198 \psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
198 \psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
199 a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
199 a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
200 \end{eqnarray*} |
200 \end{eqnarray*} |
201 \nn{need to say something somewhere about pinched boundary convention for products} |
201 \nn{need to say something somewhere about pinched boundary convention for products} |
202 We will call $\psi_{Y,J}$ an extended isotopy. |
202 We will call $\psi_{Y,J}$ an extended isotopy. |
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203 \nn{or extended homeomorphism? see below.} |
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204 \nn{maybe remark that in some examples (e.g.\ ones based on sub cell complexes) |
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205 extended isotopies are also plain isotopies, so |
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206 no extension necessary} |
203 It can be thought of as the action of the inverse of |
207 It can be thought of as the action of the inverse of |
204 a map which projects a collar neighborhood of $Y$ onto $Y$. |
208 a map which projects a collar neighborhood of $Y$ onto $Y$. |
205 (This sort of collapse map is the other sense of ``pseudo-isotopy".) |
209 (This sort of collapse map is the other sense of ``pseudo-isotopy".) |
206 \nn{need to check this} |
210 \nn{need to check this} |
207 |
211 |
212 to the identity on $\bd X$ and is pseudo-isotopic or extended isotopic (rel boundary) to the identity. |
216 to the identity on $\bd X$ and is pseudo-isotopic or extended isotopic (rel boundary) to the identity. |
213 Then $f$ acts trivially on $\cC(X)$.} |
217 Then $f$ acts trivially on $\cC(X)$.} |
214 |
218 |
215 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
219 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
216 |
220 |
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221 \smallskip |
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222 |
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223 For $A_\infty$ $n$-categories, we replace |
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224 isotopy invariance with the requirement that families of homeomorphisms act. |
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225 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
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226 |
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227 \xxpar{Families of homeomorphisms act.} |
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228 {For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes |
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229 \[ |
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230 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
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231 \] |
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232 Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$ |
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233 which fix $\bd X$. |
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234 These action maps are required to be associative up to homotopy |
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235 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
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236 a diagram like the one in Proposition \ref{CDprop} commutes. |
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237 \nn{repeat diagram here?} |
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238 \nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?}} |
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239 |
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240 We should strengthen the above axiom to apply to families of extended homeomorphisms. |
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241 To do this we need to explain extended homeomorphisms form a space. |
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242 Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
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243 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
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244 \nn{need to also say something about collaring homeomorphisms.} |
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245 \nn{this paragraph needs work.} |
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246 |
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247 Note that if take homology of chain complexes, we turn an $A_\infty$ $n$-category |
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248 into a plain $n$-category. |
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249 \nn{say more here?} |
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250 In the other direction, if we enrich over topological spaces instead of chain complexes, |
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251 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
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252 instead of $C_*(\Homeo_\bd(X))$. |
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253 Taking singular chains converts a space-type $A_\infty$ $n$-category into a chain complex |
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254 type $A_\infty$ $n$-category. |
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255 |
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256 |
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257 |
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258 |
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259 |
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260 |
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261 |
217 |
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218 |
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219 |
264 |
220 \medskip |
265 \medskip |
221 |
266 |