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 

   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 

   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?}

   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}

   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 

   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