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234 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
234 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
235 to be the union of the blobs of $b$. |
235 to be the union of the blobs of $b$. |
236 For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
236 For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
237 we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
237 we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
238 |
238 |
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239 \begin{remark} \label{blobsset-remark} \rm |
239 We note that blob diagrams in $X$ have a structure similar to that of a simplicial set, |
240 We note that blob diagrams in $X$ have a structure similar to that of a simplicial set, |
240 but with simplices replaced by a more general class of combinatorial shapes. |
241 but with simplices replaced by a more general class of combinatorial shapes. |
241 Let $P$ be the minimal set of (isomorphisms classes of) polyhedra which is closed under products |
242 Let $P$ be the minimal set of (isomorphisms classes of) polyhedra which is closed under products |
242 and cones, and which contains the point. |
243 and cones, and which contains the point. |
243 We can associate an element $p(b)$ of $P$ to each blob diagram $b$ |
244 We can associate an element $p(b)$ of $P$ to each blob diagram $b$ |
252 For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while |
253 For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while |
253 a diagram of $k$ disjoint blobs corresponds to a $k$-cube. |
254 a diagram of $k$ disjoint blobs corresponds to a $k$-cube. |
254 (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 |
255 (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 |
255 $x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cF(B_i) \to C$ is the evaluation map, |
256 $x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cF(B_i) \to C$ is the evaluation map, |
256 and $s:C \to \cF(B_i)$ is some fixed section of $e$.) |
257 and $s:C \to \cF(B_i)$ is some fixed section of $e$.) |
257 |
258 \end{remark} |
258 |
259 |