462 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
462 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
463 Boundary restrictions and gluing are again straightforward to define. |
463 Boundary restrictions and gluing are again straightforward to define. |
464 Define product morphisms via product cell decompositions. |
464 Define product morphisms via product cell decompositions. |
465 |
465 |
466 |
466 |
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 A \emph{ball decomposition} of $W$ is a |
472 A \emph{ball decomposition} of $W$ is a |
513 \end{defn} |
513 \end{defn} |
514 |
514 |
515 We will use the term `field on $W$' to refer to a point of this functor, |
515 We will use the term `field on $W$' to refer to a point of this functor, |
516 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
516 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
517 |
517 |
518 \todo{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?} |
|
519 |
518 |
520 \subsubsection{Homotopy colimits} |
519 \subsubsection{Homotopy colimits} |
521 \nn{Motivation: How can we extend an $n$-category from balls to arbitrary manifolds?} |
520 \nn{Motivation: How can we extend an $n$-category from balls to arbitrary manifolds?} |
|
521 \todo{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?} |
|
522 \nn{Explain codimension colimits here too} |
522 |
523 |
523 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
524 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
524 with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$ |
525 with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$ |
525 of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
526 of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
526 |
527 |
549 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
550 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
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 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 |
552 |
552 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of |
553 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of |
553 \begin{itemize} |
554 \begin{itemize} |
554 \item a permissible collection of $k$ embedded balls, |
555 \item a permissible collection of $k$ embedded 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 alternating signs. |
561 |
561 |
562 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. |
562 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. |
563 |
563 |
564 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. |
564 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. |
565 |
565 |