190 which is natural with respect to the actions of homeomorphisms. |
190 which is natural with respect to the actions of homeomorphisms. |
191 (When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product |
191 (When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product |
192 becomes a normal product.) |
192 becomes a normal product.) |
193 \end{lem} |
193 \end{lem} |
194 |
194 |
195 \begin{figure}[!ht] \centering |
195 \begin{figure}[t] \centering |
196 \begin{tikzpicture}[%every label/.style={green} |
196 \begin{tikzpicture}[%every label/.style={green} |
197 ] |
197 ] |
198 \node[fill=black, circle, label=below:$E$, inner sep=1.5pt](S) at (0,0) {}; |
198 \node[fill=black, circle, label=below:$E$, inner sep=1.5pt](S) at (0,0) {}; |
199 \node[fill=black, circle, label=above:$E$, inner sep=1.5pt](N) at (0,2) {}; |
199 \node[fill=black, circle, label=above:$E$, inner sep=1.5pt](N) at (0,2) {}; |
200 \draw (S) arc (-90:90:1); |
200 \draw (S) arc (-90:90:1); |
262 or if $k=n$ and we are in the $A_\infty$ case, |
262 or if $k=n$ and we are in the $A_\infty$ case, |
263 we require that $\gl_Y$ is injective. |
263 we require that $\gl_Y$ is injective. |
264 (For $k=n$ in the ordinary (non-$A_\infty$) case, see below.) |
264 (For $k=n$ in the ordinary (non-$A_\infty$) case, see below.) |
265 \end{axiom} |
265 \end{axiom} |
266 |
266 |
267 \begin{figure}[!ht] \centering |
267 \begin{figure}[t] \centering |
268 \begin{tikzpicture}[%every label/.style={green}, |
268 \begin{tikzpicture}[%every label/.style={green}, |
269 x=1.5cm,y=1.5cm] |
269 x=1.5cm,y=1.5cm] |
270 \node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {}; |
270 \node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {}; |
271 \node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {}; |
271 \node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {}; |
272 \draw (S) arc (-90:90:1); |
272 \draw (S) arc (-90:90:1); |
283 Given any splitting of a ball $B$ into smaller balls |
283 Given any splitting of a ball $B$ into smaller balls |
284 $$\bigsqcup B_i \to B,$$ |
284 $$\bigsqcup B_i \to B,$$ |
285 any sequence of gluings (in the sense of Definition \ref{defn:gluing-decomposition}, where all the intermediate steps are also disjoint unions of balls) yields the same result. |
285 any sequence of gluings (in the sense of Definition \ref{defn:gluing-decomposition}, where all the intermediate steps are also disjoint unions of balls) yields the same result. |
286 \end{axiom} |
286 \end{axiom} |
287 |
287 |
288 \begin{figure}[!ht] |
288 \begin{figure}[t] |
289 $$\mathfig{.65}{ncat/strict-associativity}$$ |
289 $$\mathfig{.65}{ncat/strict-associativity}$$ |
290 \caption{An example of strict associativity.}\label{blah6}\end{figure} |
290 \caption{An example of strict associativity.}\label{blah6}\end{figure} |
291 |
291 |
292 We'll use the notation $a\bullet b$ for the glued together field $\gl_Y(a, b)$. |
292 We'll use the notation $a\bullet b$ for the glued together field $\gl_Y(a, b)$. |
293 In the other direction, we will call the projection from $\cC(B)\trans E$ to $\cC(B_i)\trans E$ |
293 In the other direction, we will call the projection from $\cC(B)\trans E$ to $\cC(B_i)\trans E$ |
321 map from an appropriate subset (like a fibered product) |
321 map from an appropriate subset (like a fibered product) |
322 of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$, |
322 of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$, |
323 and these various $m$-fold composition maps satisfy an |
323 and these various $m$-fold composition maps satisfy an |
324 operad-type strict associativity condition (Figure \ref{fig:operad-composition}).} |
324 operad-type strict associativity condition (Figure \ref{fig:operad-composition}).} |
325 |
325 |
326 \begin{figure}[!ht] |
326 \begin{figure}[t] |
327 $$\mathfig{.8}{ncat/operad-composition}$$ |
327 $$\mathfig{.8}{ncat/operad-composition}$$ |
328 \caption{Operad composition and associativity}\label{fig:operad-composition}\end{figure} |
328 \caption{Operad composition and associativity}\label{fig:operad-composition}\end{figure} |
329 |
329 |
330 The next axiom is related to identity morphisms, though that might not be immediately obvious. |
330 The next axiom is related to identity morphisms, though that might not be immediately obvious. |
331 |
331 |
835 to be the dual Hilbert space $A(X\times F; c)$. |
835 to be the dual Hilbert space $A(X\times F; c)$. |
836 (See \S\ref{sec:constructing-a-tqft}.) |
836 (See \S\ref{sec:constructing-a-tqft}.) |
837 \end{example} |
837 \end{example} |
838 |
838 |
839 |
839 |
840 \begin{example}[The bordism $n$-category of $d$-manifolds, ordinary version] |
840 \begin{example}[te bordism $n$-category of $d$-manifolds, ordinary version] |
841 \label{ex:bord-cat} |
841 \label{ex:bord-cat} |
842 \rm |
842 \rm |
843 \label{ex:bordism-category} |
843 \label{ex:bordism-category} |
844 For a $k$-ball $X$, $k<n$, define $\Bord^{n,d}(X)$ to be the set of all $(d{-}n{+}k)$-dimensional PL |
844 For a $k$-ball $X$, $k<n$, define $\Bord^{n,d}(X)$ to be the set of all $(d{-}n{+}k)$-dimensional PL |
845 submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W = W \cap \bd X \times \Real^\infty$. |
845 submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W = W \cap \bd X \times \Real^\infty$. |
910 is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
910 is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
911 It's easy to see that with $n=0$, the corresponding system of fields is just |
911 It's easy to see that with $n=0$, the corresponding system of fields is just |
912 linear combinations of connected components of $T$, and the local relations are trivial. |
912 linear combinations of connected components of $T$, and the local relations are trivial. |
913 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
913 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
914 |
914 |
915 \begin{example}[The bordism $n$-category of $d$-manifolds, $A_\infty$ version] |
915 \begin{example}[te bordism $n$-category of $d$-manifolds, $A_\infty$ version] |
916 \rm |
916 \rm |
917 \label{ex:bordism-category-ainf} |
917 \label{ex:bordism-category-ainf} |
918 As in Example \ref{ex:bord-cat}, for $X$ a $k$-ball, $k<n$, we define $\Bord^{n,d}_\infty(X)$ |
918 As in Example \ref{ex:bord-cat}, for $X$ a $k$-ball, $k<n$, we define $\Bord^{n,d}_\infty(X)$ |
919 to be the set of all $(d{-}n{+}k)$-dimensional |
919 to be the set of all $(d{-}n{+}k)$-dimensional |
920 submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W = W \cap \bd X \times \Real^\infty$. |
920 submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W = W \cap \bd X \times \Real^\infty$. |
1043 The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
1043 The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
1044 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
1044 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
1045 See Figure \ref{partofJfig} for an example. |
1045 See Figure \ref{partofJfig} for an example. |
1046 \end{defn} |
1046 \end{defn} |
1047 |
1047 |
1048 \begin{figure}[!ht] |
1048 \begin{figure}[t] |
1049 \begin{equation*} |
1049 \begin{equation*} |
1050 \mathfig{.63}{ncat/zz2} |
1050 \mathfig{.63}{ncat/zz2} |
1051 \end{equation*} |
1051 \end{equation*} |
1052 \caption{A small part of $\cell(W)$} |
1052 \caption{A small part of $\cell(W)$} |
1053 \label{partofJfig} |
1053 \label{partofJfig} |
1274 (see Example \ref{ex:maps-with-fiber}). |
1274 (see Example \ref{ex:maps-with-fiber}). |
1275 (The union is along $N\times \bd W$.) |
1275 (The union is along $N\times \bd W$.) |
1276 %(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
1276 %(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
1277 %the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
1277 %the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
1278 |
1278 |
1279 \begin{figure}[!ht] |
1279 \begin{figure}[t] |
1280 $$\mathfig{.55}{ncat/boundary-collar}$$ |
1280 $$\mathfig{.55}{ncat/boundary-collar}$$ |
1281 \caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
1281 \caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
1282 |
1282 |
1283 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
1283 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
1284 Call such a thing a {marked $k{-}1$-hemisphere}. |
1284 Call such a thing a {marked $k{-}1$-hemisphere}. |
1340 |
1340 |
1341 We require two sorts of composition (gluing) for modules, corresponding to two ways |
1341 We require two sorts of composition (gluing) for modules, corresponding to two ways |
1342 of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
1342 of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
1343 (See Figure \ref{zzz3}.) |
1343 (See Figure \ref{zzz3}.) |
1344 |
1344 |
1345 \begin{figure}[!ht] |
1345 \begin{figure}[t] |
1346 \begin{equation*} |
1346 \begin{equation*} |
1347 \mathfig{.4}{ncat/zz3} |
1347 \mathfig{.4}{ncat/zz3} |
1348 \end{equation*} |
1348 \end{equation*} |
1349 \caption{Module composition (top); $n$-category action (bottom).} |
1349 \caption{Module composition (top); $n$-category action (bottom).} |
1350 \label{zzz3} |
1350 \label{zzz3} |
1403 |
1403 |
1404 Note that the above associativity axiom applies to mixtures of module composition, |
1404 Note that the above associativity axiom applies to mixtures of module composition, |
1405 action maps and $n$-category composition. |
1405 action maps and $n$-category composition. |
1406 See Figure \ref{zzz1b}. |
1406 See Figure \ref{zzz1b}. |
1407 |
1407 |
1408 \begin{figure}[!ht] |
1408 \begin{figure}[t] |
1409 \begin{equation*} |
1409 \begin{equation*} |
1410 \mathfig{0.49}{ncat/zz0} \mathfig{0.49}{ncat/zz1} |
1410 \mathfig{0.49}{ncat/zz0} \mathfig{0.49}{ncat/zz1} |
1411 \end{equation*} |
1411 \end{equation*} |
1412 \caption{Two examples of mixed associativity} |
1412 \caption{Two examples of mixed associativity} |
1413 \label{zzz1b} |
1413 \label{zzz1b} |