146 \xxpar{Strict associativity:}

   147 {The composition (gluing) maps above are strictly associative.

   148 It follows that given a decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball

   149 into small $k$-balls, there is a well-defined

   150 map from an appropriate subset of $\cC(B_1)\times\cdots\times\cC(B_m)$ to $\cC(B)$,

   151 and these various $m$-fold composition maps satisfy an

   152 operad-type associativity condition.}

   153 

   154 \nn{above maybe needs some work}

   155 

   156 The next axiom is related to identity morphisms, though that might not be immediately obvious.

   157 

   158 \xxpar{Product (identity) morphisms:}

   159 {Let $X$ be homeomorphic to a $k$-ball and $D$ be homeomorphic to an $m$-ball, with $k+m \le n$.

   160 Then we have a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$.

   161 If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram

   162 \[ \xymatrix{

   163 	X\times D \ar[r]^{\tilde{f}} \ar[d]^{\pi} & X'\times D' \ar[d]^{\pi} \\

   164 	X \ar[r]^{f} & X'

   165 } \]

   166 commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.}

   167 

   168 \nn{Need to say something about compatibility with gluing (of both $X$ and $D$) above.}

   169 

   170 All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories.

   171 The last axiom (below), concerning actions of 

   172 homeomorphisms in the top dimension $n$, distinguishes the two cases.

   173 

   174 We start with the plain $n$-category case.

   175 

   176 \xxpar{Isotopy invariance in dimension $n$ (preliminary version):}

   177 {Let $X$ be homeomorphic to the $n$-ball and $f: X\to X$ be a homeomorphism which restricts

   178 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity.

   179 Then $f(a) = a$ for all $a\in \cC(X)$.}

   182 

   183 

   184 

   185 \medskip

   186 

   187 \hrule

   188 

   189 \medskip

   190 

   191 \nn{to be continued...}

   192