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 %which are weak enough to include the basic examples and strong enough to support the proofs
 %of the main theorems; and (2) specify a minimal set of generators and/or axioms.
 %We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second.
 %More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.}
-We will define two variations simultaneously,  as all but one of the axioms are identical
+We will define two variations simultaneously,  as all but one of the axioms are identical in the two cases.
-in the two cases. These variations are `linear $n$-categories', where the sets associated to
+These variations are `isotopy $n$-categories', where homeomorphisms fixing the boundary
-$n$-balls with specified boundary conditions are in fact vector spaces, and `$A_\infty$ $n$-categories',
+act trivially on the sets associated to $n$-balls
-where these sets are chain complexes.
+(and these sets are usually vector spaces or more generally modules over a commutative ring)
+and `$A_\infty$ $n$-categories',  where there is a homotopy action of
+$k$-parameter families of homeomorphisms on these sets
+(which are usually chain complexes or topological spaces).
 There are five basic ingredients
 \cite{life-of-brian} of an $n$-category definition:
 $k$-morphisms (for $0\le k \le n$), domain and range, composition,
 identity morphisms, and special behavior in dimension $n$ (e.g. enrichment
 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions
 to the intersection of the boundaries of $B$ and $B_i$.
 If $k < n$,
 or if $k=n$ and we are in the $A_\infty$ case,
 we require that $\gl_Y$ is injective.
-(For $k=n$ in the linear case, see below.)
+(For $k=n$ in the isotopy $n$-category case, see below.)
 \end{axiom}
 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity}
 The gluing maps above are strictly associative.
 Given any decomposition of a ball $B$ into smaller balls
 where the first arrow is gluing with a product morphism on $Y\times_p J$ and
 the second is induced by a homeomorphism from $X\cup_Y (Y\times_p J)$ to $X$ which restricts
 to the identity on the boundary.
-\begin{axiom}[\textup{\textbf{[linear  version]}} Extended isotopy invariance in dimension $n$.]
+\begin{axiom}[\textup{\textbf{[for isotopy  $n$-categories]}} 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 $\cC(X)$.
 For the moment, assume that our $n$-morphisms are enriched over chain complexes.
 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and
 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space.
-\begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.]
+\begin{axiom}[\textup{\textbf{[for $A_\infty$ $n$-categories]}} Families of homeomorphisms act in dimension $n$.]
 \label{axiom:families}
 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes
 \[
 	C_*(\Homeo_\bd(X))\tensor \cC(X; c) \to \cC(X; c) .
 \]
 We will use the term `field on $W$' to refer to a point of this functor,
 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$.
 \subsubsection{Colimits}
-Our definition of an $n$-category is essentially a collection of functors defined on $k$-balls (and homeomorphisms)
+Recall that our definition of an $n$-category is essentially a collection of functors
+defined on the categories of homeomorphisms $k$-balls
 for $k \leq n$ satisfying certain axioms.
-It is natural to consider extending such functors to the
+It is natural to hope to extend such functors to the
 larger categories of all $k$-manifolds (again, with homeomorphisms).
-In fact, the axioms stated above explicitly require such an extension to $k$-spheres for $k<n$.
+In fact, the axioms stated above already require such an extension to $k$-spheres for $k<n$.
-The natural construction achieving this is the colimit. For a linear $n$-category $\cC$,
+The natural construction achieving this is a colimit along the poset of permissible decompositions.
+For an isotopy $n$-category $\cC$,
 we denote the extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$,
-this is defined to be the colimit of the function $\psi_{\cC;W}$.
+this is defined to be the colimit of the functor $\psi_{\cC;W}$.
 Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity}
 imply that $\cl{\cC}(X)  \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$.
 Recall that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball,
 the set $\cC(X;c)$ is a vector space. Using this, we note that for $c \in \cl{\cC}(\bdy X)$,
 for $X$ an arbitrary $n$-manifold, the set $\cl{\cC}(X;c) = \bdy^{-1} (c)$ inherits the structure of a vector space.
 The blob complex has several important special cases.
 \begin{thm}[Skein modules]
 \label{thm:skein-modules}
-Suppose $\cC$ is a linear $n$-category
+Suppose $\cC$ is an isotopy $n$-category
 The $0$-th blob homology of $X$ is the usual
 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$
 by $\cC$.
 \begin{equation*}
 H_0(\bc_*(X;\cC)) \iso \cl{\cC}(X)
 blob complexes for the $A_\infty$ $n$-categories constructed as above.
 \begin{thm}[Product formula]
 \label{thm:product}
 Let $W$ be a $k$-manifold and $Y$ be an $n{-}k$ manifold.
-Let $\cC$ be a linear $n$-category.
+Let $\cC$ be an isotopy $n$-category.
 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above.
 Then
 \[
 	\bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W).
 \]

changeset 655	71eb442b8500
parent 654	76252091abf6
child 656	28592849a474