equal
deleted
inserted
replaced
944 \] |
944 \] |
945 which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. |
945 which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. |
946 Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls |
946 Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls |
947 are glued up to yield $W$, so long as there is some (non-pathological) way to glue them. |
947 are glued up to yield $W$, so long as there is some (non-pathological) way to glue them. |
948 |
948 |
949 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ or $W$, we say that $x$ is a refinement |
949 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
950 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
950 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
951 with $\du_b Y_b = M_i$ for some $i$. |
951 with $\du_b Y_b = M_i$ for some $i$. |
952 |
952 |
953 \begin{defn} |
953 \begin{defn} |
954 The category (poset) $\cell(W)$ has objects the permissible decompositions of $W$, |
954 The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
955 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
955 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
956 See Figure \ref{partofJfig} for an example. |
956 See Figure \ref{partofJfig} for an example. |
957 \end{defn} |
957 \end{defn} |
958 |
958 |
959 \begin{figure}[!ht] |
959 \begin{figure}[!ht] |