276 \nn{these examples need to be fleshed out a bit more}

   277 

   278 Examples of plain $n$-categories:

   279 \begin{itemize}

   280 

   281 \item Let $F$ be a closed $m$-manifold (e.g.\ a point).

   282 Let $T$ be a topological space.

   283 For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\cC(X)$ to be the set of 

   284 all maps from $X\times F$ to $T$.

   285 For $X$ an $n$-ball define $\cC(X)$ to be maps from $X\times F$ to $T$ modulo

   286 homotopies fixed on $\bd X$.

   287 (Note that homotopy invariance implies isotopy invariance.)

   288 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to

   289 be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection.

   290 

   291 \item We can linearize the above example as follows.

   292 Let $\alpha$ be an $(n{+}m{+}1)$-cocycle on $T$ with values in a ring $R$

   293 (e.g.\ the trivial cocycle).

   294 For $X$ of dimension less than $n$ define $\cC(X)$ as before.

   295 For $X$ an $n$-ball and $c\in \cC(\bd X)$ define $\cC(X; c)$ to be

   296 the $R$-module of finite linear combinations of maps from $X\times F$ to $T$,

   297 modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy

   298 $h: X\times F\times I \to T$, then $a \sim \alpha(h)b$.

   299 \nn{need to say something about fundamental classes, or choose $\alpha$ carefully}

   300 

   301 \item Given a traditional $n$-category $C$ (with strong duality etc.),

   302 define $\cC(X)$ (with $\dim(X) < n$) 

   303 to be the set of all $C$-labeled sub cell complexes of $X$.

   304 For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear

   305 combinations of $C$-labeled sub cell complexes of $X$

   306 modulo the kernel of the evaluation map.

   307 Define a product morphism $a\times D$ to be the product of the cell complex of $a$ with $D$,

   308 and with the same labeling as $a$.

   309 \nn{refer elsewhere for details?}

   310 

   311 \item Variation on the above examples:

   312 We could allow $F$ to have boundary and specify boundary conditions on $(\bd X)\times F$,

   313 for example product boundary conditions or take the union over all boundary conditions.

   314 

   315 \end{itemize}

   316 

   317 

   318 Examples of $A_\infty$ $n$-categories:

   319 \begin{itemize}

   320 

   321 \item Same as in example \nn{xxxx} above (fiber $F$, target space $T$),

   322 but we define, for an $n$-ball $X$, $\cC(X; c)$ to be the chain complex 

   323 $C_*(\Maps_c(X\times F))$, where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary,

   324 and $C_*$ denotes singular chains.

   325 

   326 \item

   327 Given a plain $n$-category $C$, 

   328 define $\cC(X; c) = \bc^C_*(X\times F; c)$, where $X$ is an $n$-ball

   329 and $\bc^C_*$ denotes the blob complex based on $C$.

   330 

   331 \end{itemize}

   332 

   333 \medskip

   334 

   335 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations,

   336 a.k.a.\ actions).

   337 

   338 \medskip