+(\rad,0)+(\theta:\srad)$); +} +\filldraw[fill=white] (1) circle (\rad); +\foreach \n in {1,2} { + \fill (intersection \n of 1-small and 1) circle (2pt); +} +\node[below] at (1-small.south) {$a \times J$}; +\draw[->] ($(1.east)+(1,0)$) -- ($(2.west)+(-0.2,0)$); + +\begin{scope} +\path[clip] (2) circle (\rad); +\draw[clip] (2.east) circle (\srad); +\foreach \y in {1, 0.86, ..., -1} { + \draw[blue] ($(2)+(-1,\y) $)-- ($(2)+(1,\y)$); +} +\end{scope} +\end{tikzpicture} +\end{equation*} +\begin{equation*} +\xymatrix@C+2cm{\cC(X) \ar[r]^(0.45){\text{glue}} & \cC(X \cup \text{collar}) \ar[r]^(0.55){\text{homeo}} & \cC(X)} \end{equation*} \caption{Extended homeomorphism.}\label{glue-collar}\end{figure} @@ -345,12 +375,13 @@ The revised axiom is -\begin{axiom}[Extended isotopy invariance in dimension $n$] +\stepcounter{axiom} +\begin{axiom-numbered}{\arabic{axiom}a}{Extended isotopy invariance in dimension $n$} \label{axiom:extended-isotopies} Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity. Then $f$ acts trivially on $\cC(X)$. -\end{axiom} +\end{axiom-numbered} \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} @@ -360,7 +391,7 @@ isotopy invariance with the requirement that families of homeomorphisms act. For the moment, assume that our $n$-morphisms are enriched over chain complexes. -\begin{axiom}[Families of homeomorphisms act in dimension $n$] +\begin{axiom-numbered}{\arabic{axiom}b}{Families of homeomorphisms act in dimension $n$} For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes \[ C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . @@ -372,7 +403,7 @@ a diagram like the one in Proposition \ref{CDprop} commutes. \nn{repeat diagram here?} \nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} -\end{axiom} +\end{axiom-numbered} We should strengthen the above axiom to apply to families of extended homeomorphisms. To do this we need to explain how extended homeomorphisms form a topological space. @@ -412,13 +443,15 @@ \medskip +\subsection{Examples of $n$-categories} + \nn{these examples need to be fleshed out a bit more} -Examples of plain $n$-categories: -\begin{itemize} +We know describe several classes of examples of $n$-categories satisfying our axioms. -\item Let $F$ be a closed $m$-manifold (e.g.\ a point). -Let $T$ be a topological space. +\begin{example}{Maps to a space} +\label{ex:maps-to-a-space}% +Fix $F$ a closed $m$-manifold (keep in mind the case where $F$ is a point). Fix a `target space' $T$, any topological space. For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\cC(X)$ to be the set of all maps from $X\times F$ to $T$. For $X$ an $n$-ball define $\cC(X)$ to be maps from $X\times F$ to $T$ modulo @@ -426,8 +459,11 @@ (Note that homotopy invariance implies isotopy invariance.) For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. +\end{example} -\item We can linearize the above example as follows. +\begin{example}{Linearized, twisted, maps to a space} +\label{ex:linearized-maps-to-a-space}% +We can linearize the above example as follows. Let $\alpha$ be an $(n{+}m{+}1)$-cocycle on $T$ with values in a ring $R$ (e.g.\ the trivial cocycle). For $X$ of dimension less than $n$ define $\cC(X)$ as before. @@ -436,6 +472,11 @@ modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy $h: X\times F\times I \to T$, then $a \sim \alpha(h)b$. \nn{need to say something about fundamental classes, or choose $\alpha$ carefully} +\end{example} + +\begin{itemize} + +\item \nn{Continue converting these into examples} \item Given a traditional $n$-category $C$ (with strong duality etc.), define $\cC(X)$ (with $\dim(X) < n$) @@ -473,22 +514,25 @@ \end{itemize} -Examples of $A_\infty$ $n$-categories: -\begin{itemize} +We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex. -\item Same as in example \nn{xxxx} above (fiber $F$, target space $T$), -but we define, for an $n$-ball $X$, $\cC(X; c)$ to be the chain complex -$C_*(\Maps_c(X\times F))$, where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, +\begin{example}{Chains of maps to a space} +We can modify Example \ref{ex:maps-to-a-space} above by defining $\cC(X; c)$ for an $n$-ball $X$ to be the chain complex +$C_*(\Maps_c(X\times F \to T))$, where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, and $C_*$ denotes singular chains. +\end{example} -\item +\begin{example}{Blob complexes of balls (with a fiber)} +Fix an $m$-dimensional manifold $F$. Given a plain $n$-category $C$, -define $\cC(X; c) = \bc^C_*(X\times F; c)$, where $X$ is an $n$-ball -and $\bc^C_*$ denotes the blob complex based on $C$. +when $X$ is a $k$-ball or $k$-sphere, with $k