blob: comparison text/ncat.tex

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 the smaller balls to $X$.
 We  say that elements of $\cC(X)_\alpha$ are morphisms which are `splittable along $\alpha$'.
 In situations where the subdivision is notationally anonymous, we will write
 $\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to)
 the unnamed subdivision.
-If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cC(\bd X)_\beta)$;
+If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$;
 this can also be denoted $\cC(X)\spl$ if the context contains an anonymous
 subdivision of $\bd X$ and no competing subdivision of $X$.
 The above two composition axioms are equivalent to the following one,
 which we state in slightly vague form.
 isotopy invariance with the requirement that families of homeomorphisms act.
 For the moment, assume that our $n$-morphisms are enriched over chain complexes.
 \addtocounter{axiom}{-1}
 \begin{axiom}{\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$}
-For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes
+For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes
 \[
 	C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) .
 \]
 Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$
 which fix $\bd X$.
 \rm
 \label{ex:traditional-n-categories}
 Given a `traditional $n$-category with strong duality' $C$
 define $\cC(X)$, for $X$ a $k$-ball with $k < n$,
 to be the set of all $C$-labeled sub cell complexes of $X$ (c.f. \S \ref{sec:fields}).
-For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear
+For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X)$ to finite linear
 combinations of $C$-labeled sub cell complexes of $X$
 modulo the kernel of the evaluation map.
 Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$,
 with each cell labelled by the $m$-th iterated identity morphism of the corresponding cell for $a$.
 More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$.
 Define $\cC(X; c)$, for $X$ an $n$-ball,
 to be the dual Hilbert space $A(X\times F; c)$.
 \nn{refer elsewhere for details?}
-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.
+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.}
 \end{example}
 Finally, we describe a version of the bordism $n$-category suitable to our definitions.
 \nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example}
 %\subsection{From $n$-categories to systems of fields}
 \subsection{From balls to manifolds}
 \label{ss:ncat_fields} \label{ss:ncat-coend}
 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.
 That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension
-from $k$-balls to arbitrary $k$-manifolds.
+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.
-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}.
+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}.
 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 $\cC$.
 We will first define the `cell-decomposition' poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$.
 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.
 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).

changeset 335	9bf409eb5040
parent 334	7b632b53eb45
child 336	7a5a73ec8961