184 \nn{ |
184 \nn{ |
185 Decide if we need a friendlier, skein-module version. |
185 Decide if we need a friendlier, skein-module version. |
186 } |
186 } |
187 \subsection{The blob complex} |
187 \subsection{The blob complex} |
188 \subsubsection{Decompositions of manifolds} |
188 \subsubsection{Decompositions of manifolds} |
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189 |
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190 A {\emph ball decomposition} of $W$ is a |
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191 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
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192 $\du_a X_a$. |
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193 |
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194 If $X_a$ is some component of $M_0$, note that its image in $W$ need not be a ball; parts of $\bd X_a$ may have been glued together. |
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195 Define a {\it permissible decomposition} of $W$ to be a map |
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196 \[ |
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197 \coprod_a X_a \to W, |
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198 \] |
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199 which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. |
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200 Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls |
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201 are glued up to yield $W$, so long as there is some (non-pathological) way to glue them. |
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202 |
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203 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
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204 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
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205 with $\du_b Y_b = M_i$ for some $i$. |
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206 |
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207 \begin{defn} |
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208 The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
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209 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
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210 See Figure \ref{partofJfig} for an example. |
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211 \end{defn} |
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212 |
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213 |
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214 An $n$-category $\cC$ determines |
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215 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
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216 (possibly with additional structure if $k=n$). |
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217 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
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218 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
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219 are splittable along this decomposition. |
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220 |
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221 \begin{defn} |
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222 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
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223 For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
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224 \begin{equation} |
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225 \label{eq:psi-C} |
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226 \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
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227 \end{equation} |
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228 where the restrictions to the various pieces of shared boundaries amongst the cells |
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229 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). |
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230 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
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231 \end{defn} |
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232 |
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233 |
189 \nn{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls} |
234 \nn{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls} |
190 \subsubsection{Homotopy colimits} |
235 \subsubsection{Homotopy colimits} |
191 \nn{How can we extend an $n$-category from balls to arbitrary manifolds?} |
236 \nn{How can we extend an $n$-category from balls to arbitrary manifolds?} |
192 |
237 |
193 \nn{In practice, this gives the old definition} |
238 \nn{In practice, this gives the old definition} |
471 %% |
516 %% |
472 %% \begin{figure} |
517 %% \begin{figure} |
473 %% \caption{Almost Sharp Front}\label{afoto} |
518 %% \caption{Almost Sharp Front}\label{afoto} |
474 %% \end{figure} |
519 %% \end{figure} |
475 |
520 |
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521 |
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522 \begin{figure} |
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523 \begin{equation*} |
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524 \mathfig{.23}{ncat/zz2} |
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525 \end{equation*} |
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526 \caption{A small part of $\cell(W)$} |
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527 \label{partofJfig} |
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528 \end{figure} |
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529 |
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530 |
476 %% For Tables, put caption above table |
531 %% For Tables, put caption above table |
477 %% |
532 %% |
478 %% Table caption should start with a capital letter, continue with lower case |
533 %% Table caption should start with a capital letter, continue with lower case |
479 %% and not have a period at the end |
534 %% and not have a period at the end |
480 %% Using @{\vrule height ?? depth ?? width0pt} in the tabular preamble will |
535 %% Using @{\vrule height ?? depth ?? width0pt} in the tabular preamble will |