467 \nn{also do bordism category?} |
467 \nn{also do bordism category?} |
468 |
468 |
469 \subsection{The blob complex} |
469 \subsection{The blob complex} |
470 \subsubsection{Decompositions of manifolds} |
470 \subsubsection{Decompositions of manifolds} |
471 |
471 |
472 \nn{KW: I'm inclined to suppress all discussion of the subtleties of decompositions. |
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473 Maybe just a single remark that we are omitting some details which appear in our |
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474 longer paper.} |
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475 \nn{SM: for now I disagree: the space expense is pretty minor, and it allows us to be ``in principle" complete. Let's see how we go for length.} |
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476 \nn{KW: It's not the length I'm worried about --- I was worried about distracting the reader |
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477 with an arcane technical issue. But we can decide later.} |
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478 |
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479 A \emph{ball decomposition} of $W$ is a |
472 A \emph{ball decomposition} of $W$ is a |
480 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
473 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
481 $\du_a X_a$ and each $M_i$ is a manifold. |
474 $\du_a X_a$ and each $M_i$ is a manifold. |
482 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. |
475 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. |
483 A {\it permissible decomposition} of $W$ is a map |
476 A {\it permissible decomposition} of $W$ is a map |
536 where $g$ is the gluing map from $x_0$ to $x_1$, and $d_i \bar{x}$ denotes the $i$-th face of the simplex $\bar{x}$. |
529 where $g$ is the gluing map from $x_0$ to $x_1$, and $d_i \bar{x}$ denotes the $i$-th face of the simplex $\bar{x}$. |
537 |
530 |
538 Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the homotopy colimit via the cone-product polyhedra in $\cell(W)$. A cone-product polyhedra is obtained from a point by successively taking the cone or taking the product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, and taking product identifies the roots of several trees. The `local homotopy colimit' is then defined according to the same formula as above, but with $x$ a cone-product polyhedron in $\cell(W)$. |
531 Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the homotopy colimit via the cone-product polyhedra in $\cell(W)$. A cone-product polyhedra is obtained from a point by successively taking the cone or taking the product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, and taking product identifies the roots of several trees. The `local homotopy colimit' is then defined according to the same formula as above, but with $x$ a cone-product polyhedron in $\cell(W)$. |
539 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
532 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
540 |
533 |
541 When $\cC$ is a topological $n$-category, |
534 %When $\cC$ is a topological $n$-category, |
542 the flexibility available in the construction of a homotopy colimit allows |
535 %the flexibility available in the construction of a homotopy colimit allows |
543 us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. \todo{either need to explain why this is the same, or significantly rewrite this section} |
536 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
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537 %\todo{either need to explain why this is the same, or significantly rewrite this section} |
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538 When $\cC$ is the topological $n$-category based on string diagrams for a traditional |
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539 $n$-category $C$, |
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540 one can show \nn{cite us} that the above two constructions of the homotopy colimit |
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541 are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; C)$. |
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542 Roughly speaking, the generators of $\bc_k(W; C)$ are string diagrams on $W$ together with |
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543 a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested. |
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544 The restriction of the string diagram to innermost blobs is required to be ``null" in the sense that |
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545 it evaluates to a zero $n$-morphism of $C$. |
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546 The next few paragraphs describe this in more detail. |
544 |
547 |
545 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
548 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
546 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
549 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
547 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. |
548 |
551 |