985 We will first define the ``decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
985 We will first define the ``decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
986 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
986 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
987 and we will define $\cl{\cC}(W)$ as a suitable colimit |
987 and we will define $\cl{\cC}(W)$ as a suitable colimit |
988 (or homotopy colimit in the $A_\infty$ case) of this functor. |
988 (or homotopy colimit in the $A_\infty$ case) of this functor. |
989 We'll later give a more explicit description of this colimit. |
989 We'll later give a more explicit description of this colimit. |
990 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-balls with boundary data), |
990 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain |
991 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex). |
991 complexes to $n$-balls with boundary data), |
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992 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into |
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993 subsets according to boundary data, and each of these subsets has the appropriate structure |
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994 (e.g. a vector space or chain complex). |
992 |
995 |
993 Recall (Definition \ref{defn:gluing-decomposition}) that a {\it ball decomposition} of $W$ is a |
996 Recall (Definition \ref{defn:gluing-decomposition}) that a {\it ball decomposition} of $W$ is a |
994 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
997 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
995 $\du_a X_a$. |
998 $\du_a X_a$. |
996 Abusing notation, we let $X_a$ denote both the ball (component of $M_0$) and |
999 Abusing notation, we let $X_a$ denote both the ball (component of $M_0$) and |
1003 Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls |
1006 Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls |
1004 are glued up to yield $W$, so long as there is some (non-pathological) way to glue them. |
1007 are glued up to yield $W$, so long as there is some (non-pathological) way to glue them. |
1005 |
1008 |
1006 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
1009 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
1007 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
1010 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
1008 with $\du_b Y_b = M_i$ for some $i$. |
1011 with $\du_b Y_b = M_i$ for some $i$, |
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1012 and with $M_0,\ldots, M_i$ each being a disjoint union of balls. |
1009 |
1013 |
1010 \begin{defn} |
1014 \begin{defn} |
1011 The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
1015 The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
1012 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
1016 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
1013 See Figure \ref{partofJfig} for an example. |
1017 See Figure \ref{partofJfig} for an example. |
1034 \begin{equation} |
1038 \begin{equation} |
1035 \label{eq:psi-C} |
1039 \label{eq:psi-C} |
1036 \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
1040 \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
1037 \end{equation} |
1041 \end{equation} |
1038 where the restrictions to the various pieces of shared boundaries amongst the cells |
1042 where the restrictions to the various pieces of shared boundaries amongst the cells |
1039 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). |
1043 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n{-}1$-cells). |
1040 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
1044 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
1041 \end{defn} |
1045 \end{defn} |
1042 |
1046 |
1043 If $k=n$ in the above definition and we are enriching in some auxiliary category, |
1047 If $k=n$ in the above definition and we are enriching in some auxiliary category, |
1044 we need to say a bit more. |
1048 we need to say a bit more. |