271 The universal (colimit) construction becomes our generalized definition of blob homology. |
271 The universal (colimit) construction becomes our generalized definition of blob homology. |
272 Need to explain how it relates to the old definition.} |
272 Need to explain how it relates to the old definition.} |
273 |
273 |
274 \medskip |
274 \medskip |
275 |
275 |
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276 \nn{these examples need to be fleshed out a bit more} |
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277 |
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278 Examples of plain $n$-categories: |
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279 \begin{itemize} |
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280 |
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281 \item Let $F$ be a closed $m$-manifold (e.g.\ a point). |
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282 Let $T$ be a topological space. |
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283 For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\cC(X)$ to be the set of |
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284 all maps from $X\times F$ to $T$. |
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285 For $X$ an $n$-ball define $\cC(X)$ to be maps from $X\times F$ to $T$ modulo |
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286 homotopies fixed on $\bd X$. |
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287 (Note that homotopy invariance implies isotopy invariance.) |
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288 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
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289 be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
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290 |
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291 \item We can linearize the above example as follows. |
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292 Let $\alpha$ be an $(n{+}m{+}1)$-cocycle on $T$ with values in a ring $R$ |
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293 (e.g.\ the trivial cocycle). |
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294 For $X$ of dimension less than $n$ define $\cC(X)$ as before. |
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295 For $X$ an $n$-ball and $c\in \cC(\bd X)$ define $\cC(X; c)$ to be |
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296 the $R$-module of finite linear combinations of maps from $X\times F$ to $T$, |
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297 modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
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298 $h: X\times F\times I \to T$, then $a \sim \alpha(h)b$. |
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299 \nn{need to say something about fundamental classes, or choose $\alpha$ carefully} |
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300 |
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301 \item Given a traditional $n$-category $C$ (with strong duality etc.), |
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302 define $\cC(X)$ (with $\dim(X) < n$) |
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303 to be the set of all $C$-labeled sub cell complexes of $X$. |
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304 For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear |
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305 combinations of $C$-labeled sub cell complexes of $X$ |
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306 modulo the kernel of the evaluation map. |
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307 Define a product morphism $a\times D$ to be the product of the cell complex of $a$ with $D$, |
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308 and with the same labeling as $a$. |
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309 \nn{refer elsewhere for details?} |
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310 |
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311 \item Variation on the above examples: |
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312 We could allow $F$ to have boundary and specify boundary conditions on $(\bd X)\times F$, |
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313 for example product boundary conditions or take the union over all boundary conditions. |
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314 |
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315 \end{itemize} |
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316 |
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317 |
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318 Examples of $A_\infty$ $n$-categories: |
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319 \begin{itemize} |
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320 |
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321 \item Same as in example \nn{xxxx} above (fiber $F$, target space $T$), |
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322 but we define, for an $n$-ball $X$, $\cC(X; c)$ to be the chain complex |
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323 $C_*(\Maps_c(X\times F))$, where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
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324 and $C_*$ denotes singular chains. |
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325 |
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326 \item |
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327 Given a plain $n$-category $C$, |
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328 define $\cC(X; c) = \bc^C_*(X\times F; c)$, where $X$ is an $n$-ball |
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329 and $\bc^C_*$ denotes the blob complex based on $C$. |
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330 |
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331 \end{itemize} |
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332 |
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333 \medskip |
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334 |
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335 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
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336 a.k.a.\ actions). |
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337 |
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338 \medskip |
276 \hrule |
339 \hrule |
277 |
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278 \medskip |
340 \medskip |
279 |
341 |
280 \nn{to be continued...} |
342 \nn{to be continued...} |
281 |
343 \medskip |
282 |
344 |
283 |
345 |
284 Stuff that remains to be done (either below or in an appendix or in a separate section or in |
346 Stuff that remains to be done (either below or in an appendix or in a separate section or in |
285 a separate paper): |
347 a separate paper): |
286 \begin{itemize} |
348 \begin{itemize} |