169 \end{tikzpicture} |
170 \end{tikzpicture} |
170 $$ |
171 $$ |
171 What is the element $x'$? It should be an element of $\Hom(a \tensor c^*, b)$, and in a pivotal $2$-category this space is naturally isomorphic to $\Hom(a, b \tensor c)$, so we just choose $x'$ to be the image of $x$ under this isomorphism. |
172 What is the element $x'$? It should be an element of $\Hom(a \tensor c^*, b)$, and in a pivotal $2$-category this space is naturally isomorphic to $\Hom(a, b \tensor c)$, so we just choose $x'$ to be the image of $x$ under this isomorphism. |
172 |
173 |
173 Finally, when $Y$ is a ball, how do we interpret a string diagram on $Y$ as a $2$-morphism in $\cC$? First choose a parametrization of $Y$ as a standard bihedron; now `sweep out' the interior of $Y$. We'll build a $2$-morphism from the tensor product of the $1$-morphisms labeling the $1$-cells meeting the lower boundary to the tensor product of the $1$-morphisms labelling the upper boundary. As we pass critical points in the $1$-cells, apply a pairing or copairing map from the category. As we pass $0$-cells, modify the parametrization to match the direction we're sweeping out, and compose with the label of the $0$-cell, acting on the appropriate tensor factors. |
174 Finally, when $Y$ is a ball, how do we interpret a string diagram on $Y$ as a $2$-morphism in $\cC$? First choose a parametrization of $Y$ as a standard bihedron; now `sweep out' the interior of $Y$. We'll build a $2$-morphism from the tensor product of the $1$-morphisms labeling the $1$-cells meeting the lower boundary to the tensor product of the $1$-morphisms labelling the upper boundary. As we pass critical points in the $1$-cells, apply a pairing or copairing map from the category. As we pass $0$-cells, modify the parametrization to match the direction we're sweeping out, and compose with the label of the $0$-cell, acting on the appropriate tensor factors. |
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175 |
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176 As usual for fields based on string diagrams, the corresponding local relations are exactly the kernel of this `evaluation' map. |
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177 |
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178 \section{Axioms for fields} |
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179 |
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180 \section{TQFT from fields} |
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181 Given a system of fields and local relations $\cF, \cU$, we define the corresponding vector space valued invariant of $n$-manifolds $A$ as follows. For $X$ an $n$-manifold, write $\cU(X)$ for the subspace of $\cF(X)$ consisting of the span of the images of a gluing map $\cU(B; c) \tensor \cF(X \setminus B; c)$ for any embedded $n$-ball $B \subset X$, and boundary field $c \in \cF(\bdy B)$. We then define |
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182 $$A(X) = \cF(X) / \cU(X).$$ |
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183 It's clear that homeomorphisms of $X$ act on this space. Actually, this collapses to an action of the mapping class group: |
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184 \begin{lem} |
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185 Homeomorphisms isotopic to the identity act trivially on $A(X)$. |
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186 \end{lem} |
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187 \begin{proof} |
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188 Any $1$-parameter family of homeomorphisms is homotopic (rel boundary) to a family for which during any sufficiently short interval of time, the homeomorphism is only being modified inside a ball. The difference between a field at the beginning of such an interval and the field at the end is in $\cU(X)$, and hence zero in $A(X)$. |
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189 \end{proof} |
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190 |
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191 |
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192 If $X$ has boundary, we can choose $c \in \cF(\bdy X)$ and similarly define a vector space $A(X; c) = \cF(X; c) / \cU(X; c)$. |
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193 |
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194 This invariant also extends to manifolds of other dimensions, associating to a codimension $k$ manifold $Y$ a linear $k$-category $A(Y)$. We'll spell this out below for small values of $k$, and postpone the full story until we have our own notion of $k$-category. Thus the TQFT we obtain from fields and local relations is `fully extended'. On the other hand, often a TQFT invariant that associates vector spaces to $n$-manifolds will also associate numbers to $(n+1)$-manifolds. Such a TQFT is called `$(n+1)$-dimensional', while one that doesn't is called alternatively `$(n+\epsilon)$-dimensional', `decapitated' or `topless'. In general the TQFTs from fields and local relations are just $(n+\epsilon)$-dimensional, although with some extra conditions on the input we can produce $(n+1)$-dimensional TQFTs. This discussion is almost entirely orthogonal to the content of the blob complex paper (although c.f. \S 6.7 on the $(n+1)$-category of $n$-categories), so we won't pursue it here. |
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195 |
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196 To an $(n-1)$-dimensional manifold $Y$, we associate a $1$-category $A(Y)$. Its objects are simply $\cF(Y)$. The morphism spaces are given by $$\Hom(a,b) = A(Y \times [0,1]; a \bullet b).$$ |
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197 Composition of morphisms is via gluing then reparametrization: |
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198 $$A(Y \times [0,1]; a \bullet b) \tensor A(Y \times [0,1]; b \bullet c) \to A(Y \times [0,2]; a \bullet c) \to A(Y \times [0,1]; a \bullet c).$$ |
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199 The gluing maps themselves are strictly associative, and by the lemma above we don't have worry about the reparametrization step here breaking associativity. |
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200 |
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201 |
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202 If $Y$ itself has boundary, we have some alternatives here. One is to interpret $Y \times [0,1]$ as the `pinched product', where we collapse the copy of $[0,1]$ over each point of $\bdy Y$. The other is to fix $c \in \cF(\bdy Y)$, and to define $A(Y; c)$, with objects $\cF(Y; c)$ and in which $\Hom(a,b) = A(Y \times [0,1]; a \bullet b \bullet (c \times [0,1]))$. |
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203 |
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204 Going deeper, we associate a $2$-category $A(P)$ to an $(n-2)$-dimensional manifold $P$. The $0$-morphisms are $\cF(P)$, the $1$-morphisms are $\cF(P \times I)$, and they compose by gluing intervals together. (Note that this composition is not associative on the nose, but will be associative up to a $2$-morphism shortly.) Finally the $2$-morphisms from $a$ to $b$, each $1$-morphisms from $x$ to $y$ are given by the vector space |
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205 $$A(P \times I \times I; \tikz[baseline=11.5]{\draw (0,0) -- node[below] {$a$} (1,0) -- node[below, sloped] {$y \times I$} (1,1) -- node[above] {$b$} (0,1) (0,0) -- node[above,sloped] {$x \times I$} (0,1);})$$ |
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206 |
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207 \subsection{Gluing formulas} |
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208 Even though the definition of these TQFTs is via an abstract looking (not to mention scarily infinite-dimensional) quotient, we can prove various `gluing formulas' that allow us to compute the invariants algebraically. |
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209 |
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210 \subsubsection{Codimension 1 gluing} |
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211 |
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212 Suppose an $n$-manifold $X$ contains a copy of $Y$, an $n-1$ manifold, as a codimension $0$ submanifold of its boundary. Fix a boundary condition $c \in \cF(\bdy X \setminus Y)$. Then the collection $A(X; c \bullet d)$, as $d$ varies over $\cF(Y)$, forms a module over the $1$-category $A(Y)$. The action is via gluing a collar onto $Y$, then applying a `collaring homeomorphism' $X \cup_Y Y \times I \to X$. |
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213 |
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214 If $X$ contains two copies of $Y$, $A(X)$ is then a bimodule over $A(Y)$. Below, we'll compute the invariant of the `glued up' manifold $X \bigcup_Y \selfarrow$ as the self-tensor product of this bimodule. |
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215 |
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216 \begin{lem} |
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217 Any isotopy of $X \bigcup_Y \selfarrow$ is homotopic to a composition of `collar shift' isotopies and isotopies that are constant on $Y$ (i.e. the image of an isotopy of $X$ itself). |
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218 \end{lem} |
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219 \begin{proof} |
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220 First make the isotopy act locally. When it's acting in a small ball overlapping $Y$, conjugate by a collar shift to move it off. |
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221 \end{proof} |
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222 |
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223 \begin{thm} |
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224 $$A(X \bigcup_Y \selfarrow) \iso A(X) \Tensor_{A(Y)} \selfarrow$$ |
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225 \end{thm} |
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226 \begin{proof} |
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227 Certainly there is a map $A(X) \selfarrow \to A(X \bigcup_Y \selfarrow)$. We send an element of $A(X)$ to the corresponding `glued up' element of $A(X \bigcup_Y \selfarrow)$. This is well-defined since $\cU(X)$ maps into $\cU(X \bigcup_Y \selfarrow)$. This map descends down to a map |
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228 $$A(X) \Tensor_{A(Y)} \selfarrow \to A(X \bigcup_Y \selfarrow)$$ |
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229 since the fields $ev$ and $ve$ (here $e \in A(Y), v \in A(X)$) are isotopic on $X \bigcup_Y \selfarrow$ (see Figure \ref{fig:ev-ve}). |
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230 |
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231 \begin{figure}[!ht] |
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232 $$ |
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233 \begin{tikzpicture}[x=4cm,y=4cm] |
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234 \node (a) at (0,2) { |
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235 \begin{tikzpicture}[x=0.5cm,y=0.5cm] |
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236 \node (a1) at (-2,1.2) {}; |
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237 \node (a2) at (-2,0) {}; |
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238 \node (b1) at (2,1.2) {}; |
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239 \node (b2) at (2,0) {}; |
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240 \draw (a1) arc (270:90:1) -- +(4,0) arc (90:-90:1); |
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241 \draw (a2) arc (270:90:2.5) -- +(4,0) arc (90:-90:2.5); |
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242 |
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243 % end caps |
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244 \draw (a1) arc (90:450:0.3 and 0.6); |
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245 \draw (b1) arc (90:270:0.3 and 0.6); |
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246 \draw[dashed] (b1) arc (90:-90:0.3 and 0.6); |
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247 |
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248 % the cylinder |
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249 \draw (-1.2,1.2) arc (90:270:0.3 and 0.6); |
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250 \draw[dashed] (-1.2,0) arc (-90:90: 0.3 and 0.6); |
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251 \draw (-1.2,1.2) -- (1.2,1.2) (-1.2,0) -- (1.2,0); |
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252 \draw (1.2,0) arc (-90:270:0.3 and 0.6); |
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253 % the donut hole |
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254 \draw (-2.5,4.2) arc (-135:-45:2); |
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255 \draw (-2,3.9) arc (135:45:1.3); |
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256 |
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257 % labels |
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258 \node at (1.8,4) {\Large $v$}; |
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259 \node at (0,0.5) {\Large $e$}; |
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260 \end{tikzpicture} |
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261 }; |
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262 \node (ev) at (-1,1) { |
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263 \begin{tikzpicture}[x=0.5cm,y=0.5cm] |
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264 \node[coordinate] (a1) at (-2,1.2) {}; |
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265 \node[coordinate] (a2) at (-2,0) {}; |
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266 \node[coordinate] (b1) at (2,1.2) {}; |
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267 \node[coordinate] (b2) at (2,0) {}; |
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268 \draw (0.5,1.2) -- (a1) arc (270:90:1) -- +(4,0) arc (90:-90:1); |
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269 \draw (0.5,0) -- (a2) arc (270:90:2.5) -- +(4,0) arc (90:-90:2.5); |
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270 |
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271 % end caps |
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272 \draw (0.5,1.2) arc (90:450:0.3 and 0.6); |
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273 \draw (b1) arc (90:270:0.3 and 0.6); |
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274 \draw[dashed] (b1) arc (90:-90:0.3 and 0.6); |
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275 |
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276 % the donut hole |
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277 \draw (-2.5,4.2) arc (-135:-45:2); |
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278 \draw (-2,3.9) arc (135:45:1.3); |
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279 |
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280 % dots |
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281 \draw[dotted] (-2,0.6) ellipse (0.3 and 0.6); |
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282 |
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283 % labels |
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284 \node at (1.8,4) {\Large $ev$}; |
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285 \end{tikzpicture} |
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286 }; |
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287 \node (ve) at (1,1) { |
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288 \begin{tikzpicture}[x=0.5cm,y=0.5cm] |
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289 \node[coordinate] (a1) at (-2,1.2) {}; |
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290 \node[coordinate] (a2) at (-2,0) {}; |
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291 \node[coordinate] (b1) at (2,1.2) {}; |
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292 \node[coordinate] (b2) at (2,0) {}; |
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293 \draw (a1) arc (270:90:1) -- +(4,0) arc (90:-90:1) -- (-0.5,1.2); |
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294 \draw (a2) arc (270:90:2.5) -- +(4,0) arc (90:-90:2.5) -- (-0.5,0); |
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295 |
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296 % end caps |
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297 \draw (a1) arc (90:450:0.3 and 0.6); |
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298 \draw (-0.5,1.2) arc (90:270:0.3 and 0.6); |
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299 \draw[dashed] (-0.5,1.2) arc (90:-90:0.3 and 0.6); |
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300 |
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301 % dots |
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302 \draw[dotted] (2,0.6) ellipse (0.3 and 0.6); |
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303 |
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304 % the donut hole |
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305 \draw (-2.5,4.2) arc (-135:-45:2); |
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306 \draw (-2,3.9) arc (135:45:1.3); |
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307 |
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308 % labels |
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309 \node at (1.8,4) {\Large $ve$}; |
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310 \end{tikzpicture} |
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311 }; |
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312 \node (b) at (0,0) { |
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313 \begin{tikzpicture}[x=0.5cm,y=0.5cm] |
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314 \node[coordinate] (a1) at (-2,1.2) {}; |
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315 \node[coordinate] (a2) at (-2,0) {}; |
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316 \node[coordinate] (b1) at (2,1.2) {}; |
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317 \node[coordinate] (b2) at (2,0) {}; |
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318 \draw (a1) arc (270:90:1) -- +(4,0) arc (90:-90:1) -- (-2,1.2); |
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319 \draw (a2) arc (270:90:2.5) -- +(4,0) arc (90:-90:2.5) -- (-2,0); |
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320 |
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321 % the donut hole |
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322 \draw (-2.5,4.2) arc (-135:-45:2); |
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323 \draw (-2,3.9) arc (135:45:1.3); |
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324 |
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325 % dots |
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326 \draw[dotted] (-2,0.6) ellipse (0.3 and 0.6); |
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327 \draw[dotted] (2,0.6) ellipse (0.3 and 0.6); |
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328 |
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329 % labels |
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330 \node at (1.8,4) {$ve = ev$}; |
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331 \end{tikzpicture} |
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332 }; |
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333 \draw[->] (a) -- (ev); |
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334 \draw[->] (a) -- (ve); |
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335 \draw[->] (ev) -- (b); |
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336 \draw[->] (ve) -- (b); |
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337 \end{tikzpicture} |
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338 $$ |
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339 \caption{Isotopic fields on the glued manifold} |
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340 \label{fig:ev-ve} |
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341 \end{figure} |
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342 |
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343 There is a map the other way, too. There isn't quite a map $\cF(X \bigcup_Y \selfarrow) \to \cF(X)$, since a field on $X \bigcup_Y \selfarrow$ need not be splittable along $Y$. Nevertheless, every field is isotopic to one that is splittable along $Y$, and combining this with the lemma above we obtain a map $\cF(X \bigcup_Y \selfarrow) / (\text{isotopy}) \to A(X) \Tensor_{A(Y)} \selfarrow$. We now need to show that this descends to a map from $A(X \bigcup_Y \selfarrow)$. Consider an field of the form $u \bullet f$, for some ball $B$ embedded in $X \bigcup_Y \selfarrow$ and $u \in \cU(B), f \in \cF(X \bigcup_Y \selfarrow \setminus B)$. Now $B$ might cross $Y$, but we can choose an isotopy of $X \bigcup_Y \selfarrow$ so that it doesn't. Thus $u \bullet f$ is sent to a field in $\cU(X)$, and is zero in $A(X) \Tensor_{A(Y)} \selfarrow$. |
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344 |
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345 It's not too hard to see that these maps are mutual inverses. |
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346 \end{proof} |
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347 |
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348 \subsubsection{Codimension 2 gluing} |
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349 |
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350 \section{$n$-categories and fields} |
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351 Roughly, the data of a system of fields and local relations and the data of a disklike $n$-category (from \S 6) are intended to be equivalent. |
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352 |
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353 You essentially recover the axioms for a disklike $n$-category by just remembering everything about $\cF(X) / \cU(X)$ for $X$ a ball. |
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354 Almost equivalently, $A(\bullet)$ gives a disklike $n$-category. |
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355 |
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356 Going the other direction, we've already sketch one method of producing a system of fields from an $n$-category (string diagrams). In \S 6.3 we give another (although not explicitly), based on ball decompositions, which are roughly generalized pasting diagrams. |
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357 |
174 \end{document} |
358 \end{document} |