--- a/blob to-do Tue Jun 28 18:19:16 2011 -0700
+++ b/blob to-do Wed Jun 29 23:06:18 2011 -0700
@@ -4,13 +4,17 @@
* reconcile splittability with A-inf/families of maps examples
* better discussion of systems of fields from disk-like n-cats
+** splittability axiom for fields
+** topology on fields, topology on morphisms (used in construction of BT)
* need to fix fam-o-homeo argument per discussion with Rob
* need to change module axioms to follow changes in n-cat axioms; search for and destroy all the "Homeo_\bd"'s, add a v-cone axiom
* probably should go through and refer to new splitting axiom when we need to choose refinements etc.
-
+** in the proof that gluing in dimension < n is injective
+** in the proof that D(a) is acyclic
+** in the small blobs lemma
* framings and duality -- work out what's going on! (alternatively, vague-ify current statement)
--- a/text/a_inf_blob.tex Tue Jun 28 18:19:16 2011 -0700
+++ b/text/a_inf_blob.tex Wed Jun 29 23:06:18 2011 -0700
@@ -8,10 +8,10 @@
We will show below
in Corollary \ref{cor:new-old}
-that when $\cC$ is obtained from a system of fields $\cD$
+that when $\cC$ is obtained from a system of fields $\cE$
as the blob complex of an $n$-ball (see Example \ref{ex:blob-complexes-of-balls}),
$\cl{\cC}(M)$ is homotopy equivalent to
-our original definition of the blob complex $\bc_*(M;\cD)$.
+our original definition of the blob complex $\bc_*(M;\cE)$.
%\medskip
@@ -51,7 +51,7 @@
First we define a map
\[
- \psi: \cl{\cC_F}(Y) \to \bc_*(Y\times F;C) .
+ \psi: \cl{\cC_F}(Y) \to \bc_*(Y\times F;\cE) .
\]
On 0-simplices of the hocolimit
we just glue together the various blob diagrams on $X_i\times F$
@@ -60,7 +60,7 @@
For simplices of dimension 1 and higher we define the map to be zero.
It is easy to check that this is a chain map.
-In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;C)$
+In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;\cE)$
and a map
\[
\phi: G_* \to \cl{\cC_F}(Y) .
@@ -69,9 +69,9 @@
Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding
decomposition of $Y\times F$ into the pieces $X_i\times F$.
-Let $G_*\sub \bc_*(Y\times F;C)$ be the subcomplex generated by blob diagrams $a$ such that there
+Let $G_*\sub \bc_*(Y\times F;\cE)$ be the subcomplex generated by blob diagrams $a$ such that there
exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$.
-It follows from Lemma \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$
+It follows from Lemma \ref{thm:small-blobs} that $\bc_*(Y\times F; \cE)$
is homotopic to a subcomplex of $G_*$.
(If the blobs of $a$ are small with respect to a sufficiently fine cover then their
projections to $Y$ are contained in some disjoint union of balls.)
@@ -106,7 +106,7 @@
We want to find 1-simplices which connect $K$ and $K'$.
We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily
the case.
-(Consider the $x$-axis and the graph of $y = x^2\sin(1/x)$ in $\r^2$.)
+(Consider the $x$-axis and the graph of $y = x^2\sin(1/x)$ in $\r^2$.) \scott{Why the $x^2$ here?}
However, we {\it can} find another decomposition $L$ such that $L$ shares common
refinements with both $K$ and $K'$.
Let $KL$ and $K'L$ denote these two refinements.
@@ -412,7 +412,7 @@
\begin{proof}
The proof is again similar to that of Theorem \ref{thm:product}.
-We begin by constructing chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$.
+We begin by constructing a chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$.
Recall that
the 0-simplices of the homotopy colimit $\cB^\cT(M)$
--- a/text/evmap.tex Tue Jun 28 18:19:16 2011 -0700
+++ b/text/evmap.tex Wed Jun 29 23:06:18 2011 -0700
@@ -98,6 +98,7 @@
$\sbc_0(X) = \bc_0(X)$.
(This is true for all of the examples presented in this paper.)
Accordingly, we define $h_0 = 0$.
+\nn{Since we now have an axiom providing this, we should use it. (At present, the axiom is only for morphisms, not fields.)}
Next we define $h_1$.
Let $b\in C_1$ be a 1-blob diagram.
@@ -222,12 +223,12 @@
\item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous.
\item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous,
where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on
-$\bc_0(B)$ comes from the generating set $\BD_0(B)$.
+$\bc_0(B)$ comes from the generating set $\BD_0(B)$. \nn{This topology is implicitly part of the data of a system of fields, but never mentioned. It should be!}
\end{itemize}
We can summarize the above by saying that in the typical continuous family
$P\to \BD_k(X)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ and $r(p)$ are induced by a map
-$P\to \Homeo(X)$, with the twig blob labels $u_i(p)$ varying independently.
+$P\to \Homeo(X)$, with the twig blob labels $u_i(p)$ varying independently. \nn{``varying independently'' means that \emph{after} you pull back via the family of homeomorphisms to the original twig blob, you see a continuous family of labels, right? We should say this. --- Scott}
We note that while we've decided not to allow the blobs $B_i(p)$ to vary independently of the field $r(p)$,
if we did allow this it would not affect the truth of the claims we make below.
In particular, such a definition of $\btc_*(X)$ would result in a homotopy equivalent complex.
@@ -350,7 +351,7 @@
of blob diagrams that are small with respect to $\cU$.
(If $f:P \to \BD_k$ is the family then for all $p\in P$ we have that $f(p)$ is a diagram in which the blobs are small.)
This is done as in the proof of Lemma \ref{small-blobs-b}; the technique of the proof works in families.
-Each such family is homotopic to a sum families which can be a ``lifted" to $\Homeo(X)$.
+Each such family is homotopic to a sum of families which can be a ``lifted" to $\Homeo(X)$.
That is, $f:P \to \BD_k$ has the form $f(p) = g(p)(b)$ for some $g:P\to \Homeo(X)$ and $b\in \BD_k$.
(We are ignoring a complication related to twig blob labels, which might vary
independently of $g$, but this complication does not affect the conclusion we draw here.)
--- a/text/ncat.tex Tue Jun 28 18:19:16 2011 -0700
+++ b/text/ncat.tex Wed Jun 29 23:06:18 2011 -0700
@@ -34,9 +34,8 @@
The axioms for an $n$-category are spread throughout this section.
Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms},
-\ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:vcones} and
-\ref{axiom:extended-isotopies}.
-For an enriched $n$-category we add \ref{axiom:enriched}.
+\ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:extended-isotopies} and \ref{axiom:vcones}.
+For an enriched $n$-category we add Axiom \ref{axiom:enriched}.
For an $A_\infty$ $n$-category, we replace
Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}.
@@ -207,8 +206,8 @@
We do not insist on surjectivity of the gluing map, since this is not satisfied by all of the examples
we are trying to axiomatize.
If our $k$-morphisms $\cC(X)$ are labeled cell complexes embedded in $X$ (c.f. Example \ref{ex:traditional-n-categories} below), then a $k$-morphism is
-in the image of the gluing map precisely which the cell complex is in general position
-with respect to $E$. On the other hand, in categories based on maps to a target space (c.f. Example \ref{ex:maps-to-a-space} below) the gluing map is always surjective
+in the image of the gluing map precisely when the cell complex is in general position
+with respect to $E$. On the other hand, in categories based on maps to a target space (c.f. Example \ref{ex:maps-to-a-space} below) the gluing map is always surjective.
If $S$ is a 0-sphere (the case $k=1$ above), then $S$ can be identified with the {\it disjoint} union
of two 0-balls $B_1$ and $B_2$ and the colimit construction $\cl{\cC}(S)$ can be identified
@@ -579,7 +578,7 @@
Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which
acts trivially on the restriction $\bd b$ of $b$ to $\bd X$.
(Keep in mind the important special case where $f$ restricted to $\bd X$ is the identity.)
-Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which
+Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which act
trivially on $\bd b$.
Then $f(b) = b$.
In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially on
@@ -654,7 +653,7 @@
The revised axiom is
%\addtocounter{axiom}{-1}
-\begin{axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$]
+\begin{axiom}[Extended isotopy invariance in dimension $n$]
\label{axiom:extended-isotopies}
Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which
acts trivially on the restriction $\bd b$ of $b$ to $\bd X$.
@@ -876,7 +875,7 @@
or more generally an appropriate sort of $\infty$-category,
we can modify the extended isotopy axiom \ref{axiom:extended-isotopies}
to require that families of homeomorphisms act
-and obtain an $A_\infty$ $n$-category.
+and obtain what we shall call an $A_\infty$ $n$-category.
\noop{
We believe that abstract definitions should be guided by diverse collections
@@ -892,7 +891,7 @@
and let $\cJ$ be an $\infty$-functor from topological spaces to $\cS$
(e.g.\ the singular chain functor $C_*$).
-\begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.]
+\begin{axiom}[\textup{\textbf{[$A_\infty$ replacement for Axiom \ref{axiom:extended-isotopies}]}} Families of homeomorphisms act in dimension $n$.]
\label{axiom:families}
For each pair of $n$-balls $X$ and $X'$ and each pair $c\in \cl{\cC}(\bd X)$ and $c'\in \cl{\cC}(\bd X')$ we have an $\cS$-morphism
\[
@@ -913,12 +912,12 @@
We now describe the topology on $\Coll(X; c)$.
We retain notation from the above definition of collar map.
Each collaring homeomorphism $X \cup (Y\times J) \to X$ determines a map from points $p$ of $\bd X$ to
-(possibly zero-width) embedded intervals in $X$ terminating at $p$.
+(possibly length zero) embedded intervals in $X$ terminating at $p$.
If $p \in Y$ this interval is the image of $\{p\}\times J$.
-If $p \notin Y$ then $p$ is assigned the zero-width interval $\{p\}$.
+If $p \notin Y$ then $p$ is assigned the length zero interval $\{p\}$.
Such collections of intervals have a natural topology, and $\Coll(X; c)$ inherits its topology from this.
-Note in particular that parts of the collar are allowed to shrink continuously to zero width.
-(This is the real content; if nothing shrinks to zero width then the action of families of collar
+Note in particular that parts of the collar are allowed to shrink continuously to zero length.
+(This is the real content; if nothing shrinks to zero length then the action of families of collar
maps follows from the action of families of homeomorphisms and compatibility with gluing.)
The $k=n$ case of Axiom \ref{axiom:morphisms} posits a {\it strictly} associative action of {\it sets}
@@ -1000,7 +999,7 @@
\item in the $A_\infty$ case, actions of the topological spaces of homeomorphisms preserving boundary conditions
and collar maps (Axiom \ref{axiom:families}).
\end{itemize}
-The above data must satisfy the following conditions:
+The above data must satisfy the following conditions.
\begin{itemize}
\item The gluing maps are compatible with actions of homeomorphisms and boundary
restrictions (Axiom \ref{axiom:composition}).
@@ -1118,9 +1117,9 @@
The case $n=d$ captures the $n$-categorical nature of bordisms.
The case $n > 2d$ captures the full symmetric monoidal $n$-category structure.
\end{example}
-\begin{remark}
+\begin{rem}
Working with the smooth bordism category would require careful attention to either collars, corners or halos.
-\end{remark}
+\end{rem}
%\nn{the next example might be an unnecessary distraction. consider deleting it.}
@@ -1344,7 +1343,7 @@
Let $\du_a X_a = M_0\to\cdots\to M_m = W$ be a ball decomposition compatible with $x$.
Let $\bd M_i = N_i \cup Y_i \cup Y'_i$, where $Y_i$ and $Y'_i$ are glued together to produce $M_{i+1}$.
-We will define $\psi_{\cC;W}(x)$ be be the subset of $\prod_a \cC(X_a)$ which satisfies a series of conditions
+We will define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies a series of conditions
related to the gluings $M_{i-1} \to M_i$, $1\le i \le m$.
By Axiom \ref{nca-boundary}, we have a map
\[
@@ -1361,7 +1360,7 @@
along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\cl\cC(Y_1)$ and $\cl\cC(Y'_1)$ agree
(with respect to the identification of $Y_1$ and $Y'_1$ provided by the gluing map).
The $i$-th condition is defined similarly.
-Note that these conditions depend on on the boundaries of elements of $\prod_a \cC(X_a)$.
+Note that these conditions depend on the boundaries of elements of $\prod_a \cC(X_a)$.
We define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies the
above conditions for all $i$ and also all
@@ -1440,7 +1439,7 @@
of the splitting along $\bd Y$, and this implies that the combined decomposition $\du_{ia} X_{ia}$
is permissible.
We can now define the gluing $y_1\bullet y_2$ in the obvious way, and a further application of Axiom \ref{axiom:vcones}
-shows that this is independebt of the choices of representatives of $y_i$.
+shows that this is independent of the choices of representatives of $y_i$.
\medskip
@@ -1454,7 +1453,7 @@
\]
where $x$ runs through decompositions of $W$, and $\sim$ is the obvious equivalence relation
induced by refinement and gluing.
-If $\cC$ is enriched over vector spaces and $W$ is an $n$-manifold,
+If $\cC$ is enriched over, for example, vector spaces and $W$ is an $n$-manifold,
we can take
\begin{equation*}
\cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K,
@@ -2411,7 +2410,7 @@
This will allow us to define $\cS(X; c)$ independently of the choice of $E$.
First we must define ``inner product", ``non-degenerate" and ``compatible".
-Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image.
+Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ its mirror image.
(We assume we are working in the unoriented category.)
Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$
along their common boundary.
--- a/text/tqftreview.tex Tue Jun 28 18:19:16 2011 -0700
+++ b/text/tqftreview.tex Wed Jun 29 23:06:18 2011 -0700
@@ -368,7 +368,7 @@
Local relations are subspaces $U(B; c)\sub \cC(B; c)$ of the fields on balls which form an ideal under gluing.
Again, we give the examples first.
-\addtocounter{prop}{-2}
+\addtocounter{subsection}{-2}
\begin{example}[contd.]
For maps into spaces, $U(B; c)$ is generated by fields of the form $a-b \in \lf(B; c)$,
where $a$ and $b$ are maps (fields) which are homotopic rel boundary.
@@ -379,6 +379,8 @@
$\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into
domain and range.
\end{example}
+\addtocounter{subsection}{2}
+\addtocounter{prop}{-2}
These motivate the following definition.