blob: comparison text/ncat.tex

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 \medskip
 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{nca-boundary}, \ref{axiom:composition},  \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:extended-isotopies} and  \ref{axiom:vcones}.
-\ref{axiom:extended-isotopies}.
+For an enriched $n$-category we add Axiom \ref{axiom:enriched}.
-For an enriched $n$-category we add \ref{axiom:enriched}.
 For an $A_\infty$ $n$-category, we replace
 Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}.
 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms
 for $k{-}1$-morphisms.
 \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$]
 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
 all of $\cC(X)$.
 \end{axiom}
 isotopic (rel boundary) to the identity {\it extended isotopy}.
 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 $\bd X$.
 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which
 act trivially on $\bd b$.
 When the enriching category $\cS$ is chain complexes or topological spaces,
 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
 of concrete examples, and a lack of diversity in our present collection of examples of $A_\infty$ $n$-categories
 makes us reluctant to commit to an all-encompassing general definition.
 Note that the morphisms $\Homeo(X,c; X', c')$ from $(X, c)$ to $(X', c')$ form a topological space.
 Let $\cS$ be an appropriate $\infty$-category (e.g.\ chain complexes)
 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
 \[
 	\cJ(\Homeo(X,c; X', c')) \ot \cC(X; c) \to \cC(X'; c') .
 \]
 \end{axiom}
 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.
+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 width then the action of families of collar
+(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}
 $\Homeo(X,c; X', c') \times \cC(X; c) \to \cC(X'; c')$, and at first it might seem that this would force the above
 action of $\cJ(\Homeo(X,c; X', c'))$ to be strictly associative as well (assuming the two actions are compatible).
 $W \to W'$ which restricts to the identity on the boundary.
 For $n=1$ we have the familiar bordism 1-category of $d$-manifolds.
 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.}
 %\begin{example}[Variation on the above examples]
 %We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$,
 We also assume, inductively, that we have gluing and restriction maps for colimits of $k{-}1$-manifolds.
 Gluing and restriction maps for colimits of $k$-manifolds will be defined later in this subsection.
 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
 \[
 	\prod_a \cC(X_a) \to \cl\cC(\bd M_0) .
 \]
 $\cl\cC(N_0) \to \cl\cC(\bd M_1)$.
 The second condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_1)$ is splittable
 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
 ball decompositions compatible with $x$.
 (If $x$ is a nice, non-pathological cell decomposition, then it is easy to see that gluing
 However, using the fact that $\bd y_i$ splits along $\bd Y$ and applying Axiom \ref{axiom:vcones},
 we can choose the decomposition $\du_{a} X_{ia}$ so that its restriction to $\bd W_i$ is a refinement
 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
 We now give more concrete descriptions of the above colimits.
 \[
 	\cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) \Bigg/ \sim ,
 \]
 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,
 \end{equation*}
 where $K$ is the vector space spanned by elements $a - g(a)$, with

changeset 833	f40f726d6cca
parent 826	979c947d0b82
child 837	f38558decd51