blob: comparison text/ncat.tex

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 such that $E$ is a $k{+}m$-ball, $X$ is a $k$-ball ($m\ge 1$), and $\pi$ is locally modeled
 on a standard iterated degeneracy map
 \[
 	d: \Delta^{k+m}\to\Delta^k .
 \]
-In other words, \nn{each point has a neighborhood blah blah...}
 (We thank Kevin Costello for suggesting this approach.)
 Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball,
 and for for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension
 $l \le m$, with $l$ depending on $x$.
 The last axiom (below), concerning actions of
 homeomorphisms in the top dimension $n$, distinguishes the two cases.
 We start with the plain $n$-category case.
-\begin{axiom}[Isotopy invariance in dimension $n$]{\textup{\textbf{[preliminary]}}}
+\begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$]
 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity.
 Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$.
 \end{axiom}
 isotopic (rel boundary) to the identity {\it extended isotopy}.
 The revised axiom is
 \addtocounter{axiom}{-1}
-\begin{axiom}{\textup{\textbf{[topological  version]}} Extended isotopy invariance in dimension $n$.}
+\begin{axiom}[\textup{\textbf{[topological  version]}} 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 isotopic (rel boundary) to the identity.
 Then $f$ acts trivially on $\cC(X)$.
 In addition, collar maps act trivially on $\cC(X)$.
 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and
 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space.
 \addtocounter{axiom}{-1}
-\begin{axiom}{\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.}
+\begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.]
 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes
 \[
 	C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) .
 \]
 These action maps are required to be associative up to homotopy
 To do this we need to explain how collar maps form a topological space.
 Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology,
 and we can replace the class of all intervals $J$ with intervals contained in $\r$.
 Having chains on the space of collar maps act gives rise to coherence maps involving
 weak identities.
-We will not pursue this in this draft of the paper.
+We will not pursue this in detail here.
 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category
 into a plain $n$-category (enriched over graded groups).
 In a different direction, if we enrich over topological spaces instead of chain complexes,
 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting
 We will first define the ``cell-decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$.
 An $n$-category $\cC$ provides a functor from this poset to the category of sets,
 and we  will define $\cC(W)$ as a suitable colimit
 (or homotopy colimit in the $A_\infty$ case) of this functor.
 We'll later give a more explicit description of this colimit.
-In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data),
+In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-balls with boundary data),
 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex).
 \begin{defn}
 Say that a ``permissible decomposition" of $W$ is a cell decomposition
 \[
 (Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.)
 Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we
 fix a field on $\bd W$
 (i.e. fix an element of the colimit associated to $\bd W$).
-Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$.
+Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$.
 \begin{defn}[System of fields functor]
 \label{def:colim-fields}
 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$.
 That is, for each decomposition $x$ there is a map
 Then we kill the extra homology we just introduced with mapping
 cylinders between the mapping cylinders (filtration degree 2), and so on.
 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}.
-\todo{This next sentence is circular: these maps are an axiom, not a consequence of anything. -S} It is easy to see that
+It is easy to see that
 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps
 comprise a natural transformation of functors.
 \begin{lem}
 \label{lem:colim-injective}
 We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$
 for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs
 $(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all
 such maps modulo homotopies fixed on $\bdy B \setminus N$.
 This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}.
+\end{example}
 Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and
 \ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to
 Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains.
-\end{example}
 \subsection{Modules as boundary labels (colimits for decorated manifolds)}
 \label{moddecss}
 Fix a topological $n$-category or $A_\infty$ $n$-category  $\cC$.

changeset 420	257066702f60
parent 418	a96f3d2ef852
child 421	a896ec294254