21 For examples of a more purely algebraic origin, one would typically need the combinatorial |
21 For examples of a more purely algebraic origin, one would typically need the combinatorial |
22 results that we have avoided here. |
22 results that we have avoided here. |
23 |
23 |
24 \medskip |
24 \medskip |
25 |
25 |
26 There are many existing definitions of $n$-categories, with various intended uses. In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. Generally, these sets are indexed by instances of a certain typical chape. |
26 There are many existing definitions of $n$-categories, with various intended uses. In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. Generally, these sets are indexed by instances of a certain typical shape. |
27 Some $n$-category definitions model $k$-morphisms on the standard bihedrons (interval, bigon, and so on). |
27 Some $n$-category definitions model $k$-morphisms on the standard bihedrons (interval, bigon, and so on). |
28 Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
28 Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
29 a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
29 a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
30 and so on. |
30 and so on. |
31 (This allows for strict associativity.) |
31 (This allows for strict associativity.) |
32 Still other definitions (see, for example, \cite{MR2094071}) |
32 Still other definitions (see, for example, \cite{MR2094071}) |
33 model the $k$-morphisms on more complicated combinatorial polyhedra. |
33 model the $k$-morphisms on more complicated combinatorial polyhedra. |
34 |
34 |
35 For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball: |
35 For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball. Thus we expect to associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |
36 |
36 to the standard $k$-ball. By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
37 \begin{axiom}[Morphisms]{\textup{\textbf{[preliminary]}}} |
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38 For any $k$-manifold $X$ homeomorphic |
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39 to the standard $k$-ball, we have a set of $k$-morphisms |
|
40 $\cC_k(X)$. |
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41 \end{axiom} |
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42 |
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43 By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
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44 standard $k$-ball. |
37 standard $k$-ball. |
45 We {\it do not} assume that it is equipped with a |
38 We {\it do not} assume that it is equipped with a |
46 preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
39 preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
47 |
40 |
48 |
|
49 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
41 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
50 the boundary), we want a corresponding |
42 the boundary), we want a corresponding |
51 bijection of sets $f:\cC(X)\to \cC(Y)$. |
43 bijection of sets $f:\cC(X)\to \cC(Y)$. |
52 (This will imply ``strong duality", among other things.) |
44 (This will imply ``strong duality", among other things.) Putting these together, we have |
53 So we replace the above with |
45 |
54 |
|
55 \addtocounter{axiom}{-1} |
|
56 \begin{axiom}[Morphisms] |
46 \begin{axiom}[Morphisms] |
57 \label{axiom:morphisms} |
47 \label{axiom:morphisms} |
58 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
48 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
59 the category of $k$-balls and |
49 the category of $k$-balls and |
60 homeomorphisms to the category of sets and bijections. |
50 homeomorphisms to the category of sets and bijections. |
105 For each $1 \le k \le n$, we have a functor $\cC_{k-1}$ from |
95 For each $1 \le k \le n$, we have a functor $\cC_{k-1}$ from |
106 the category of $k{-}1$-spheres and |
96 the category of $k{-}1$-spheres and |
107 homeomorphisms to the category of sets and bijections. |
97 homeomorphisms to the category of sets and bijections. |
108 \end{prop} |
98 \end{prop} |
109 |
99 |
110 We postpone the proof of this result until after we've actually given all the axioms. Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, along with the data described in the other Axioms at lower levels. |
100 We postpone the proof \todo{} of this result until after we've actually given all the axioms. Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, along with the data described in the other Axioms at lower levels. |
111 |
101 |
112 %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
102 %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
113 |
103 |
114 \begin{axiom}[Boundaries]\label{nca-boundary} |
104 \begin{axiom}[Boundaries]\label{nca-boundary} |
115 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cC_{k-1}(\bd X)$. |
105 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cC_{k-1}(\bd X)$. |
477 Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex |
467 Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex |
478 type $A_\infty$ $n$-category. |
468 type $A_\infty$ $n$-category. |
479 |
469 |
480 \medskip |
470 \medskip |
481 |
471 |
482 The alert reader will have already noticed that our definition of (plain) $n$-category |
472 The alert reader will have already noticed that our definition of a (plain) $n$-category |
483 is extremely similar to our definition of topological fields. |
473 is extremely similar to our definition of a topological system of fields. |
484 The main difference is that for the $n$-category definition we restrict our attention to balls |
474 There are two essential differences. |
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475 First, for the $n$-category definition we restrict our attention to balls |
485 (and their boundaries), while for fields we consider all manifolds. |
476 (and their boundaries), while for fields we consider all manifolds. |
486 (A minor difference is that in the category definition we directly impose isotopy |
477 Second, in category definition we directly impose isotopy |
487 invariance in dimension $n$, while in the fields definition we have non-isotopy-invariant fields |
478 invariance in dimension $n$, while in the fields definition we have do not expect isotopy invariance on fields |
488 but then mod out by local relations which imply isotopy invariance.) |
479 but instead remember a subspace of local relations which contain differences of isotopic fields. (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
489 Thus a system of fields determines an $n$-category simply by restricting our attention to |
480 Thus a system of fields and local relations $(\cF,\cU)$ determines an $n$-category $\cC_ {\cF,\cU}$ simply by restricting our attention to |
490 balls. |
481 balls and, at level $n$, quotienting out by the local relations: |
|
482 \begin{align*} |
|
483 \cC_{\cF,\cU}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / \cU(B) & \text{when $k=n$.}\end{cases} |
|
484 \end{align*} |
491 This $n$-category can be thought of as the local part of the fields. |
485 This $n$-category can be thought of as the local part of the fields. |
492 Conversely, given an $n$-category we can construct a system of fields via |
486 Conversely, given a topological $n$-category we can construct a system of fields via |
493 a colimit construction; see \S \ref{ss:ncat_fields} below. |
487 a colimit construction; see \S \ref{ss:ncat_fields} below. |
494 |
|
495 %\nn{Next, say something about $A_\infty$ $n$-categories and ``homological" systems |
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496 %of fields. |
|
497 %The universal (colimit) construction becomes our generalized definition of blob homology. |
|
498 %Need to explain how it relates to the old definition.} |
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499 |
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500 \medskip |
|
501 |
488 |
502 \subsection{Examples of $n$-categories} |
489 \subsection{Examples of $n$-categories} |
503 \label{ss:ncat-examples} |
490 \label{ss:ncat-examples} |
504 |
491 |
505 |
492 |
615 When $X$ is an $k$-ball, |
602 When $X$ is an $k$-ball, |
616 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
603 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
617 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
604 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
618 \end{example} |
605 \end{example} |
619 |
606 |
620 This example will be essential for Theorem \ref{product_thm} below, which allows us to compute the blob complex of a product. Notice that with $F$ a point, the above example is a construction turning a topological $n$-category into an $A_\infty$ $n$-category. We think of this as providing a `free resolution' of the topological $n$-category. \todo{Say more here!} In fact, there is also a trivial way to do this: we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, and take $\CD{B}$ to act trivially. |
607 This example will be essential for Theorem \ref{product_thm} below, which allows us to compute the blob complex of a product. Notice that with $F$ a point, the above example is a construction turning a topological $n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. We think of this as providing a `free resolution' of the topological $n$-category. \todo{Say more here!} In fact, there is also a trivial, but mostly uninteresting, way to do this: we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, and take $\CD{B}$ to act trivially. |
621 |
608 |
622 Be careful that the `free resolution' of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. It's easy to see that with $n=0$, the corresponding system of fields is just linear combinations of connected components of $T$, and the local relations are trivial. There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
609 Be careful that the `free resolution' of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. It's easy to see that with $n=0$, the corresponding system of fields is just linear combinations of connected components of $T$, and the local relations are trivial. There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
623 |
610 |
624 \begin{example}[The bordism $n$-category, $A_\infty$ version] |
611 \begin{example}[The bordism $n$-category, $A_\infty$ version] |
625 \rm |
612 \rm |
631 \begin{example}[$E_n$ algebras] |
618 \begin{example}[$E_n$ algebras] |
632 \rm |
619 \rm |
633 \label{ex:e-n-alg} |
620 \label{ex:e-n-alg} |
634 Let $\cE\cB_n$ be the operad of smooth embeddings of $k$ (little) |
621 Let $\cE\cB_n$ be the operad of smooth embeddings of $k$ (little) |
635 copies of the standard $n$-ball $B^n$ into another (big) copy of $B^n$. |
622 copies of the standard $n$-ball $B^n$ into another (big) copy of $B^n$. |
636 $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad. |
623 The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad. |
637 (By shrining the little balls, we see that both are homotopic to the space of $k$ framed points |
624 (By peeling the little balls, we see that both are homotopic to the space of $k$ framed points |
638 in $B^n$.) |
625 in $B^n$.) |
639 |
626 |
640 Let $A$ be an $\cE\cB_n$-algebra. |
627 Let $A$ be an $\cE\cB_n$-algebra. |
641 We will define an $A_\infty$ $n$-category $\cC^A$. |
628 We will define an $A_\infty$ $n$-category $\cC^A$. |
642 \nn{...} |
629 \nn{...} |
648 |
635 |
649 |
636 |
650 %\subsection{From $n$-categories to systems of fields} |
637 %\subsection{From $n$-categories to systems of fields} |
651 \subsection{From balls to manifolds} |
638 \subsection{From balls to manifolds} |
652 \label{ss:ncat_fields} \label{ss:ncat-coend} |
639 \label{ss:ncat_fields} \label{ss:ncat-coend} |
653 In this section we describe how to extend an $n$-category as described above (of either the plain or $A_\infty$ variety) to a system of fields. |
640 In this section we describe how to extend an $n$-category $\cC$ as described above (of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. This extension is a certain colimit, and we've chosen the notation to remind you of this. |
654 That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension |
641 That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension |
655 from $k$-balls to arbitrary $k$-manifolds. |
642 from $k$-balls to arbitrary $k$-manifolds. |
656 In the case of plain $n$-categories, this is just the usual construction of a TQFT |
643 In the case of plain $n$-categories, this construction factors into a construction of a system of fields and local relations, followed by the usual TQFT definition of a vector space invariant of manifolds of Definition \ref{defn:TQFT-invariant}. |
657 from an $n$-category. |
644 For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. Recall that we can take a plain $n$-category $\cC$ and pass to the `free resolution', an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls (recall Example \ref{ex:blob-complexes-of-balls} above). We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant 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 $\cC$. |
658 For $A_\infty$ $n$-categories, this gives an alternate (and |
645 |
659 somewhat more canonical/tautological) construction of the blob complex. |
646 We will first define the `cell-decomposition' poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
660 \nn{though from this point of view it seems more natural to just add some |
|
661 adjective to ``TQFT" rather than coining a completely new term like ``blob complex".} |
|
662 |
|
663 We will first define the `cell-decomposition' poset $\cJ(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
|
664 An $n$-category $\cC$ provides a functor from this poset to the category of sets, and we will define $\cC(W)$ as a suitable colimit (or homotopy colimit in the $A_\infty$ case) of this functor. |
647 An $n$-category $\cC$ provides a functor from this poset to the category of sets, and we will define $\cC(W)$ as a suitable colimit (or homotopy colimit in the $A_\infty$ case) of this functor. |
665 We'll later give a more explicit description of this colimit. In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data), then the resulting system of fields 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). |
648 We'll later give a more explicit description of this colimit. In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data), 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). |
666 |
649 |
667 \begin{defn} |
650 \begin{defn} |
668 Say that a `permissible decomposition' of $W$ is a cell decomposition |
651 Say that a `permissible decomposition' of $W$ is a cell decomposition |
669 \[ |
652 \[ |
670 W = \bigcup_a X_a , |
653 W = \bigcup_a X_a , |
672 where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
655 where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
673 |
656 |
674 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
657 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
675 of $y$, or write $x \le y$, if each $k$-ball of $y$ is a union of $k$-balls of $x$. |
658 of $y$, or write $x \le y$, if each $k$-ball of $y$ is a union of $k$-balls of $x$. |
676 |
659 |
677 The category $\cJ(W)$ has objects the permissible decompositions of $W$, and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
660 The category $\cell(W)$ has objects the permissible decompositions of $W$, and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
678 See Figure \ref{partofJfig} for an example. |
661 See Figure \ref{partofJfig} for an example. |
679 \end{defn} |
662 \end{defn} |
680 |
663 |
681 \begin{figure}[!ht] |
664 \begin{figure}[!ht] |
682 \begin{equation*} |
665 \begin{equation*} |
683 \mathfig{.63}{ncat/zz2} |
666 \mathfig{.63}{ncat/zz2} |
684 \end{equation*} |
667 \end{equation*} |
685 \caption{A small part of $\cJ(W)$} |
668 \caption{A small part of $\cell(W)$} |
686 \label{partofJfig} |
669 \label{partofJfig} |
687 \end{figure} |
670 \end{figure} |
688 |
671 |
689 |
672 |
690 |
673 |
691 An $n$-category $\cC$ determines |
674 An $n$-category $\cC$ determines |
692 a functor $\psi_{\cC;W}$ from $\cJ(W)$ to the category of sets |
675 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
693 (possibly with additional structure if $k=n$). |
676 (possibly with additional structure if $k=n$). |
694 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
677 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
695 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
678 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
696 are splittable along this decomposition. |
679 are splittable along this decomposition. |
697 %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. |
680 %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. |
698 |
681 |
699 \begin{defn} |
682 \begin{defn} |
700 Define the functor $\psi_{\cC;W} : \cJ(W) \to \Set$ as follows. |
683 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
701 For a decomposition $x = \bigcup_a X_a$ in $\cJ(W)$, $\psi_{\cC;W}(x)$ is the subset |
684 For a decomposition $x = \bigcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
702 \begin{equation} |
685 \begin{equation} |
703 \label{eq:psi-C} |
686 \label{eq:psi-C} |
704 \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
687 \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
705 \end{equation} |
688 \end{equation} |
706 where the restrictions to the various pieces of shared boundaries amongst the cells |
689 where the restrictions to the various pieces of shared boundaries amongst the cells |
707 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). |
690 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). |
708 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 $\cC$. |
691 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 $\cC$. |
709 \end{defn} |
692 \end{defn} |
710 |
693 |
711 When the $n$-category $\cC$ is enriched in some monoidal category $(A,\boxtimes)$, and $W$ is a |
694 When the $n$-category $\cC$ is enriched in some symmetric monoidal category $(A,\boxtimes)$, and $W$ is a |
712 closed $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$ and |
695 closed $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$ and |
713 we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$. (Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.) |
696 we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$. (Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.) |
714 Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we |
697 Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we |
715 fix a field on $\bd W$ |
698 fix a field on $\bd W$ |
716 (i.e. fix an element of the colimit associated to $\bd W$). |
699 (i.e. fix an element of the colimit associated to $\bd W$). |
740 |
723 |
741 In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit |
724 In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit |
742 is more involved. |
725 is more involved. |
743 %\nn{should probably rewrite this to be compatible with some standard reference} |
726 %\nn{should probably rewrite this to be compatible with some standard reference} |
744 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
727 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
745 Such sequences (for all $m$) form a simplicial set in $\cJ(W)$. |
728 Such sequences (for all $m$) form a simplicial set in $\cell(W)$. |
746 Define $V$ as a vector space via |
729 Define $V$ as a vector space via |
747 \[ |
730 \[ |
748 V = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
731 V = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
749 \] |
732 \] |
750 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. (Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$, the complex $U[m]$ is concentrated in degree $m$.) |
733 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. (Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$, the complex $U[m]$ is concentrated in degree $m$.) |
1072 \mathfig{.4}{ncat/mblabel} |
1055 \mathfig{.4}{ncat/mblabel} |
1073 \end{equation*}\caption{A permissible decomposition of a manifold |
1056 \end{equation*}\caption{A permissible decomposition of a manifold |
1074 whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure} |
1057 whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure} |
1075 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
1058 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
1076 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
1059 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
1077 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
1060 This defines a partial ordering $\cell(W)$, which we will think of as a category. |
1078 (The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
1061 (The objects of $\cell(D)$ are permissible decompositions of $W$, and there is a unique |
1079 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
1062 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
1080 |
1063 |
1081 The collection of modules $\cN$ determines |
1064 The collection of modules $\cN$ determines |
1082 a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets |
1065 a functor $\psi_\cN$ from $\cell(W)$ to the category of sets |
1083 (possibly with additional structure if $k=n$). |
1066 (possibly with additional structure if $k=n$). |
1084 For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset |
1067 For a decomposition $x = (X_a, M_{ib})$ in $\cell(W)$, define $\psi_\cN(x)$ to be the subset |
1085 \[ |
1068 \[ |
1086 \psi_\cN(x) \sub \left(\prod_a \cC(X_a)\right) \times \left(\prod_{ib} \cN_i(M_{ib})\right) |
1069 \psi_\cN(x) \sub \left(\prod_a \cC(X_a)\right) \times \left(\prod_{ib} \cN_i(M_{ib})\right) |
1087 \] |
1070 \] |
1088 such that the restrictions to the various pieces of shared boundaries amongst the |
1071 such that the restrictions to the various pieces of shared boundaries amongst the |
1089 $X_a$ and $M_{ib}$ all agree. |
1072 $X_a$ and $M_{ib}$ all agree. |