210 Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
210 Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
211 We wish to imitate this strategy in higher categories. |
211 We wish to imitate this strategy in higher categories. |
212 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
212 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
213 a product of $k$ intervals \nn{cf xxxx} but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic |
213 a product of $k$ intervals \nn{cf xxxx} but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic |
214 to the standard $k$-ball $B^k$. |
214 to the standard $k$-ball $B^k$. |
215 \nn{maybe add that in addition we want funtoriality} |
215 \nn{maybe add that in addition we want functoriality} |
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216 |
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217 \nn{say something about different flavors of balls; say it here? later?} |
216 |
218 |
217 \begin{axiom}[Morphisms] |
219 \begin{axiom}[Morphisms] |
218 \label{axiom:morphisms} |
220 \label{axiom:morphisms} |
219 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
221 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
220 the category of $k$-balls and |
222 the category of $k$-balls and |
221 homeomorphisms to the category of sets and bijections. |
223 homeomorphisms to the category of sets and bijections. |
222 \end{axiom} |
224 \end{axiom} |
223 |
225 |
224 |
226 Note that the functoriality in the above axiom allows us to operate via |
225 |
227 |
226 \begin{lem} |
228 Next we consider domains and ranges of $k$-morphisms. |
227 \label{lem:spheres} |
229 Because we assume strong duality, it doesn't make much sense to subdivide the boundary of a morphism |
228 For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from |
230 into domain and range --- the duality operations can convert domain to range and vice-versa. |
229 the category of $k{-}1$-spheres and |
231 Instead, we will use a unified domain/range, which we will call a ``boundary". |
230 homeomorphisms to the category of sets and bijections. |
232 |
231 \end{lem} |
233 In order to state the axiom for boundaries, we need to extend the functors $\cC_k$ |
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234 of $k$-balls to functors $\cl{\cC}_{k-1}$ of $k$-spheres. |
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235 This extension is described in xxxx below. |
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236 |
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237 %\begin{lem} |
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238 %\label{lem:spheres} |
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239 %For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from |
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240 %the category of $k{-}1$-spheres and |
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241 %homeomorphisms to the category of sets and bijections. |
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242 %\end{lem} |
232 |
243 |
233 \begin{axiom}[Boundaries]\label{nca-boundary} |
244 \begin{axiom}[Boundaries]\label{nca-boundary} |
234 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
245 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
235 These maps, for various $X$, comprise a natural transformation of functors. |
246 These maps, for various $X$, comprise a natural transformation of functors. |
236 \end{axiom} |
247 \end{axiom} |
358 \todo{maps to a space, string diagrams} |
369 \todo{maps to a space, string diagrams} |
359 |
370 |
360 \subsection{The blob complex} |
371 \subsection{The blob complex} |
361 \subsubsection{Decompositions of manifolds} |
372 \subsubsection{Decompositions of manifolds} |
362 |
373 |
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374 \nn{KW: I'm inclined to suppress all discussion of the subtleties of decompositions. |
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375 Maybe just a single remark that we are omitting some details which appear in our |
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376 longer paper.} |
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377 |
363 A \emph{ball decomposition} of $W$ is a |
378 A \emph{ball decomposition} of $W$ is a |
364 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
379 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
365 $\du_a X_a$ and each $M_i$ is a manifold. |
380 $\du_a X_a$ and each $M_i$ is a manifold. |
366 If $X_a$ is some component of $M_0$, its image in $W$ need not be a ball; $\bd X_a$ may have been glued to itself. |
381 If $X_a$ is some component of $M_0$, its image in $W$ need not be a ball; $\bd X_a$ may have been glued to itself. |
367 A {\it permissible decomposition} of $W$ is a map |
382 A {\it permissible decomposition} of $W$ is a map |
440 |
455 |
441 \section{Properties of the blob complex} |
456 \section{Properties of the blob complex} |
442 \subsection{Formal properties} |
457 \subsection{Formal properties} |
443 \label{sec:properties} |
458 \label{sec:properties} |
444 The blob complex enjoys the following list of formal properties. |
459 The blob complex enjoys the following list of formal properties. |
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460 |
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461 The proofs of the first three properties are immediate from the definitions. |
445 |
462 |
446 \begin{property}[Functoriality] |
463 \begin{property}[Functoriality] |
447 \label{property:functoriality}% |
464 \label{property:functoriality}% |
448 The blob complex is functorial with respect to homeomorphisms. |
465 The blob complex is functorial with respect to homeomorphisms. |
449 That is, |
466 That is, |
489 associated by the system of fields $\cF$ to balls. |
506 associated by the system of fields $\cF$ to balls. |
490 \begin{equation*} |
507 \begin{equation*} |
491 \xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & A_\cF(B^n)} |
508 \xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & A_\cF(B^n)} |
492 \end{equation*} |
509 \end{equation*} |
493 \end{property} |
510 \end{property} |
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511 |
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512 \begin{proof}(Sketch) |
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513 For $k\ge 1$, the contracting homotopy sends a $k$-blob diagram to the $(k{+}1)$-blob diagram |
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514 obtained by adding an outer $(k{+}1)$-st blob consisting of all $B^n$. |
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515 For $k=0$ we choose a splitting $s: H_0(\bc_*(B^n)) \to \bc_0(B^n)$ and send |
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516 $x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$. |
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517 \end{proof} |
494 |
518 |
495 \nn{Properties \ref{property:functoriality} will be immediate from the definition given in |
519 \nn{Properties \ref{property:functoriality} will be immediate from the definition given in |
496 \S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there. |
520 \S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there. |
497 Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and |
521 Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and |
498 \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.} |
522 \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.} |