125 |
125 |
126 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
126 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
127 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
127 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
128 all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category |
128 all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category |
129 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
129 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
|
130 \nn{actually, need both disj-union/sub and product/tensor-product; what's the name for this sort of cat?} |
130 and all the structure maps of the $n$-category should be compatible with the auxiliary |
131 and all the structure maps of the $n$-category should be compatible with the auxiliary |
131 category structure. |
132 category structure. |
132 Note that this auxiliary structure is only in dimension $n$; |
133 Note that this auxiliary structure is only in dimension $n$; |
133 $\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. |
134 $\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. |
134 |
135 |
842 (We require that the interiors of the little balls be disjoint, but their |
843 (We require that the interiors of the little balls be disjoint, but their |
843 boundaries are allowed to meet. |
844 boundaries are allowed to meet. |
844 Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely |
845 Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely |
845 the embeddings of a ``little" ball with image all of the big ball $B^n$. |
846 the embeddings of a ``little" ball with image all of the big ball $B^n$. |
846 \nn{should we warn that the inclusion of this copy of $\Diff(B^n)$ is not a homotopy equivalence?}) |
847 \nn{should we warn that the inclusion of this copy of $\Diff(B^n)$ is not a homotopy equivalence?}) |
847 The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad. |
848 The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad: |
848 By shrinking the little balls (precomposing them with dilations), |
849 by shrinking the little balls (precomposing them with dilations), |
849 we see that both operads are homotopic to the space of $k$ framed points |
850 we see that both operads are homotopic to the space of $k$ framed points |
850 in $B^n$. |
851 in $B^n$. |
851 It is easy to see that $n$-fold loop spaces $\Omega^n(T)$ have |
852 It is easy to see that $n$-fold loop spaces $\Omega^n(T)$ have |
852 an action of $\cE\cB_n$. |
853 an action of $\cE\cB_n$. |
853 \nn{add citation for this operad if we can find one} |
854 \nn{add citation for this operad if we can find one} |
911 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
912 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
912 for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the same as the original blob complex for $M$ with coefficients in $\cC$. |
913 for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the same as the original blob complex for $M$ with coefficients in $\cC$. |
913 |
914 |
914 We will first define the ``cell-decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
915 We will first define the ``cell-decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
915 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
916 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
916 and we will define $\cC(W)$ as a suitable colimit |
917 and we will define $\cl{\cC}(W)$ as a suitable colimit |
917 (or homotopy colimit in the $A_\infty$ case) of this functor. |
918 (or homotopy colimit in the $A_\infty$ case) of this functor. |
918 We'll later give a more explicit description of this colimit. |
919 We'll later give a more explicit description of this colimit. |
919 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data), |
920 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data), |
920 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). |
921 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). |
921 |
922 |
922 \begin{defn} |
923 Define a {\it permissible decomposition} of $W$ to be a cell decomposition |
923 Say that a ``permissible decomposition" of $W$ is a cell decomposition |
|
924 \[ |
924 \[ |
925 W = \bigcup_a X_a , |
925 W = \bigcup_a X_a , |
926 \] |
926 \] |
927 where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
927 where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
|
928 \nn{need to define this more carefully} |
928 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
929 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
929 of $y$, or write $x \le y$, if each $k$-ball of $y$ is a union of $k$-balls of $x$. |
930 of $y$, or write $x \le y$, if each $k$-ball of $y$ is a union of $k$-balls of $x$. |
930 |
931 |
931 The category $\cell(W)$ has objects the permissible decompositions of $W$, |
932 \begin{defn} |
|
933 The category (poset) $\cell(W)$ has objects the permissible decompositions of $W$, |
932 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
934 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
933 See Figure \ref{partofJfig} for an example. |
935 See Figure \ref{partofJfig} for an example. |
934 \end{defn} |
936 \end{defn} |
935 |
937 |
936 \begin{figure}[!ht] |
938 \begin{figure}[!ht] |
939 \end{equation*} |
941 \end{equation*} |
940 \caption{A small part of $\cell(W)$} |
942 \caption{A small part of $\cell(W)$} |
941 \label{partofJfig} |
943 \label{partofJfig} |
942 \end{figure} |
944 \end{figure} |
943 |
945 |
944 |
|
945 |
|
946 An $n$-category $\cC$ determines |
946 An $n$-category $\cC$ determines |
947 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
947 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
948 (possibly with additional structure if $k=n$). |
948 (possibly with additional structure if $k=n$). |
949 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
949 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
950 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
950 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
951 are splittable along this decomposition. |
951 are splittable along this decomposition. |
952 %For a $k$-cell $X$ in a cell composition of $W$, we can consider the ``splittable fields" $\cC(X)_{\bdy X}$, the subset of $\cC(X)$ consisting of fields which are splittable with respect to each boundary $k-1$-cell. |
|
953 |
952 |
954 \begin{defn} |
953 \begin{defn} |
955 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
954 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
956 For a decomposition $x = \bigcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
955 For a decomposition $x = \bigcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
957 \begin{equation} |
956 \begin{equation} |
961 where the restrictions to the various pieces of shared boundaries amongst the cells |
960 where the restrictions to the various pieces of shared boundaries amongst the cells |
962 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). |
961 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). |
963 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 $\cC$. |
962 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 $\cC$. |
964 \end{defn} |
963 \end{defn} |
965 |
964 |
966 When the $n$-category $\cC$ is enriched in some symmetric monoidal category $(A,\boxtimes)$, and $W$ is a |
965 If $k=n$ in the above definition and we are enriching in some auxiliary category, |
967 closed $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$ and |
966 we need to say a bit more. |
968 we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$. |
967 We can rewrite Equation \ref{eq:psi-C} as |
969 (Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.) |
968 \begin{equation} \label{eq:psi-CC} |
970 Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we |
969 \psi_{\cC;W}(x) \deq \coprod_\beta \prod_a \cC(X_a; \beta) , |
971 fix a field on $\bd W$ |
970 \end{equation} |
972 (i.e. fix an element of the colimit associated to $\bd W$). |
971 where $\beta$ runs through labelings of the $k{-}1$-skeleton of the decomposition |
|
972 (which are compatible when restricted to the $k{-}2$-skeleton), and $\cC(X_a; \beta)$ |
|
973 means the subset of $\cC(X_a)$ whose restriction to $\bd X_a$ agress with $\beta$. |
|
974 If we are enriching over $\cS$ and $k=n$, then $\cC(X_a; \beta)$ is an object in |
|
975 $\cS$ and the coproduct and product in Equation \ref{eq:psi-CC} should be replaced by the approriate |
|
976 operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect). |
973 |
977 |
974 Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
978 Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
975 |
979 |
976 \begin{defn}[System of fields functor] |
980 \begin{defn}[System of fields functor] |
977 \label{def:colim-fields} |
981 \label{def:colim-fields} |