--- a/text/ncat.tex Sat Jun 26 17:22:53 2010 -0700
+++ b/text/ncat.tex Sun Jun 27 12:28:06 2010 -0700
@@ -105,7 +105,7 @@
homeomorphisms to the category of sets and bijections.
\end{lem}
-We postpone the proof \todo{} of this result until after we've actually given all the axioms.
+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.
@@ -152,7 +152,7 @@
domain and range, but the converse meets with our approval.
That is, given compatible domain and range, we should be able to combine them into
the full boundary of a morphism.
-The following lemma follows from the colimit construction used to define $\cl{\cC}_{k-1}$
+The following lemma will follow from the colimit construction used to define $\cl{\cC}_{k-1}$
on spheres.
\begin{lem}[Boundary from domain and range]
@@ -163,7 +163,7 @@
two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$.
Then we have an injective map
\[
- \gl_E : \cC(B_1) \times_{\\cl{cC}(E)} \cC(B_2) \into \cl{\cC}(S)
+ \gl_E : \cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2) \into \cl{\cC}(S)
\]
which is natural with respect to the actions of homeomorphisms.
(When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product
@@ -184,10 +184,10 @@
$$
\caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure}
-Note that we insist on injectivity above.
+Note that we insist on injectivity above. \todo{Make sure we prove this, as a consequence of the next axiom, later.}
Let $\cl{\cC}(S)_E$ denote the image of $\gl_E$.
-We will refer to elements of $\\cl{cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$".
+We will refer to elements of $\cl{\cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$".
If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$
as above, then we define $\cC(X)_E = \bd^{-1}(\cl{\cC}(\bd X)_E)$.
@@ -884,7 +884,7 @@
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
+Thus we show that functors $\cC_k$ satisfying the axioms above have a canonical extension
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,
@@ -912,7 +912,6 @@
W = \bigcup_a X_a ,
\]
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 $k$-ball of $y$ is a union of $k$-balls of $x$.
@@ -962,26 +961,26 @@
Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$.
\begin{defn}[System of fields functor]
-If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cC(W)$ is the usual colimit of the functor $\psi_{\cC;W}$.
+If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$.
That is, for each decomposition $x$ there is a map
-$\psi_{\cC;W}(x)\to \cC(W)$, these maps are compatible with the refinement maps
-above, and $\cC(W)$ is universal with respect to these properties.
+$\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps
+above, and $\cl{\cC}(W)$ is universal with respect to these properties.
\end{defn}
\begin{defn}[System of fields functor, $A_\infty$ case]
-When $\cC$ is an $A_\infty$ $n$-category, $\cC(W)$ for $W$ a $k$-manifold with $k < n$
+When $\cC$ is an $A_\infty$ $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$
is defined as above, as the colimit of $\psi_{\cC;W}$.
-When $W$ is an $n$-manifold, the chain complex $\cC(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$.
+When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$.
\end{defn}
-We can specify boundary data $c \in \cC(\bdy W)$, and define functors $\psi_{\cC;W,c}$
+We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$
with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$.
We now give a more concrete description of the colimit in each case.
If $\cC$ is enriched over vector spaces, and $W$ is an $n$-manifold,
-we can take the vector space $\cC(W,c)$ to be the direct sum over all permissible decompositions of $W$
+we can take the vector space $\cl{\cC}(W,c)$ to be the direct sum over all permissible decompositions of $W$
\begin{equation*}
- \cC(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K
+ \cl{\cC}(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K
\end{equation*}
where $K$ is the vector space spanned by elements $a - g(a)$, with
$a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x)
@@ -992,17 +991,17 @@
%\nn{should probably rewrite this to be compatible with some standard reference}
Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$.
Such sequences (for all $m$) form a simplicial set in $\cell(W)$.
-Define $V$ as a vector space via
+Define $\cl{\cC}(W)$ as a vector space via
\[
- V = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] ,
+ \cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] ,
\]
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 $m$.)
-We endow $V$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$
+We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$
summands plus another term using the differential of the simplicial set of $m$-sequences.
More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$
-summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define
+summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define
\[
\bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) ,
\]
@@ -1021,12 +1020,14 @@
Then we kill the extra homology we just introduced with mapping
cylinders between the mapping cylinders (filtration degree 2), and so on.
-$\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds.
+$\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}.
-It is easy to see that
+\todo{This next sentence is circular: these maps are an axiom, not a consequence of anything. -S} It is easy to see that
there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps
comprise a natural transformation of functors.
+\todo{Explicitly say somewhere: `this proves Lemma \ref{lem:domain-and-range}'}
+
\nn{need to finish explaining why we have a system of fields;
need to say more about ``homological" fields?
(actions of homeomorphisms);