diff -r 36bfe7c2eecc -r f40f726d6cca text/ncat.tex --- a/text/ncat.tex Wed Jun 29 11:51:35 2011 -0700 +++ b/text/ncat.tex Wed Jun 29 12:02:47 2011 -0700 @@ -34,9 +34,8 @@ The axioms for an $n$-category are spread throughout this section. Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, -\ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:vcones} and -\ref{axiom:extended-isotopies}. -For an enriched $n$-category we add \ref{axiom:enriched}. +\ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:extended-isotopies} and \ref{axiom:vcones}. +For an enriched $n$-category we add Axiom \ref{axiom:enriched}. For an $A_\infty$ $n$-category, we replace Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. @@ -579,7 +578,7 @@ Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. (Keep in mind the important special case where $f$ restricted to $\bd X$ is the identity.) -Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which +Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which act trivially on $\bd b$. Then $f(b) = b$. In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially on @@ -654,7 +653,7 @@ The revised axiom is %\addtocounter{axiom}{-1} -\begin{axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$] +\begin{axiom}[Extended isotopy invariance in dimension $n$] \label{axiom:extended-isotopies} Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. @@ -876,7 +875,7 @@ or more generally an appropriate sort of $\infty$-category, we can modify the extended isotopy axiom \ref{axiom:extended-isotopies} to require that families of homeomorphisms act -and obtain an $A_\infty$ $n$-category. +and obtain what we shall call an $A_\infty$ $n$-category. \noop{ We believe that abstract definitions should be guided by diverse collections @@ -892,7 +891,7 @@ and let $\cJ$ be an $\infty$-functor from topological spaces to $\cS$ (e.g.\ the singular chain functor $C_*$). -\begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] +\begin{axiom}[\textup{\textbf{[$A_\infty$ replacement for Axiom \ref{axiom:extended-isotopies}]}} Families of homeomorphisms act in dimension $n$.] \label{axiom:families} For each pair of $n$-balls $X$ and $X'$ and each pair $c\in \cl{\cC}(\bd X)$ and $c'\in \cl{\cC}(\bd X')$ we have an $\cS$-morphism \[ @@ -913,12 +912,12 @@ We now describe the topology on $\Coll(X; c)$. We retain notation from the above definition of collar map. Each collaring homeomorphism $X \cup (Y\times J) \to X$ determines a map from points $p$ of $\bd X$ to -(possibly zero-width) embedded intervals in $X$ terminating at $p$. +(possibly length zero) embedded intervals in $X$ terminating at $p$. If $p \in Y$ this interval is the image of $\{p\}\times J$. -If $p \notin Y$ then $p$ is assigned the zero-width interval $\{p\}$. +If $p \notin Y$ then $p$ is assigned the length zero interval $\{p\}$. Such collections of intervals have a natural topology, and $\Coll(X; c)$ inherits its topology from this. -Note in particular that parts of the collar are allowed to shrink continuously to zero width. -(This is the real content; if nothing shrinks to zero width then the action of families of collar +Note in particular that parts of the collar are allowed to shrink continuously to zero length. +(This is the real content; if nothing shrinks to zero length then the action of families of collar maps follows from the action of families of homeomorphisms and compatibility with gluing.) The $k=n$ case of Axiom \ref{axiom:morphisms} posits a {\it strictly} associative action of {\it sets} @@ -1118,9 +1117,9 @@ The case $n=d$ captures the $n$-categorical nature of bordisms. The case $n > 2d$ captures the full symmetric monoidal $n$-category structure. \end{example} -\begin{remark} +\begin{rem} Working with the smooth bordism category would require careful attention to either collars, corners or halos. -\end{remark} +\end{rem} %\nn{the next example might be an unnecessary distraction. consider deleting it.} @@ -1344,7 +1343,7 @@ Let $\du_a X_a = M_0\to\cdots\to M_m = W$ be a ball decomposition compatible with $x$. Let $\bd M_i = N_i \cup Y_i \cup Y'_i$, where $Y_i$ and $Y'_i$ are glued together to produce $M_{i+1}$. -We will define $\psi_{\cC;W}(x)$ be be the subset of $\prod_a \cC(X_a)$ which satisfies a series of conditions +We will define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies a series of conditions related to the gluings $M_{i-1} \to M_i$, $1\le i \le m$. By Axiom \ref{nca-boundary}, we have a map \[ @@ -1361,7 +1360,7 @@ along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\cl\cC(Y_1)$ and $\cl\cC(Y'_1)$ agree (with respect to the identification of $Y_1$ and $Y'_1$ provided by the gluing map). The $i$-th condition is defined similarly. -Note that these conditions depend on on the boundaries of elements of $\prod_a \cC(X_a)$. +Note that these conditions depend on the boundaries of elements of $\prod_a \cC(X_a)$. We define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies the above conditions for all $i$ and also all @@ -1440,7 +1439,7 @@ of the splitting along $\bd Y$, and this implies that the combined decomposition $\du_{ia} X_{ia}$ is permissible. We can now define the gluing $y_1\bullet y_2$ in the obvious way, and a further application of Axiom \ref{axiom:vcones} -shows that this is independebt of the choices of representatives of $y_i$. +shows that this is independent of the choices of representatives of $y_i$. \medskip @@ -1454,7 +1453,7 @@ \] where $x$ runs through decompositions of $W$, and $\sim$ is the obvious equivalence relation induced by refinement and gluing. -If $\cC$ is enriched over vector spaces and $W$ is an $n$-manifold, +If $\cC$ is enriched over, for example, vector spaces and $W$ is an $n$-manifold, we can take \begin{equation*} \cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K,