43 Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic 

    43 Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic 

    44 to the standard $k$-ball.

    44 to the standard $k$-ball.

    45 By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the 

    45 By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the 

    46 standard $k$-ball.

    46 standard $k$-ball.

    47 We {\it do not} assume that it is equipped with a 

    47 We {\it do not} assume that it is equipped with a 

    49 

    53 

    50 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on 

    54 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on 

    51 the boundary), we want a corresponding

    55 the boundary), we want a corresponding

    52 bijection of sets $f:\cC(X)\to \cC(Y)$.

    56 bijection of sets $f:\cC(X)\to \cC(Y)$.

    53 (This will imply ``strong duality", among other things.) Putting these together, we have

    57 (This will imply ``strong duality", among other things.) Putting these together, we have

   216 (For example, vertical and horizontal composition of 2-morphisms.)

   220 (For example, vertical and horizontal composition of 2-morphisms.)

   217 In the presence of strong duality, these $k$ distinct compositions are subsumed into 

   221 In the presence of strong duality, these $k$ distinct compositions are subsumed into 

   218 one general type of composition which can be in any ``direction".

   222 one general type of composition which can be in any ``direction".

   219 

   223 

   220 \begin{axiom}[Composition]

   224 \begin{axiom}[Composition]

   221 Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$)

   226 Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$)

   222 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}).

   227 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}).

   223 Let $E = \bd Y$, which is a $k{-}2$-sphere.

   228 Let $E = \bd Y$, which is a $k{-}2$-sphere.

   224 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$.

   229 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$.

   225 We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$.

   230 We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$.

   465 same (traditional) $i$-morphism as the corresponding codimension $i$ cell $c$.

   470 same (traditional) $i$-morphism as the corresponding codimension $i$ cell $c$.

   466 

   471 

   467 

   472 

   468 %\addtocounter{axiom}{-1}

   473 %\addtocounter{axiom}{-1}

   469 \begin{axiom}[Product (identity) morphisms]

   474 \begin{axiom}[Product (identity) morphisms]

   470 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$),

   476 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$),

   471 there is a map $\pi^*:\cC(X)\to \cC(E)$.

   477 there is a map $\pi^*:\cC(X)\to \cC(E)$.

   472 These maps must satisfy the following conditions.

   478 These maps must satisfy the following conditions.

   473 \begin{enumerate}

   479 \begin{enumerate}

   474 \item

   480 \item

   610 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space.

   616 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space.

   611 

   617 

   612 

   618 

   613 %\addtocounter{axiom}{-1}

   619 %\addtocounter{axiom}{-1}

   614 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.]

   620 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.]

   615 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes

   622 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes

   616 \[

   623 \[

   617 	C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) .

   624 	C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) .

   618 \]

   625 \]

   619 These action maps are required to be associative up to homotopy,

   626 These action maps are required to be associative up to homotopy,

  1829 %For the time being, let's say they are.}

  1836 %For the time being, let's say they are.}

  1830 A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, 

  1837 A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, 

  1831 where $B^j$ is the standard $j$-ball.

  1838 where $B^j$ is the standard $j$-ball.

  1832 A 1-marked $k$-ball can be decomposed in various ways into smaller balls, which are either 

  1839 A 1-marked $k$-ball can be decomposed in various ways into smaller balls, which are either 

  1833 (a) smaller 1-marked $k$-balls, (b) 0-marked $k$-balls, or (c) plain $k$-balls.

  1840 (a) smaller 1-marked $k$-balls, (b) 0-marked $k$-balls, or (c) plain $k$-balls.

  1835 We now proceed as in the above module definitions.

  1842 We now proceed as in the above module definitions.

  1836 

  1843 

  1837 \begin{figure}[t] \centering

  1844 \begin{figure}[t] \centering

  1838 \begin{tikzpicture}[baseline,line width = 2pt]

  1845 \begin{tikzpicture}[baseline,line width = 2pt]

  1839 \draw[blue][fill=blue!15!white] (0,0) circle (2);

  1846 \draw[blue][fill=blue!15!white] (0,0) circle (2);

  1845 	\draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3);

  1852 	\draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3);

  1846 }

  1853 }

  1847 \end{tikzpicture}

  1854 \end{tikzpicture}

  1848 \caption{Cone on a marked circle, the prototypical 1-marked ball}

  1855 \caption{Cone on a marked circle, the prototypical 1-marked ball}

  1849 \label{feb21d}

  1856 \label{feb21d}

  1850 \end{figure}

  1892 \end{figure}

  1851 

  1893 

  1852 A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with

  1894 A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with

  1853 \[

  1895 \[

  1854 	\cD(X) \deq \cM(X\times C(S)) .

  1896 	\cD(X) \deq \cM(X\times C(S)) .

  2211 \end{proof}

  2253 \end{proof}

  2212 

  2254 

  2213 For $n=1$ we have to check an additional ``global" relations corresponding to 

  2255 For $n=1$ we have to check an additional ``global" relations corresponding to 

  2214 rotating the 0-sphere $E$ around the 1-sphere $\bd X$.

  2256 rotating the 0-sphere $E$ around the 1-sphere $\bd X$.

  2215 But if $n=1$, then we are in the case of ordinary algebroids and bimodules,

  2257 But if $n=1$, then we are in the case of ordinary algebroids and bimodules,

  2218 

  2259 

  2219 \medskip

  2260 \medskip

  2220 

  2261 

  2221 We have now defined $\cS(X; c)$ for any $n{+}1$-ball $X$ with boundary decoration $c$.

  2262 We have now defined $\cS(X; c)$ for any $n{+}1$-ball $X$ with boundary decoration $c$.

  2222 We must also define, for any homeomorphism $X\to X'$, an action $f: \cS(X; c) \to \cS(X', f(c))$.

  2263 We must also define, for any homeomorphism $X\to X'$, an action $f: \cS(X; c) \to \cS(X', f(c))$.