135 %For the disjoint blobs, reversing the ordering of $B_1$ and $B_2$ introduces a minus sign |
136 %For the disjoint blobs, reversing the ordering of $B_1$ and $B_2$ introduces a minus sign |
136 %(rather than a new, linearly independent, 2-blob diagram). |
137 %(rather than a new, linearly independent, 2-blob diagram). |
137 |
138 |
138 \medskip |
139 \medskip |
139 |
140 |
140 Roughly, $\bc_k(X)$ is generated by configurations of $k$ blobs, pairwise disjoint or nested, along with fields on all the components that the blobs divide $X$ into. Blobs which have no other blobs inside are called `twig blobs', and the fields on the twig blobs must be local relations. |
141 Roughly, $\bc_k(X)$ is generated by configurations of $k$ blobs, pairwise disjoint or nested, |
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142 along with fields on all the components that the blobs divide $X$ into. |
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143 Blobs which have no other blobs inside are called `twig blobs', |
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144 and the fields on the twig blobs must be local relations. |
141 The boundary is the alternating sum of erasing one of the blobs. |
145 The boundary is the alternating sum of erasing one of the blobs. |
142 In order to describe this general case in full detail, we must give a more precise description of |
146 In order to describe this general case in full detail, we must give a more precise description of |
143 which configurations of balls inside $X$ we permit. |
147 which configurations of balls inside $X$ we permit. |
144 These configurations are generated by two operations: |
148 These configurations are generated by two operations: |
145 \begin{itemize} |
149 \begin{itemize} |
160 A & = [0,1] \times [0,1] \times [0,1] \\ |
164 A & = [0,1] \times [0,1] \times [0,1] \\ |
161 B & = [0,1] \times [-1,0] \times [0,1] \\ |
165 B & = [0,1] \times [-1,0] \times [0,1] \\ |
162 C & = [-1,0] \times \setc{(y,z)}{z \sin(1/z) \leq y \leq 1, z \in [0,1]} \\ |
166 C & = [-1,0] \times \setc{(y,z)}{z \sin(1/z) \leq y \leq 1, z \in [0,1]} \\ |
163 D & = [-1,0] \times \setc{(y,z)}{-1 \leq y \leq z \sin(1/z), z \in [0,1]}. |
167 D & = [-1,0] \times \setc{(y,z)}{-1 \leq y \leq z \sin(1/z), z \in [0,1]}. |
164 \end{align*} |
168 \end{align*} |
165 Here $A \cup B = [0,1] \times [-1,1] \times [0,1]$ and $C \cup D = [-1,0] \times [-1,1] \times [0,1]$. Now, $\{A\}$ is a valid configuration of blobs in $A \cup B$, and $\{C\}$ is a valid configuration of blobs in $C \cup D$, so we must allow $\{A, C\}$ as a configuration of blobs in $[-1,1]^2 \times [0,1]$. Note however that the complement is not a manifold. |
169 Here $A \cup B = [0,1] \times [-1,1] \times [0,1]$ and $C \cup D = [-1,0] \times [-1,1] \times [0,1]$. |
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170 Now, $\{A\}$ is a valid configuration of blobs in $A \cup B$, |
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171 and $\{C\}$ is a valid configuration of blobs in $C \cup D$, |
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172 so we must allow $\{A, C\}$ as a configuration of blobs in $[-1,1]^2 \times [0,1]$. |
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173 Note however that the complement is not a manifold. |
166 \end{example} |
174 \end{example} |
167 |
175 |
168 \begin{defn} |
176 \begin{defn} |
169 \label{defn:gluing-decomposition} |
177 \label{defn:gluing-decomposition} |
170 A \emph{gluing decomposition} of an $n$-manifold $X$ is a sequence of manifolds |
178 A \emph{gluing decomposition} of an $n$-manifold $X$ is a sequence of manifolds |
171 $M_0 \to M_1 \to \cdots \to M_m = X$ such that each $M_k$ is obtained from $M_{k-1}$ |
179 $M_0 \to M_1 \to \cdots \to M_m = X$ such that each $M_k$ is obtained from $M_{k-1}$ |
172 by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$. |
180 by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$. |
173 If, in addition, $M_0$ is a disjoint union of balls, we call it a \emph{ball decomposition}. |
181 If, in addition, $M_0$ is a disjoint union of balls, we call it a \emph{ball decomposition}. |
174 \end{defn} |
182 \end{defn} |
175 Given a gluing decomposition $M_0 \to M_1 \to \cdots \to M_m = X$, we say that a field is splittable along it if it is the image of a field on $M_0$. |
183 Given a gluing decomposition $M_0 \to M_1 \to \cdots \to M_m = X$, we say that a field is |
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184 splittable along it if it is the image of a field on $M_0$. |
176 |
185 |
177 In the example above, note that |
186 In the example above, note that |
178 \[ |
187 \[ |
179 A \sqcup B \sqcup C \sqcup D \to (A \cup B) \sqcup (C \cup D) \to A \cup B \cup C \cup D |
188 A \sqcup B \sqcup C \sqcup D \to (A \cup B) \sqcup (C \cup D) \to A \cup B \cup C \cup D |
180 \] |
189 \] |
201 that for any two blobs in a configuration of blobs in $X$, |
210 that for any two blobs in a configuration of blobs in $X$, |
202 they either have disjoint interiors, or one blob is contained in the other. |
211 they either have disjoint interiors, or one blob is contained in the other. |
203 We describe these as disjoint blobs and nested blobs. |
212 We describe these as disjoint blobs and nested blobs. |
204 Note that nested blobs may have boundaries that overlap, or indeed coincide. |
213 Note that nested blobs may have boundaries that overlap, or indeed coincide. |
205 Blobs may meet the boundary of $X$. |
214 Blobs may meet the boundary of $X$. |
206 Further, note that blobs need not actually be embedded balls in $X$, since parts of the boundary of the ball $M_r'$ may have been glued together. |
215 Further, note that blobs need not actually be embedded balls in $X$, since parts of the |
207 |
216 boundary of the ball $M_r'$ may have been glued together. |
208 Note that often the gluing decomposition for a configuration of blobs may just be the trivial one: if the boundaries of all the blobs cut $X$ into pieces which are all manifolds, we can just take $M_0$ to be these pieces, and $M_1 = X$. |
217 |
209 |
218 Note that often the gluing decomposition for a configuration of blobs may just be the trivial one: |
210 In the informal description above, in the definition of a $k$-blob diagram we asked for any collection of $k$ balls which were pairwise disjoint or nested. We now further insist that the balls are a configuration in the sense of Definition \ref{defn:configuration}. Also, we asked for a local relation on each twig blob, and a field on the complement of the twig blobs; this is unsatisfactory because that complement need not be a manifold. Thus, the official definitions are |
219 if the boundaries of all the blobs cut $X$ into pieces which are all manifolds, |
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220 we can just take $M_0$ to be these pieces, and $M_1 = X$. |
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221 |
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222 In the informal description above, in the definition of a $k$-blob diagram we asked for any |
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223 collection of $k$ balls which were pairwise disjoint or nested. |
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224 We now further insist that the balls are a configuration in the sense of Definition \ref{defn:configuration}. |
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225 Also, we asked for a local relation on each twig blob, and a field on the complement of the twig blobs; |
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226 this is unsatisfactory because that complement need not be a manifold. Thus, the official definitions are |
211 \begin{defn} |
227 \begin{defn} |
212 \label{defn:blob-diagram} |
228 \label{defn:blob-diagram} |
213 A $k$-blob diagram on $X$ consists of |
229 A $k$-blob diagram on $X$ consists of |
214 \begin{itemize} |
230 \begin{itemize} |
215 \item a configuration $\{B_1, \ldots B_k\}$ of $k$ blobs in $X$, |
231 \item a configuration $\{B_1, \ldots B_k\}$ of $k$ blobs in $X$, |
216 \item and a field $r \in \cF(X)$ which is splittable along some gluing decomposition compatible with that configuration, |
232 \item and a field $r \in \cF(X)$ which is splittable along some gluing decomposition compatible with that configuration, |
217 \end{itemize} |
233 \end{itemize} |
218 such that |
234 such that |
219 the restriction $u_i$ of $r$ to each twig blob $B_i$ lies in the subspace $U(B_i) \subset \cF(B_i)$. (See Figure \ref{blobkdiagram}.) More precisely, each twig blob $B_i$ is the image of some ball $M_r'$ as above, and it is really the restriction to $M_r'$ that must lie in the subspace $U(M_r')$. |
235 the restriction $u_i$ of $r$ to each twig blob $B_i$ lies in the subspace |
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236 $U(B_i) \subset \cF(B_i)$. |
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237 (See Figure \ref{blobkdiagram}.) |
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238 More precisely, each twig blob $B_i$ is the image of some ball $M_r'$ as above, |
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239 and it is really the restriction to $M_r'$ that must lie in the subspace $U(M_r')$. |
220 \end{defn} |
240 \end{defn} |
221 \begin{figure}[t]\begin{equation*} |
241 \begin{figure}[t]\begin{equation*} |
222 \mathfig{.7}{definition/k-blobs} |
242 \mathfig{.7}{definition/k-blobs} |
223 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
243 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
224 and |
244 and |
225 \begin{defn} |
245 \begin{defn} |
226 \label{defn:blobs} |
246 \label{defn:blobs} |
227 The $k$-th vector space $\bc_k(X)$ of the \emph{blob complex} of $X$ is the direct sum over all configurations of $k$ blobs in $X$ of the vector space of $k$-blob diagrams with that configuration, modulo identifying the vector spaces for configurations that only differ by a permutation of the balls by the sign of that permutation. The differential $\bc_k(X) \to \bc_{k-1}(X)$ is, as above, the signed sum of ways of forgetting one blob from the configuration, preserving the field $r$: |
247 The $k$-th vector space $\bc_k(X)$ of the \emph{blob complex} of $X$ is the direct sum over all |
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248 configurations of $k$ blobs in $X$ of the vector space of $k$-blob diagrams with that configuration, |
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249 modulo identifying the vector spaces for configurations that only differ by a permutation of the balls |
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250 by the sign of that permutation. |
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251 The differential $\bc_k(X) \to \bc_{k-1}(X)$ is, as above, the signed sum of ways of |
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252 forgetting one blob from the configuration, preserving the field $r$: |
228 \begin{equation*} |
253 \begin{equation*} |
229 \bdy(\{B_1, \ldots B_k\}, r) = \sum_{i=1}^{k} (-1)^{i+1} (\{B_1, \ldots, \widehat{B_i}, \ldots, B_k\}, r) |
254 \bdy(\{B_1, \ldots B_k\}, r) = \sum_{i=1}^{k} (-1)^{i+1} (\{B_1, \ldots, \widehat{B_i}, \ldots, B_k\}, r) |
230 \end{equation*} |
255 \end{equation*} |
231 \end{defn} |
256 \end{defn} |
232 We readily see that if a gluing decomposition is compatible with some configuration of blobs, then it is also compatible with any configuration obtained by forgetting some blobs, ensuring that the differential in fact lands in the space of $k{-}1$-blob diagrams. |
257 We readily see that if a gluing decomposition is compatible with some configuration of blobs, |
233 A slight compensation to the complication of the official definition arising from attention to splitting is that the differential now just preserves the entire field $r$ without having to say anything about gluing together fields on smaller components. |
258 then it is also compatible with any configuration obtained by forgetting some blobs, |
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259 ensuring that the differential in fact lands in the space of $k{-}1$-blob diagrams. |
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260 A slight compensation to the complication of the official definition arising from attention |
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261 to splitting is that the differential now just preserves the entire field $r$ without |
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262 having to say anything about gluing together fields on smaller components. |
234 |
263 |
235 Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, |
264 Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, |
236 is immediately obvious from the definition. |
265 is immediately obvious from the definition. |
237 A homeomorphism acts in an obvious way on blobs and on fields. |
266 A homeomorphism acts in an obvious way on blobs and on fields. |
238 |
267 |
255 \item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which |
284 \item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which |
256 encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root). |
285 encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root). |
257 \end{itemize} |
286 \end{itemize} |
258 For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while |
287 For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while |
259 a diagram of $k$ disjoint blobs corresponds to a $k$-cube. |
288 a diagram of $k$ disjoint blobs corresponds to a $k$-cube. |
260 (When the fields come from an $n$-category, this correspondence works best if we think of each twig label $u_i$ as having the form |
289 (When the fields come from an $n$-category, this correspondence works best if we think of each |
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290 twig label $u_i$ as having the form |
261 $x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cF(B_i) \to C$ is the evaluation map, |
291 $x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cF(B_i) \to C$ is the evaluation map, |
262 and $s:C \to \cF(B_i)$ is some fixed section of $e$.) |
292 and $s:C \to \cF(B_i)$ is some fixed section of $e$.) |
263 |
293 |
264 For lack of a better name, we'll call elements of $P$ cone-product polyhedra, |
294 For lack of a better name, |
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295 \nn{can we think of a better name?} |
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296 we'll call elements of $P$ cone-product polyhedra, |
265 and say that blob diagrams have the structure of a cone-product set (analogous to simplicial set). |
297 and say that blob diagrams have the structure of a cone-product set (analogous to simplicial set). |
266 \end{remark} |
298 \end{remark} |
267 |
299 |