413 The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
413 The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
414 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
414 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
415 See Figure \ref{partofJfig} for an example. |
415 See Figure \ref{partofJfig} for an example. |
416 \end{defn} |
416 \end{defn} |
417 |
417 |
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418 This poset in fact has more structure, since we can glue together permissible decompositions of $W_1$ and $W_2$ to obtain a permissible decomposition of $W_1 \sqcup W_2$. |
418 |
419 |
419 An $n$-category $\cC$ determines |
420 An $n$-category $\cC$ determines |
420 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
421 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
421 (possibly with additional structure if $k=n$). |
422 (possibly with additional structure if $k=n$). |
422 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
423 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
441 \todo{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?} |
442 \todo{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?} |
442 |
443 |
443 \subsubsection{Homotopy colimits} |
444 \subsubsection{Homotopy colimits} |
444 \nn{Motivation: How can we extend an $n$-category from balls to arbitrary manifolds?} |
445 \nn{Motivation: How can we extend an $n$-category from balls to arbitrary manifolds?} |
445 |
446 |
446 We now define the blob complex $\bc_*(W; \cC)$ of an $n$-manifold $W$ |
447 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
447 with coefficients in the $n$-category $\cC$ to be the homotopy colimit |
448 with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$ |
448 of the functor $\psi_{\cC; W}$ described above. |
449 of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
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450 |
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451 An explicit realization of the homotopy colimit is provided by the simplices of the functor $\psi_{\cC; W}$. That is, $$\clh{\cC}(W) = \DirectSum_{x} \psi_{\cC; W}(x_0)[m],$$ where $x = x_0 \leq \cdots \leq x_m$ is a simplex in $\cell(W)$. The differential acts on \todo{finish this} |
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452 |
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453 Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the homotopy colimit as the cone-product polyhedra of the functor $\psi_{\cC;W}$. (A cone-product polyhedra is obtained from a point by successively taking the cone or taking the product with another cone-product polyhedron.) A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
449 |
454 |
450 When $\cC$ is a topological $n$-category, |
455 When $\cC$ is a topological $n$-category, |
451 the flexibility available in the construction of a homotopy colimit allows |
456 the flexibility available in the construction of a homotopy colimit allows |
452 us to give a much more explicit description of the blob complex. |
457 us to give a much more explicit description of the blob complex. We'll write $\bc_*(W; \cC)$ for this more explicit version. |
453 |
458 |
454 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
459 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
455 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
460 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
456 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. |
461 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. |
457 |
462 |
463 \end{itemize} |
468 \end{itemize} |
464 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 $B$'. |
469 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 $B$'. |
465 |
470 |
466 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. |
471 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. |
467 |
472 |
468 \todo{Say why this really is the homotopy colimit} |
473 We now spell this out for some small values of $k$. For $k=0$, the $0$-blob group is simply fields on $W$. For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field to that ball is a null field. The differential simply forgets the ball. Thus we see that $H_0$ of the blob complex is the quotient of fields by fields which are null on some ball. |
469 |
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470 We now spell this out for some small values of $k$. For $k=0$, the $0$-blob group is simply fields on $W$. For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field that that ball is a null field. The differential simply forgets the ball. Thus we see that $H_0$ of the blob complex is the quotient of fields by null fields. |
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471 |
474 |
472 For $k=2$, we have a two types of generators; they each consists of a field $f$ on $W$, and two balls $B_1$ and $B_2$. In the first case, the balls are disjoint, and $f$ restricted to either of the $B_i$ is a null field. In the second case, the balls are properly nested, say $B_1 \subset B_2$, and $f$ restricted to $B_1$ is null. Note that this implies that $f$ restricted to $B_2$ is also null, by the associativity of the gluing operation. This ensures that the differential is well-defined. |
475 For $k=2$, we have a two types of generators; they each consists of a field $f$ on $W$, and two balls $B_1$ and $B_2$. In the first case, the balls are disjoint, and $f$ restricted to either of the $B_i$ is a null field. In the second case, the balls are properly nested, say $B_1 \subset B_2$, and $f$ restricted to $B_1$ is null. Note that this implies that $f$ restricted to $B_2$ is also null, by the associativity of the gluing operation. This ensures that the differential is well-defined. |
473 |
476 |
474 \section{Properties of the blob complex} |
477 \section{Properties of the blob complex} |
475 \subsection{Formal properties} |
478 \subsection{Formal properties} |
628 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
631 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
629 Let $\cC$ be an $n$-category. |
632 Let $\cC$ be an $n$-category. |
630 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology. |
633 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology. |
631 Then |
634 Then |
632 \[ |
635 \[ |
633 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
636 \bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). |
634 \] |
637 \] |
635 \end{thm} |
638 \end{thm} |
636 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
639 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
637 (see \cite[\S7.1]{1009.5025}). |
640 (see \cite[\S7.1]{1009.5025}). |
638 |
641 |