211 \] |
211 \] |
212 in the case that $X = X_1 \du X_2$, with $x_i \in \cC(X_i)$. |
212 in the case that $X = X_1 \du X_2$, with $x_i \in \cC(X_i)$. |
213 |
213 |
214 \medskip |
214 \medskip |
215 |
215 |
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216 |
216 Using the functoriality and product field properties above, together |
217 Using the functoriality and product field properties above, together |
217 with boundary collar homeomorphisms of manifolds, we can define |
218 with boundary collar homeomorphisms of manifolds, we can define |
218 {\it collar maps} $\cC(M)\to \cC(M)$. |
219 {\it collar maps} $\cC(M)\to \cC(M)$. |
219 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
220 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
220 of $\bd M$. |
221 of $\bd M$. |
229 can be thought of (informally) as the limit of homeomorphisms |
230 can be thought of (informally) as the limit of homeomorphisms |
230 which expand an infinitesimally thin collar neighborhood of $Y$ to a thicker |
231 which expand an infinitesimally thin collar neighborhood of $Y$ to a thicker |
231 collar neighborhood. |
232 collar neighborhood. |
232 |
233 |
233 |
234 |
234 % all this linearizing stuff is unnecessary, I think |
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235 \noop{ |
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236 |
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237 \nn{the following discussion of linearizing fields is kind of lame. |
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238 maybe just assume things are already linearized.} |
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239 |
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240 \nn{remark that if top dimensional fields are not already linear |
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241 then we will soon linearize them(?)} |
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242 |
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243 For top dimensional ($n$-dimensional) manifolds, we're actually interested |
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244 in the linearized space of fields. |
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245 By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is |
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246 the vector space of finite |
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247 linear combinations of fields on $X$. |
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248 If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$. |
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249 Thus the restriction (to boundary) maps are well defined because we never |
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250 take linear combinations of fields with differing boundary conditions. |
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251 |
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252 In some cases we don't linearize the default way; instead we take the |
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253 spaces $\lf(X; a)$ to be part of the data for the system of fields. |
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254 In particular, for fields based on linear $n$-category pictures we linearize as follows. |
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255 Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
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256 obvious relations on 0-cell labels. |
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257 More specifically, let $L$ be a cell decomposition of $X$ |
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258 and let $p$ be a 0-cell of $L$. |
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259 Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
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260 $\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
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261 Then the subspace $K$ is generated by things of the form |
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262 $\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader |
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263 to infer the meaning of $\alpha_{\lambda c + d}$. |
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264 Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. |
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265 |
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266 \nn{Maybe comment further: if there's a natural basis of morphisms, then no need; |
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267 will do something similar below; in general, whenever a label lives in a linear |
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268 space we do something like this; ? say something about tensor |
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269 product of all the linear label spaces? Yes:} |
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270 |
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271 For top dimensional ($n$-dimensional) manifolds, we linearize as follows. |
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272 Define an ``almost-field" to be a field without labels on the 0-cells. |
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273 (Recall that 0-cells are labeled by $n$-morphisms.) |
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274 To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
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275 space determined by the labeling of the link of the 0-cell. |
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276 (If the 0-cell were labeled, the label would live in this space.) |
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277 We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
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278 We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the |
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279 above tensor products. |
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280 |
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281 } % end \noop |
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282 |
235 |
283 |
236 |
284 \subsection{Systems of fields from \texorpdfstring{$n$}{n}-categories} |
237 \subsection{Systems of fields from \texorpdfstring{$n$}{n}-categories} |
285 \label{sec:example:traditional-n-categories(fields)} |
238 \label{sec:example:traditional-n-categories(fields)} |
286 We now describe in more detail Example \ref{ex:traditional-n-categories(fields)}, |
239 We now describe in more detail Example \ref{ex:traditional-n-categories(fields)}, |