blob: comparison text/ncat.tex

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 \nn{Almost certainly we need a little more than the above axiom.
 More specifically, in order to bootstrap our way from the top dimension
 properties of identity morphisms to low dimensions, we need regular products,
 pinched products and even half-pinched products.
-I'm not sure what the best way to cleanly axiomatize the properties of these various is.
+I'm not sure what the best way to cleanly axiomatize the properties of these various
+products is.
 For the moment, I'll assume that all flavors of the product are at
 our disposal, and I'll plan on revising the axioms later.}
 \nn{current idea for fixing this: make the above axiom a ``preliminary version"
 (as we have already done with some of the other axioms), then state the official
 We define a map
 \begin{eqnarray*}
 	\psi_{Y,J}: \cC(X) &\to& \cC(X) \\
 	a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) .
 \end{eqnarray*}
-\nn{need to say something somewhere about pinched boundary convention for products}
+(See Figure \ref{glue-collar}.)
+\begin{figure}[!ht]\begin{equation*}
+\mathfig{.9}{tempkw/glue-collar}
+\end{equation*}\caption{Extended homeomorphism.}\label{glue-collar}\end{figure}
 We will call $\psi_{Y,J}$ an extended isotopy.
 \nn{or extended homeomorphism?  see below.}
 \nn{maybe remark that in some examples (e.g.\ ones based on sub cell complexes)
 extended isotopies are also plain isotopies, so
 no extension necessary}
 \medskip
 The alert reader will have already noticed that our definition of (plain) $n$-category
 is extremely similar to our definition of topological fields.
-The only difference is that for the $n$-category definition we restrict our attention to balls
+The main difference is that for the $n$-category definition we restrict our attention to balls
 (and their boundaries), while for fields we consider all manifolds.
-\nn{also: difference at the top dimension; fix this}
+(A minor difference is that in the category definition we directly impose isotopy
+invariance in dimension $n$, while in the fields definition we have non-isotopy-invariant fields
+but then mod out by local relations which imply isotopy invariance.)
 Thus a system of fields determines an $n$-category simply by restricting our attention to
 balls.
-The $n$-category can be thought of as the local part of the fields.
+This $n$-category can be thought of as the local part of the fields.
 Conversely, given an $n$-category we can construct a system of fields via
-\nn{gluing, or a universal construction}
+a colimit construction; see below.
-\nn{see subsection below}
+%\nn{Next, say something about $A_\infty$ $n$-categories and ``homological" systems
-\nn{Next, say something about $A_\infty$ $n$-categories and ``homological" systems
+%of fields.
-of fields.
+%The universal (colimit) construction becomes our generalized definition of blob homology.
-The universal (colimit) construction becomes our generalized definition of blob homology.
+%Need to explain how it relates to the old definition.}
-Need to explain how it relates to the old definition.}
 \medskip
 \nn{these examples need to be fleshed out a bit more}
 \nn{need to say something about fundamental classes, or choose $\alpha$ carefully}
 \item Given a traditional $n$-category $C$ (with strong duality etc.),
 define $\cC(X)$ (with $\dim(X) < n$)
 to be the set of all $C$-labeled sub cell complexes of $X$.
+(See Subsection \ref{sec:fields}.)
 For $X$ an $n$-ball and $c\in \cC(\bd X)$, 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 $a\times D$ to be the product of the cell complex of $a$ with $D$,
 and with the same labeling as $a$.
 \nn{refer elsewhere for details?}
 \item Variation on the above examples:
 We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$,
 for example product boundary conditions or take the union over all boundary conditions.
-\nn{maybe should not emphasize this case, since it's ``better" in some sense
+%\nn{maybe should not emphasize this case, since it's ``better" in some sense
-to think of these guys as affording a representation
+%to think of these guys as affording a representation
-of the $n{+}1$-category associated to $\bd F$.}
+%of the $n{+}1$-category associated to $\bd F$.}
-\item \nn{should add bordism $n$-cat}
+\item Here's our version of the bordism $n$-category.
+For a $k$-ball $X$, $k<n$, define $\cC(X)$ to be the set of all $k$-dimensional
+submanifolds $W$ of $X\times \r^\infty$ such that the projection $W \to X$ is transverse
+to $\bd X$.
+For $k=n$ define $\cC(X)$ to be homeomorphism classes (rel boundary) of such submanifolds;
+we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism
+$W\to W'$ which restricts to the identity on the boundary.
 \end{itemize}
 Examples of $A_\infty$ $n$-categories:
 (If $k = n$ and our $k$-categories are enriched, then
 $\cC(W)$ will have additional structure; see below.)
 $\cC(W)$ will be the colimit of a functor defined on a category $\cJ(W)$,
 which we define next.
-Define a permissible decomposition of $W$ to be a decomposition
+Define a permissible decomposition of $W$ to be a cell decomposition
 \[
 	W = \bigcup_a X_a ,
 \]
-where each $X_a$ is a $k$-ball.
+where each closed top-dimensional cell $X_a$ is an embedded $k$-ball.
 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement
 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$.
 This defines a partial ordering $\cJ(W)$, which we will think of as a category.
 (The objects of $\cJ(W)$ are permissible decompositions of $W$, and there is a unique
 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.
 (If $\dim(W) = n$ then we need to also make use of the monoidal
 product in the enriching category.
 \nn{should probably be more explicit here})
 Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$.
-In the plain (non-$A_\infty$) case, this means that
+When $k<n$ or $k=n$ and we are in the plain (non-$A_\infty$) case, this means that
 for each decomposition $x$ there is a map
 $\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps
 above, and $\cC(W)$ is universal with respect to these properties.
-In the $A_\infty$ case, it means
+When $k=n$ and we are in the $A_\infty$ case, it means
-\nn{.... need to check if there is a def in the literature before writing this down;
+homotopy colimit.
-homotopy colimit I think}
 More concretely, in the plain case enriched over vector spaces, and with $\dim(W) = n$, we can take
 \[
 	\cC(W) = \left( \oplus_x \psi_\cC(x)\right) \big/ K
 \]
 $a\in \psi_\cC(x)$ for some decomposition $x$, $x\le y$, and $g: \psi_\cC(x)
 \to \psi_\cC(y)$ is value of $\psi_\cC$ on the antirefinement $x\to y$.
 In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit
 is as follows.
-\nn{should probably rewrite this to be compatible with some standard reference}
+%\nn{should probably rewrite this to be compatible with some standard reference}
 Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions.
 Such sequences (for all $m$) form a simplicial set.
 Let
 \[
 	V = \bigoplus_{(x_i)} \psi_\cC(x_0) ,
 \medskip
 Examples of modules:
 \begin{itemize}
-\item
+\item \nn{examples from TQFTs}
+\item \nn{for maps to $T$, can restrict to subspaces of $T$;}
 \end{itemize}
 \subsection{Modules as boundary labels}
 \label{moddecss}
 	W = (\bigcup_a X_a) \cup (\bigcup_{i,b} M_{ib}) ,
 \]
 where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and
 each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$,
 with $M_{ib}\cap\bd_i W$ being the marking.
+\nn{need figure}
 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement
 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$.
 This defines a partial ordering $\cJ(W)$, which we will think of as a category.
 (The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique
 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.)

changeset 142	16539d77fb37
parent 141	e1d24be683bb
child 143	c46b2a01e789