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536 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
536 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
537 %\todo{either need to explain why this is the same, or significantly rewrite this section} |
537 %\todo{either need to explain why this is the same, or significantly rewrite this section} |
538 When $\cC$ is the topological $n$-category based on string diagrams for a traditional |
538 When $\cC$ is the topological $n$-category based on string diagrams for a traditional |
539 $n$-category $C$, |
539 $n$-category $C$, |
540 one can show \nn{cite us} that the above two constructions of the homotopy colimit |
540 one can show \nn{cite us} that the above two constructions of the homotopy colimit |
541 are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; C)$. |
541 are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; \cC)$. |
542 Roughly speaking, the generators of $\bc_k(W; C)$ are string diagrams on $W$ together with |
542 Roughly speaking, the generators of $\bc_k(W; \cC)$ are string diagrams on $W$ together with |
543 a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested. |
543 a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested. |
544 The restriction of the string diagram to innermost blobs is required to be ``null" in the sense that |
544 The restriction of the string diagram to innermost blobs is required to be ``null" in the sense that |
545 it evaluates to a zero $n$-morphism of $C$. |
545 it evaluates to a zero $n$-morphism of $C$. |
546 The next few paragraphs describe this in more detail. |
546 The next few paragraphs describe this in more detail. |
547 |
547 |
550 each $B_i$ appears as a connected component of one of the $M_j$. Note that this allows the balls to be pairwise either disjoint or nested. Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. These pieces need not be manifolds, but they do automatically have permissible decompositions. |
550 each $B_i$ appears as a connected component of one of the $M_j$. Note that this allows the balls to be pairwise either disjoint or nested. Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. These pieces need not be manifolds, but they do automatically have permissible decompositions. |
551 |
551 |
552 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of |
552 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of |
553 \begin{itemize} |
553 \begin{itemize} |
554 \item a permissible collection of $k$ embedded balls, |
554 \item a permissible collection of $k$ embedded balls, |
555 \item an ordering of the balls, and |
555 \item an ordering of the balls, and \nn{what about reordering?} |
556 \item for each resulting piece of $W$, a field, |
556 \item for each resulting piece of $W$, a field, |
557 \end{itemize} |
557 \end{itemize} |
558 such that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. We call such a field a `null field on $B$'. |
558 such that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. We call such a field a `null field on $B$'. |
559 |
559 |
560 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with signs given by the ordering. |
560 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with signs given by the ordering. |