258 the smaller balls to $X$. |
258 the smaller balls to $X$. |
259 We say that elements of $\cC(X)_\alpha$ are morphisms which are `splittable along $\alpha$'. |
259 We say that elements of $\cC(X)_\alpha$ are morphisms which are `splittable along $\alpha$'. |
260 In situations where the subdivision is notationally anonymous, we will write |
260 In situations where the subdivision is notationally anonymous, we will write |
261 $\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
261 $\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
262 the unnamed subdivision. |
262 the unnamed subdivision. |
263 If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cC(\bd X)_\beta)$; |
263 If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$; |
264 this can also be denoted $\cC(X)\spl$ if the context contains an anonymous |
264 this can also be denoted $\cC(X)\spl$ if the context contains an anonymous |
265 subdivision of $\bd X$ and no competing subdivision of $X$. |
265 subdivision of $\bd X$ and no competing subdivision of $X$. |
266 |
266 |
267 The above two composition axioms are equivalent to the following one, |
267 The above two composition axioms are equivalent to the following one, |
268 which we state in slightly vague form. |
268 which we state in slightly vague form. |
436 isotopy invariance with the requirement that families of homeomorphisms act. |
436 isotopy invariance with the requirement that families of homeomorphisms act. |
437 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
437 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
438 |
438 |
439 \addtocounter{axiom}{-1} |
439 \addtocounter{axiom}{-1} |
440 \begin{axiom}{\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$} |
440 \begin{axiom}{\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$} |
441 For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes |
441 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
442 \[ |
442 \[ |
443 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
443 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
444 \] |
444 \] |
445 Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$ |
445 Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$ |
446 which fix $\bd X$. |
446 which fix $\bd X$. |
533 \rm |
533 \rm |
534 \label{ex:traditional-n-categories} |
534 \label{ex:traditional-n-categories} |
535 Given a `traditional $n$-category with strong duality' $C$ |
535 Given a `traditional $n$-category with strong duality' $C$ |
536 define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
536 define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
537 to be the set of all $C$-labeled sub cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
537 to be the set of all $C$-labeled sub cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
538 For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear |
538 For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X)$ to finite linear |
539 combinations of $C$-labeled sub cell complexes of $X$ |
539 combinations of $C$-labeled sub cell complexes of $X$ |
540 modulo the kernel of the evaluation map. |
540 modulo the kernel of the evaluation map. |
541 Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, |
541 Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, |
542 with each cell labelled by the $m$-th iterated identity morphism of the corresponding cell for $a$. |
542 with each cell labelled by the $m$-th iterated identity morphism of the corresponding cell for $a$. |
543 More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
543 More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
546 Define $\cC(X; c)$, for $X$ an $n$-ball, |
546 Define $\cC(X; c)$, for $X$ an $n$-ball, |
547 to be the dual Hilbert space $A(X\times F; c)$. |
547 to be the dual Hilbert space $A(X\times F; c)$. |
548 \nn{refer elsewhere for details?} |
548 \nn{refer elsewhere for details?} |
549 |
549 |
550 |
550 |
551 Recall we described a system of fields and local relations based on a `traditional $n$-category' $C$ in Example \ref{ex:traditional-n-categories(fields)} above. Constructing a system of fields from $\cC$ recovers that example. |
551 Recall we described a system of fields and local relations based on a `traditional $n$-category' $C$ in Example \ref{ex:traditional-n-categories(fields)} above. Constructing a system of fields from $\cC$ recovers that example. \todo{Except that it doesn't: pasting diagrams v.s. string diagrams.} |
552 \end{example} |
552 \end{example} |
553 |
553 |
554 Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
554 Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
555 |
555 |
556 \nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example} |
556 \nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example} |
637 %\subsection{From $n$-categories to systems of fields} |
637 %\subsection{From $n$-categories to systems of fields} |
638 \subsection{From balls to manifolds} |
638 \subsection{From balls to manifolds} |
639 \label{ss:ncat_fields} \label{ss:ncat-coend} |
639 \label{ss:ncat_fields} \label{ss:ncat-coend} |
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. |
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. |
641 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 |
642 from $k$-balls to arbitrary $k$-manifolds. |
642 from $k$-balls to arbitrary $k$-manifolds. Recall that we've already anticipated this construction in the previous section, inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
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}. |
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 given as Definition \ref{defn:TQFT-invariant}. |
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 $\cC$. |
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 $\cC$. |
645 |
645 |
646 We will first define the `cell-decomposition' poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
646 We will first define the `cell-decomposition' poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
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. |
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. |
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). |
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). |