282 {Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
282 {Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
283 into small $k$-balls, there is a |
283 into small $k$-balls, there is a |
284 map from an appropriate subset (like a fibered product) |
284 map from an appropriate subset (like a fibered product) |
285 of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$, |
285 of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$, |
286 and these various $m$-fold composition maps satisfy an |
286 and these various $m$-fold composition maps satisfy an |
287 operad-type strict associativity condition (Figure \ref{blah7}).} |
287 operad-type strict associativity condition (Figure \ref{fig:operad-composition}).} |
288 |
288 |
289 \begin{figure}[!ht] |
289 \begin{figure}[!ht] |
290 $$\mathfig{.8}{tempkw/blah7}$$ |
290 $$\mathfig{.8}{ncat/operad-composition}$$ |
291 \caption{Operad composition and associativity}\label{blah7}\end{figure} |
291 \caption{Operad composition and associativity}\label{fig:operad-composition}\end{figure} |
292 |
292 |
293 The next axiom is related to identity morphisms, though that might not be immediately obvious. |
293 The next axiom is related to identity morphisms, though that might not be immediately obvious. |
294 |
294 |
295 \begin{axiom}[Product (identity) morphisms, preliminary version] |
295 \begin{axiom}[Product (identity) morphisms, preliminary version] |
296 For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, |
296 For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, |