751 to $\cM(M)$, |
751 to $\cM(M)$, |
752 and these various multifold composition maps satisfy an |
752 and these various multifold composition maps satisfy an |
753 operad-type strict associativity condition.} |
753 operad-type strict associativity condition.} |
754 |
754 |
755 (The above operad-like structure is analogous to the swiss cheese operad |
755 (The above operad-like structure is analogous to the swiss cheese operad |
756 \nn{need citation}.) |
756 \cite{MR1718089}.) |
757 \nn{need to double-check that this is true.} |
757 \nn{need to double-check that this is true.} |
758 |
758 |
759 \xxpar{Module product (identity) morphisms:} |
759 \xxpar{Module product (identity) morphisms:} |
760 {Let $M$ be a marked $k$-ball and $D$ be a plain $m$-ball, with $k+m \le n$. |
760 {Let $M$ be a marked $k$-ball and $D$ be a plain $m$-ball, with $k+m \le n$. |
761 Then we have a map $\cM(M)\to \cM(M\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cM(M)$. |
761 Then we have a map $\cM(M)\to \cM(M\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cM(M)$. |
852 with $M_{ib}\cap Y_i$ being the marking. |
852 with $M_{ib}\cap Y_i$ being the marking. |
853 (See Figure \ref{mblabel}.) |
853 (See Figure \ref{mblabel}.) |
854 \begin{figure}[!ht]\begin{equation*} |
854 \begin{figure}[!ht]\begin{equation*} |
855 \mathfig{.9}{tempkw/mblabel} |
855 \mathfig{.9}{tempkw/mblabel} |
856 \end{equation*}\caption{A permissible decomposition of a manifold |
856 \end{equation*}\caption{A permissible decomposition of a manifold |
857 whose boundary components are labeled my $\cC$ modules $\{\cN_i\}$.}\label{mblabel}\end{figure} |
857 whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$.}\label{mblabel}\end{figure} |
858 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
858 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
859 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
859 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
860 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
860 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
861 (The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
861 (The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
862 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
862 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |