269 We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
269 We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
270 |
270 |
271 More generally, let $\alpha$ be a subdivision of a ball $X$ into smaller balls. |
271 More generally, let $\alpha$ be a subdivision of a ball $X$ into smaller balls. |
272 Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
272 Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
273 the smaller balls to $X$. |
273 the smaller balls to $X$. |
274 We say that elements of $\cC(X)_\alpha$ are morphisms which are `splittable along $\alpha$'. |
274 We say that elements of $\cC(X)_\alpha$ are morphisms which are ``splittable along $\alpha$". |
275 In situations where the subdivision is notationally anonymous, we will write |
275 In situations where the subdivision is notationally anonymous, we will write |
276 $\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
276 $\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
277 the unnamed subdivision. |
277 the unnamed subdivision. |
278 If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$; |
278 If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$; |
279 this can also be denoted $\cC(X)\spl$ if the context contains an anonymous |
279 this can also be denoted $\cC(X)\spl$ if the context contains an anonymous |
665 We now describe several classes of examples of $n$-categories satisfying our axioms. |
665 We now describe several classes of examples of $n$-categories satisfying our axioms. |
666 |
666 |
667 \begin{example}[Maps to a space] |
667 \begin{example}[Maps to a space] |
668 \rm |
668 \rm |
669 \label{ex:maps-to-a-space}% |
669 \label{ex:maps-to-a-space}% |
670 Fix a `target space' $T$, any topological space. |
670 Fix a ``target space" $T$, any topological space. |
671 We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
671 We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
672 For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
672 For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
673 all continuous maps from $X$ to $T$. |
673 all continuous maps from $X$ to $T$. |
674 For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
674 For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
675 homotopies fixed on $\bd X$. |
675 homotopies fixed on $\bd X$. |
702 modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
702 modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
703 $h: X\times F\times I \to T$, then $a = \alpha(h)b$. |
703 $h: X\times F\times I \to T$, then $a = \alpha(h)b$. |
704 \nn{need to say something about fundamental classes, or choose $\alpha$ carefully} |
704 \nn{need to say something about fundamental classes, or choose $\alpha$ carefully} |
705 \end{example} |
705 \end{example} |
706 |
706 |
707 The next example is only intended to be illustrative, as we don't specify which definition of a `traditional $n$-category' we intend. |
707 The next example is only intended to be illustrative, as we don't specify which definition of a ``traditional $n$-category" we intend. |
708 Further, most of these definitions don't even have an agreed-upon notion of `strong duality', which we assume here. |
708 Further, most of these definitions don't even have an agreed-upon notion of ``strong duality", which we assume here. |
709 \begin{example}[Traditional $n$-categories] |
709 \begin{example}[Traditional $n$-categories] |
710 \rm |
710 \rm |
711 \label{ex:traditional-n-categories} |
711 \label{ex:traditional-n-categories} |
712 Given a `traditional $n$-category with strong duality' $C$ |
712 Given a ``traditional $n$-category with strong duality" $C$ |
713 define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
713 define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
714 to be the set of all $C$-labeled embedded cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
714 to be the set of all $C$-labeled embedded cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
715 For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X; c)$ to be finite linear |
715 For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X; c)$ to be finite linear |
716 combinations of $C$-labeled embedded cell complexes of $X$ |
716 combinations of $C$-labeled embedded cell complexes of $X$ |
717 modulo the kernel of the evaluation map. |
717 modulo the kernel of the evaluation map. |
723 to be the set of all $C$-labeled embedded cell complexes of $X\times F$. |
723 to be the set of all $C$-labeled embedded cell complexes of $X\times F$. |
724 Define $\cC(X; c)$, for $X$ an $n$-ball, |
724 Define $\cC(X; c)$, for $X$ an $n$-ball, |
725 to be the dual Hilbert space $A(X\times F; c)$. |
725 to be the dual Hilbert space $A(X\times F; c)$. |
726 \nn{refer elsewhere for details?} |
726 \nn{refer elsewhere for details?} |
727 |
727 |
728 Recall we described a system of fields and local relations based on a `traditional $n$-category' |
728 Recall we described a system of fields and local relations based on a ``traditional $n$-category" |
729 $C$ in Example \ref{ex:traditional-n-categories(fields)} above. |
729 $C$ in Example \ref{ex:traditional-n-categories(fields)} above. |
730 \nn{KW: We already refer to \S \ref{sec:fields} above} |
730 \nn{KW: We already refer to \S \ref{sec:fields} above} |
731 Constructing a system of fields from $\cC$ recovers that example. |
731 Constructing a system of fields from $\cC$ recovers that example. |
732 \todo{Except that it doesn't: pasting diagrams v.s. string diagrams.} |
732 \todo{Except that it doesn't: pasting diagrams v.s. string diagrams.} |
733 \nn{KW: but the above example is all about string diagrams. the only difference is at the top level, |
733 \nn{KW: but the above example is all about string diagrams. the only difference is at the top level, |
792 \end{example} |
792 \end{example} |
793 |
793 |
794 This example will be essential for Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
794 This example will be essential for Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
795 Notice that with $F$ a point, the above example is a construction turning a topological |
795 Notice that with $F$ a point, the above example is a construction turning a topological |
796 $n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. |
796 $n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. |
797 We think of this as providing a `free resolution' |
797 We think of this as providing a ``free resolution" |
798 \nn{`cofibrant replacement'?} |
798 \nn{``cofibrant replacement"?} |
799 of the topological $n$-category. |
799 of the topological $n$-category. |
800 \todo{Say more here!} |
800 \todo{Say more here!} |
801 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
801 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
802 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
802 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
803 and take $\CD{B}$ to act trivially. |
803 and take $\CD{B}$ to act trivially. |
804 |
804 |
805 Be careful that the `free resolution' of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
805 Be careful that the ``free resolution" of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
806 It's easy to see that with $n=0$, the corresponding system of fields is just |
806 It's easy to see that with $n=0$, the corresponding system of fields is just |
807 linear combinations of connected components of $T$, and the local relations are trivial. |
807 linear combinations of connected components of $T$, and the local relations are trivial. |
808 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
808 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
809 |
809 |
810 \begin{example}[The bordism $n$-category, $A_\infty$ version] |
810 \begin{example}[The bordism $n$-category, $A_\infty$ version] |
893 so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
893 so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
894 In the case of plain $n$-categories, this construction factors into a construction of a |
894 In the case of plain $n$-categories, this construction factors into a construction of a |
895 system of fields and local relations, followed by the usual TQFT definition of a |
895 system of fields and local relations, followed by the usual TQFT definition of a |
896 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
896 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
897 For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
897 For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
898 Recall that we can take a plain $n$-category $\cC$ and pass to the `free resolution', |
898 Recall that we can take a plain $n$-category $\cC$ and pass to the ``free resolution", |
899 an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls (recall Example \ref{ex:blob-complexes-of-balls} above). |
899 an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls (recall Example \ref{ex:blob-complexes-of-balls} above). |
900 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
900 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
901 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 $\cC$. |
901 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 $\cC$. |
902 |
902 |
903 We will first define the `cell-decomposition' poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
903 We will first define the ``cell-decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
904 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
904 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
905 and we will define $\cC(W)$ as a suitable colimit |
905 and we will define $\cC(W)$ as a suitable colimit |
906 (or homotopy colimit in the $A_\infty$ case) of this functor. |
906 (or homotopy colimit in the $A_\infty$ case) of this functor. |
907 We'll later give a more explicit description of this colimit. |
907 We'll later give a more explicit description of this colimit. |
908 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data), |
908 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data), |
909 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). |
909 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). |
910 |
910 |
911 \begin{defn} |
911 \begin{defn} |
912 Say that a `permissible decomposition' of $W$ is a cell decomposition |
912 Say that a ``permissible decomposition" of $W$ is a cell decomposition |
913 \[ |
913 \[ |
914 W = \bigcup_a X_a , |
914 W = \bigcup_a X_a , |
915 \] |
915 \] |
916 where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
916 where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
917 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
917 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
936 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
936 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
937 (possibly with additional structure if $k=n$). |
937 (possibly with additional structure if $k=n$). |
938 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
938 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
939 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
939 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
940 are splittable along this decomposition. |
940 are splittable along this decomposition. |
941 %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. |
941 %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. |
942 |
942 |
943 \begin{defn} |
943 \begin{defn} |
944 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
944 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
945 For a decomposition $x = \bigcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
945 For a decomposition $x = \bigcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
946 \begin{equation} |
946 \begin{equation} |
1738 If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are. |
1738 If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are. |
1739 Given a decomposition of a $0$-marked $k$-ball $X$ into smaller balls $X_i$, we have |
1739 Given a decomposition of a $0$-marked $k$-ball $X$ into smaller balls $X_i$, we have |
1740 morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) |
1740 morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) |
1741 or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side) |
1741 or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side) |
1742 or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball). |
1742 or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball). |
1743 Corresponding to this decomposition we have a composition (or `gluing') map |
1743 Corresponding to this decomposition we have a composition (or ``gluing") map |
1744 from the product (fibered over the boundary data) of these various sets into $\cM_k(X)$. |
1744 from the product (fibered over the boundary data) of these various sets into $\cM_k(X)$. |
1745 |
1745 |
1746 \medskip |
1746 \medskip |
1747 |
1747 |
1748 Part of the structure of an $n$-category 0-sphere module $\cM$ is captured by saying it is |
1748 Part of the structure of an $n$-category 0-sphere module $\cM$ is captured by saying it is |