--- a/blob1.tex Mon Jul 05 10:27:45 2010 -0700
+++ b/blob1.tex Mon Jul 05 10:27:51 2010 -0700
@@ -16,7 +16,7 @@
\maketitle
-[revision $\ge$ 393; $\ge$ 24 June 2010]
+[revision $\ge$ 418; $\ge$ 5 July 2010]
{\color[rgb]{.9,.5,.2} \large \textbf{Draft version, read with caution.}}
We're in the midst of revising this, and hope to have a version on the arXiv soon.
@@ -25,24 +25,13 @@
\paragraph{To do list}
\begin{itemize}
-\item[1] (K) tweak intro
-\item[2] (S) needs explanation that this will be superseded by the n-cat
-definitions in \S 7.
-\item[2] (S) incorporate improvements from later
-\item[2.3] (S) foreshadow generalising; quotient to resolution
-\item[3] (K) look over blob homology section again
-\item[4] (S) basic properties, not much to do
-\item[5] (K) finish the lemmas in the Hochschild section
\item[6] (K) proofs need finishing, then (S) needs to confirm details and try
to make more understandable
-\item[7] (S) do some work here -- identity morphisms are still imperfect. Say something about the cobordism and stabilization hypotheses \cite{MR1355899} in this setting? Say something about $E_n$ algebras?
+\item[7] (S) do some work here -- identity morphisms are still imperfect. Say something about the cobordism and stabilization hypotheses \cite{MR1355899} in this setting?
\item[7.6] is new! (S) read
\item[8] improve the beginning, finish proof for products,
check the argument about maps
\item[9] (K) proofs trail off
-\item[10] (S) read what's already here
-\item[A] may need to weaken statement to get boundaries working (K) finish
-\item[B] (S) look at this, decide what to keep
\item Work in the references Chris Douglas gave us on the classification of local field theories, \cite{BDH-seminar,DSP-seminar,schommer-pries-thesis,0905.0465}.
\nn{KW: Do we need to do this? We don't really classify field theories.
@@ -64,9 +53,10 @@
} % end \noop
+
+
\tableofcontents
-
\input{text/intro}
\input{text/tqftreview}
--- a/build.xml Mon Jul 05 10:27:45 2010 -0700
+++ b/build.xml Mon Jul 05 10:27:51 2010 -0700
@@ -63,6 +63,11 @@
</target>
<target name="copy-pdf" depends="pdf">
+ <exec executable="svn" dir="../../Sites/tqft.net/papers/">
+ <arg value="up"/>
+ <arg value="--accept"/>
+ <arg value="theirs-full"/>
+ </exec>
<copy file="blob1.pdf" tofile="../../Sites/tqft.net/papers/blobs.pdf"/>
<exec executable="svn" dir="../../Sites/tqft.net/papers/">
<arg value="commit"/>
--- a/text/a_inf_blob.tex Mon Jul 05 10:27:45 2010 -0700
+++ b/text/a_inf_blob.tex Mon Jul 05 10:27:51 2010 -0700
@@ -2,18 +2,13 @@
\section{The blob complex for $A_\infty$ $n$-categories}
\label{sec:ainfblob}
+Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the anticlimactically tautological definition of the blob
+complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of Section \ref{ss:ncat_fields}.
-Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we define the blob
-complex $\bc_*(M)$ to the be the homotopy colimit $\cC(M)$ of Section \ref{sec:ncats}.
-\nn{say something about this being anticlimatically tautological?}
We will show below
in Corollary \ref{cor:new-old}
-that this agrees (up to homotopy) with our original definition of the blob complex
-in the case of plain $n$-categories.
-When we need to distinguish between the new and old definitions, we will refer to the
-new-fangled and old-fashioned blob complex.
-
-\medskip
+that when $\cC$ is obtained from a topological $n$-category $\cD$ as the blob complex of a point, this agrees (up to homotopy) with our original definition of the blob complex
+for $\cD$.
An important technical tool in the proofs of this section is provided by the idea of `small blobs'.
Fix $\cU$, an open cover of $M$.
@@ -42,27 +37,27 @@
\nn{need to settle on notation; proof and statement are inconsistent}
-\begin{thm} \label{product_thm}
+\begin{thm} \label{thm:product}
Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from
-Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $C^{\times F}$ defined by
+Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $\bc_*(F; C)$ defined by
\begin{equation*}
-C^{\times F}(B) = \cB_*(B \times F, C).
+\bc_*(F; C) = \cB_*(B \times F, C).
\end{equation*}
Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned'
-blob complex for $Y \times F$ with coefficients in $C$ and the `new-fangled'
-(i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $C^{\times F}$:
+blob complex for $Y \times F$ with coefficients in $C$ and the ``new-fangled"
+(i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $\bc_*(F; C)$:
\begin{align*}
-\cB_*(Y \times F, C) & \htpy \cB_*(Y, C^{\times F})
+\cB_*(Y \times F; C) & \htpy \cl{\bc_*(F; C)}(Y)
\end{align*}
\end{thm}
-\begin{proof}%[Proof of Theorem \ref{product_thm}]
+\begin{proof}
We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}.
First we define a map
\[
- \psi: \bc_*^\cF(Y) \to \bc_*^C(Y\times F) .
+ \psi: \cl{\bc_*(F; C)}(Y) \to \bc_*(Y\times F;C) .
\]
In filtration degree 0 we just glue together the various blob diagrams on $X_i\times F$
(where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on
@@ -70,25 +65,25 @@
In filtration degrees 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 $G_*\sub \bc_*^C(Y\times F)$
+In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;C)$
and a map
\[
- \phi: G_* \to \bc_*^\cF(Y) .
+ \phi: G_* \to \cl{\bc_*(F; C)}(Y) .
\]
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 $G_*\sub \bc_*^C(Y\times F)$ be the subcomplex generated by blob diagrams $a$ such that there
+Let $G_*\sub \bc_*(Y\times F;C)$ 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 Proposition \ref{thm:small-blobs} that $\bc_*^C(Y\times F)$ is homotopic to a subcomplex of $G_*$.
+It follows from Proposition \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ 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.)
Note that the image of $\psi$ is equal to $G_*$.
-We will define $\phi: G_* \to \bc_*^\cF(Y)$ using the method of acyclic models.
+We will define $\phi: G_* \to \cl{\bc_*(F; C)}(Y)$ using the method of acyclic models.
Let $a$ be a generator of $G_*$.
-Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(b, \ol{K})$
+Let $D(a)$ denote the subcomplex of $\cl{\bc_*(F; C)}(Y)$ generated by all $(b, \ol{K})$
such that $a$ splits along $K_0\times F$ and $b$ is a generator appearing
in an iterated boundary of $a$ (this includes $a$ itself).
(Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions;
@@ -194,13 +189,13 @@
\end{proof}
We are now in a position to apply the method of acyclic models to get a map
-$\phi:G_* \to \bc_*^\cF(Y)$.
+$\phi:G_* \to \cl{\bc_*(F; C)}(Y)$.
We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is in filtration degree zero
and $r$ has filtration degree greater than zero.
We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity.
-$\psi\circ\phi$ is the identity on the nose:
+First, $\psi\circ\phi$ is the identity on the nose:
\[
\psi(\phi(a)) = \psi((a,K)) + \psi(r) = a + 0.
\]
@@ -208,31 +203,31 @@
$\psi$ glues those pieces back together, yielding $a$.
We have $\psi(r) = 0$ since $\psi$ is zero in positive filtration degrees.
-$\phi\circ\psi$ is the identity up to homotopy by another MoAM argument.
+Second, $\phi\circ\psi$ is the identity up to homotopy by another argument based on the method of acyclic models.
To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above.
Both the identity map and $\phi\circ\psi$ are compatible with this
-collection of acyclic subcomplexes, so by the usual MoAM argument these two maps
+collection of acyclic subcomplexes, so by the usual method of acyclic models argument these two maps
are homotopic.
-This concludes the proof of Theorem \ref{product_thm}.
+This concludes the proof of Theorem \ref{thm:product}.
\end{proof}
\nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category}
\medskip
-\todo{rephrase this}
\begin{cor}
\label{cor:new-old}
-The new-fangled and old-fashioned blob complexes are homotopic.
+The blob complex of a manifold $M$ with coefficients in a topological $n$-category $\cC$ is homotopic to the homotopy colimit invariant of $M$ defined using the $A_\infty$ $n$-category obtained by applying the blob complex to a point:
+$$\bc_*(M; \cC) \htpy \cl{\bc_*(pt; \cC)}(M).$$
\end{cor}
\begin{proof}
-Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point.
+Apply Theorem \ref{thm:product} with the fiber $F$ equal to a point.
\end{proof}
\medskip
-Theorem \ref{product_thm} extends to the case of general fiber bundles
+Theorem \ref{thm:product} extends to the case of general fiber bundles
\[
F \to E \to Y .
\]
@@ -247,13 +242,13 @@
Let $\cF_E$ denote this $k$-category over $Y$.
We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to
get a chain complex $\cF_E(Y)$.
-The proof of Theorem \ref{product_thm} goes through essentially unchanged
+The proof of Theorem \ref{thm:product} goes through essentially unchanged
to show that
\[
\bc_*(E) \simeq \cF_E(Y) .
\]
-
+\nn{remark further that this still works when the map is not even a fibration?}
\nn{put this later}
@@ -266,7 +261,8 @@
}
\nn{There is a version of this last construction for arbitrary maps $E \to Y$
-(not necessarily a fibration).}
+(not necessarily a fibration).
+In fact, there is also a version of the first construction for non-fibrations.}
@@ -298,9 +294,9 @@
\begin{proof}
\nn{for now, just prove $k=0$ case.}
-The proof is similar to that of Theorem \ref{product_thm}.
+The proof is similar to that of Theorem \ref{thm:product}.
We give a short sketch with emphasis on the differences from
-the proof of Theorem \ref{product_thm}.
+the proof of Theorem \ref{thm:product}.
Let $\cT$ denote the chain complex $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
Recall that this is a homotopy colimit based on decompositions of the interval $J$.
@@ -316,17 +312,15 @@
a subcomplex of $G_*$.
Next we define a map $\phi:G_*\to \cT$ using the method of acyclic models.
-As in the proof of Theorem \ref{product_thm}, we assign to a generator $a$ of $G_*$
+As in the proof of Theorem \ref{thm:product}, we assign to a generator $a$ of $G_*$
an acyclic subcomplex which is (roughly) $\psi\inv(a)$.
The proof of acyclicity is easier in this case since any pair of decompositions of $J$ have
a common refinement.
The proof that these two maps are inverse to each other is the same as in
-Theorem \ref{product_thm}.
+Theorem \ref{thm:product}.
\end{proof}
-This establishes Property \ref{property:gluing}.
-
\noop{
Let $\cT$ denote the $n{-}k$-category $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
Let $D$ be an $n{-}k$-ball.
@@ -337,13 +331,14 @@
on $D\times X$, a decomposition of $J$ such that $b$ splits on the corresponding
decomposition of $D\times X$.
The proof that these two maps are inverse to each other is the same as in
-Theorem \ref{product_thm}.
+Theorem \ref{thm:product}.
}
\medskip
\subsection{Reconstructing mapping spaces}
+\label{sec:map-recon}
The next theorem shows how to reconstruct a mapping space from local data.
Let $T$ be a topological space, let $M$ be an $n$-manifold,
@@ -353,7 +348,8 @@
want to know about spaces of maps of $k$-balls into $T$ ($k\le n$).
To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$.
-\begin{thm} \label{thm:map-recon}
+\begin{thm}
+\label{thm:map-recon}
The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$
is quasi-isomorphic to singular chains on maps from $M$ to $T$.
$$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$
@@ -369,7 +365,7 @@
\end{rem}
\begin{proof}
-The proof is again similar to that of Theorem \ref{product_thm}.
+The proof is again similar to that of Theorem \ref{thm:product}.
We begin by constructing chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$.
@@ -411,7 +407,7 @@
Define $D(a)$ to be the subcomplex of $\cB^\cT(M)$ generated by all
pairs $(b, \ol{K})$, where $b$ is a generator appearing in an iterated boundary of $a$
and $\ol{K}$ is an index of the homotopy colimit $\cB^\cT(M)$.
-(See the proof of Theorem \ref{product_thm} for more details.)
+(See the proof of Theorem \ref{thm:product} for more details.)
The same proof as of Lemma \ref{lem:d-a-acyclic} shows that $D(a)$ is acyclic.
By the usual acyclic models nonsense, there is a (unique up to homotopy)
map $\phi:G_*\to \cB^\cT(M)$ such that $\phi(a)\in D(a)$.
@@ -423,7 +419,7 @@
It is now easy to see that $\psi\circ\phi$ is the identity on the nose.
Another acyclic models argument shows that $\phi\circ\psi$ is homotopic to the identity.
-(See the proof of Theorem \ref{product_thm} for more details.)
+(See the proof of Theorem \ref{thm:product} for more details.)
\end{proof}
\noop{
--- a/text/appendixes/comparing_defs.tex Mon Jul 05 10:27:45 2010 -0700
+++ b/text/appendixes/comparing_defs.tex Mon Jul 05 10:27:51 2010 -0700
@@ -293,4 +293,4 @@
as required (c.f. \cite[p. 6]{MR1854636}).
\todo{then the general case.}
We won't describe a reverse construction (producing a topological $A_\infty$ category
-from a `conventional' $A_\infty$ category), but we presume that this will be easy for the experts.
\ No newline at end of file
+from a ``conventional" $A_\infty$ category), but we presume that this will be easy for the experts.
\ No newline at end of file
--- a/text/appendixes/smallblobs.tex Mon Jul 05 10:27:45 2010 -0700
+++ b/text/appendixes/smallblobs.tex Mon Jul 05 10:27:51 2010 -0700
@@ -7,7 +7,7 @@
\begin{lem}
\label{lem:CH-small-blobs}
-Fix an open cover $\cU$, and a sequence $\cV_k$ of open covers which are each strictly subordinate to $\cU$. For a given $k$, consider $\cG_k$ the subspace of $C_k(\Homeo(M)) \tensor \bc_*(M)$ spanned by $f \tensor b$, where $f:P^k \times M \to M$ is a $k$-parameter family of homeomorphisms such that for each $p \in P$, $f(p, -)$ makes $b$ small with respect to $\cV_k$. We can choose an up-to-homotopy representative $\ev$ of the chain map of Property \ref{property:evaluation} which gives the action of families of homeomorphisms, which restricts to give a map
+Fix an open cover $\cU$, and a sequence $\cV_k$ of open covers which are each strictly subordinate to $\cU$. For a given $k$, consider $\cG_k$ the subspace of $C_k(\Homeo(M)) \tensor \bc_*(M)$ spanned by $f \tensor b$, where $f:P^k \times M \to M$ is a $k$-parameter family of homeomorphisms such that for each $p \in P$, $f(p, -)$ makes $b$ small with respect to $\cV_k$. We can choose an up-to-homotopy representative $\ev$ of the chain map of Theorem \ref{thm:evaluation} which gives the action of families of homeomorphisms, which restricts to give a map
$$\ev : \cG_k \subset C_k(\Homeo(M)) \tensor \bc_*(M) \to \bc^{\cU}_*(M)$$
for each $k$.
\end{lem}
@@ -30,9 +30,9 @@
But as noted above, maybe it's best to ignore this.}
Nevertheless, we'll begin introducing nomenclature at this point: for configuration $\beta$ of disjoint embedded balls in $M$ we'll associate a one parameter family of homeomorphisms $\phi_\beta : \Delta^1 \to \Homeo(M)$ (here $\Delta^m$ is the standard simplex $\setc{\mathbf{x} \in \Real^{m+1}}{\sum_{i=0}^m x_i = 1}$). For $0$-blobs, where $\beta = \eset$, all these homeomorphisms are just the identity.
-When $\beta$ is a collection of disjoint embedded balls in $M$, we say that a homeomorphism of $M$ `makes $\beta$ small' if the image of each ball in $\beta$ under the homeomorphism is contained in some open set of $\cU$. Further, we'll say a homeomorphism `makes $\beta$ $\epsilon$-small' if the image of each ball is contained in some open ball of radius $\epsilon$.
+When $\beta$ is a collection of disjoint embedded balls in $M$, we say that a homeomorphism of $M$ ``makes $\beta$ small" if the image of each ball in $\beta$ under the homeomorphism is contained in some open set of $\cU$. Further, we'll say a homeomorphism ``makes $\beta$ $\epsilon$-small" if the image of each ball is contained in some open ball of radius $\epsilon$.
-On a $1$-blob $b$, with ball $\beta$, $s$ is defined as the sum of two terms. Essentially, the first term `makes $\beta$ small', while the other term `gets the boundary right'. First, pick a one-parameter family $\phi_\beta : \Delta^1 \to \Homeo(M)$ of homeomorphisms, so $\phi_\beta(1,0)$ is the identity and $\phi_\beta(0,1)$ makes the ball $\beta$ small --- in fact, not just small with respect to $\cU$, but $\epsilon/2$-small, where $\epsilon > 0$ is such that every $\epsilon$-ball is contained in some open set of $\cU$. Next, pick a two-parameter family $\phi_{\eset \prec \beta} : \Delta^2 \to \Homeo(M)$ so that $\phi_{\eset \prec \beta}(0,x_1,x_2)$ makes the ball $\beta$ $\frac{3\epsilon}{4}$-small for all $x_1+x_2=1$, while $\phi_{\eset \prec \beta}(x_0,0,x_2) = \phi_\beta(x_0,x_2)$ and $\phi_{\eset \prec \beta}(x_0,x_1,0) = \phi_\eset(x_0,x_1)$. (It's perhaps not obvious that this is even possible --- see Lemma \ref{lem:extend-small-homeomorphisms} below.) We now define $s$ by
+On a $1$-blob $b$, with ball $\beta$, $s$ is defined as the sum of two terms. Essentially, the first term ``makes $\beta$ small", while the other term ``gets the boundary right". First, pick a one-parameter family $\phi_\beta : \Delta^1 \to \Homeo(M)$ of homeomorphisms, so $\phi_\beta(1,0)$ is the identity and $\phi_\beta(0,1)$ makes the ball $\beta$ small --- in fact, not just small with respect to $\cU$, but $\epsilon/2$-small, where $\epsilon > 0$ is such that every $\epsilon$-ball is contained in some open set of $\cU$. Next, pick a two-parameter family $\phi_{\eset \prec \beta} : \Delta^2 \to \Homeo(M)$ so that $\phi_{\eset \prec \beta}(0,x_1,x_2)$ makes the ball $\beta$ $\frac{3\epsilon}{4}$-small for all $x_1+x_2=1$, while $\phi_{\eset \prec \beta}(x_0,0,x_2) = \phi_\beta(x_0,x_2)$ and $\phi_{\eset \prec \beta}(x_0,x_1,0) = \phi_\eset(x_0,x_1)$. (It's perhaps not obvious that this is even possible --- see Lemma \ref{lem:extend-small-homeomorphisms} below.) We now define $s$ by
$$s(b) = \restrict{\phi_\beta}{x_0=0}(b) - \restrict{\phi_{\eset \prec \beta}}{x_0=0}(\bdy b).$$
Here, $\restrict{\phi_\beta}{x_0=0} = \phi_\beta(0,1)$ is just a homeomorphism, which we apply to $b$, while $\restrict{\phi_{\eset \prec \beta}}{x_0=0}$ is a one parameter family of homeomorphisms which acts on the $0$-blob $\bdy b$ to give a $1$-blob. To be precise, this action is via the chain map identified in Lemma \ref{lem:CH-small-blobs} with $\cV_0$ the open cover by $\epsilon/2$-balls and $\cV_1$ the open cover by $\frac{3\epsilon}{4}$-balls. From this, it is immediate that $s(b) \in \bc^{\cU}_1(M)$, as desired.
@@ -57,7 +57,7 @@
In order to define $s$ on arbitrary blob diagrams, we first fix a sequence of strictly subordinate covers for $\cU$. First choose an $\epsilon > 0$ so every $\epsilon$ ball is contained in some open set of $\cU$. For $k \geq 1$, let $\cV_{k}$ be the open cover of $M$ by $\epsilon (1-2^{-k})$ balls, and $\cV_0 = \cU$. Certainly $\cV_k$ is strictly subordinate to $\cU$. We now chose the chain map $\ev$ provided by Lemma \ref{lem:CH-small-blobs} for the open covers $\cV_k$ strictly subordinate to $\cU$. Note that $\cV_1$ and $\cV_2$ have already implicitly appeared in the description above.
-Next, we choose a `shrinking system' for $\left(\cU,\{\cV_k\}_{k \geq 1}\right)$, namely for each increasing sequence of blob configurations
+Next, we choose a ``shrinking system" for $\left(\cU,\{\cV_k\}_{k \geq 1}\right)$, namely for each increasing sequence of blob configurations
$\beta_1 \prec \cdots \prec \beta_n$, an $n$ parameter family of diffeomorphisms
$\phi_{\beta_1 \prec \cdots \prec \beta_n} : \Delta^{n+1} \to \Diff{M}$, such that
\begin{itemize}
@@ -119,7 +119,7 @@
It may be useful to look at Figure \ref{fig:erectly-a-tent-badly} to help understand the arrangement. The red, blue and orange $2$-cells there correspond to the $m=0$, $m=1$ and $m=2$ terms respectively, while the $3$-cells (only one of each type is shown) correspond to the terms in the homotopy $h$.
\begin{figure}[!ht]
$$\mathfig{0.5}{smallblobs/tent}$$
-\caption{``Erecting a tent badly.'' We know where we want to send a simplex, and each of the iterated boundary components. However, these do not agree, and we need to stitch the pieces together. Note that these diagrams don't exactly match the situation in the text: a $k$-simplex has $k+1$ boundary components, while a $k$-blob has $k$ boundary terms.}
+\caption{``Erecting a tent badly.'' We know where we want to send a simplex, and each of the iterated boundary components. However, these do not agree, and we need to stitch the pieces together. Note that these diagrams don't exactly match the situation in the text: a $k$-simplex has $k+1$ boundary components, while a $k$-blob has $k$ boundary terms. \nn{turn upside?}}
\label{fig:erectly-a-tent-badly}
\end{figure}
--- a/text/basic_properties.tex Mon Jul 05 10:27:45 2010 -0700
+++ b/text/basic_properties.tex Mon Jul 05 10:27:51 2010 -0700
@@ -87,31 +87,31 @@
$r$ be the restriction of $b$ to $X\setminus S$.
Note that $S$ is a disjoint union of balls.
Assign to $b$ the acyclic (in positive degrees) subcomplex $T(b) \deq r\bullet\bc_*(S)$.
-note that if a diagram $b'$ is part of $\bd b$ then $T(B') \sub T(b)$.
+Note that if a diagram $b'$ is part of $\bd b$ then $T(B') \sub T(b)$.
Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models),
-so $f$ and the identity map are homotopic.
+so $f$ and the identity map are homotopic. \nn{We should actually have a section with a definition of ``compatible" and this statement as a lemma}
\end{proof}
For the next proposition we will temporarily restore $n$-manifold boundary
conditions to the notation.
-Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$.
+Let $X$ be an $n$-manifold, $\bd X = Y \cup Y \cup Z$.
Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$
with boundary $Z\sgl$.
-Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$,
+Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $Y$ and $Z$,
we have the blob complex $\bc_*(X; a, b, c)$.
-If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on
+If $b = a$, then we can glue up blob diagrams on
$X$ to get blob diagrams on $X\sgl$.
This proves Property \ref{property:gluing-map}, which we restate here in more detail.
\textbf{Property \ref{property:gluing-map}.}\emph{
There is a natural chain map
\eq{
- \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl).
+ \gl: \bigoplus_a \bc_*(X; a, a, c) \to \bc_*(X\sgl; c\sgl).
}
The sum is over all fields $a$ on $Y$ compatible at their
($n{-}2$-dimensional) boundaries with $c$.
-`Natural' means natural with respect to the actions of diffeomorphisms.
+``Natural" means natural with respect to the actions of diffeomorphisms.
}
This map is very far from being an isomorphism, even on homology.
--- a/text/blobdef.tex Mon Jul 05 10:27:45 2010 -0700
+++ b/text/blobdef.tex Mon Jul 05 10:27:51 2010 -0700
@@ -58,7 +58,7 @@
Note that the skein space $A(X)$
is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.
-This is Property \ref{property:skein-modules}, and also used in the second
+This is Theorem \ref{thm:skein-modules}, and also used in the second
half of Property \ref{property:contractibility}.
Next, we want the vector space $\bc_2(X)$ to capture `the space of all relations
@@ -87,7 +87,7 @@
A nested 2-blob diagram consists of
\begin{itemize}
-\item A pair of nested balls (blobs) $B_1 \sub B_2 \sub X$.
+\item A pair of nested balls (blobs) $B_1 \subseteq B_2 \subseteq X$.
\item A field $r' \in \cC(B_2 \setminus B_1; c_1, c_2)$
(for some $c_1 \in \cC(\bdy B_1)$ and $c_2 \in \cC(\bdy B_2)$).
\item A field $r \in \cC(X \setminus B_2; c_2)$.
@@ -109,19 +109,35 @@
\begin{eqnarray*}
\bc_2(X) & \deq &
\left(
- \bigoplus_{B_1, B_2 \text{disjoint}} \bigoplus_{c_1, c_2}
+ \bigoplus_{B_1, B_2\; \text{disjoint}} \bigoplus_{c_1, c_2}
U(B_1; c_1) \otimes U(B_2; c_2) \otimes \lf(X\setmin (B_1\cup B_2); c_1, c_2)
- \right) \\
- && \bigoplus \left(
+ \right) \bigoplus \\
+ && \quad\quad \left(
\bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2}
U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1) \tensor \cC(X \setminus B_2; c_2)
\right) .
\end{eqnarray*}
For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign
(rather than a new, linearly independent 2-blob diagram).
+\noop{
\nn{Hmm, I think we should be doing this for nested blobs too --
we shouldn't force the linear indexing of the blobs to have anything to do with
the partial ordering by inclusion -- this is what happens below}
+\nn{KW: I think adding that detail would only add distracting clutter, and the statement is true as written (in the sense that it yields a vector space isomorphic to the general def below}
+}
+
+Before describing the general case we should say more precisely what we mean by
+disjoint and nested blobs.
+Disjoint will mean disjoint interiors.
+Nested blobs are allowed to coincide, or to have overlapping boundaries.
+Blob are allowed to intersect $\bd X$.
+However, we require of any collection of blobs $B_1,\ldots,B_k \subseteq X$ that
+$X$ is decomposable along the union of the boundaries of the blobs.
+\nn{need to say more here. we want to be able to glue blob diagrams, but avoid pathological
+behavior}
+\nn{need to allow the case where $B\to X$ is not an embedding
+on $\bd B$. this is because any blob diagram on $X_{cut}$ should give rise to one on $X_{gl}$
+and blobs are allowed to meet $\bd X$.}
Now for the general case.
A $k$-blob diagram consists of
@@ -132,9 +148,6 @@
(The case $B_i = B_j$ is allowed.
If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.)
If a blob has no other blobs strictly contained in it, we call it a twig blob.
-\nn{need to allow the case where $B\to X$ is not an embedding
-on $\bd B$. this is because any blob diagram on $X_{cut}$ should give rise to one on $X_{gl}$
-and blobs are allowed to meet $\bd X$.}
\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$.
(These are implied by the data in the next bullets, so we usually
suppress them from the notation.)
--- a/text/comm_alg.tex Mon Jul 05 10:27:45 2010 -0700
+++ b/text/comm_alg.tex Mon Jul 05 10:27:51 2010 -0700
@@ -105,7 +105,7 @@
\medskip
-In view of \ref{hochthm}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$,
+In view of Theorem \ref{thm:hochschild}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$,
and that the cyclic homology of $k[t]$ is related to the action of rotations
on $C_*(\Sigma^\infty(S^1), k)$.
\nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section}
--- a/text/deligne.tex Mon Jul 05 10:27:45 2010 -0700
+++ b/text/deligne.tex Mon Jul 05 10:27:51 2010 -0700
@@ -8,7 +8,7 @@
the proof of a higher dimensional version of the Deligne conjecture
about the action of the little disks operad on Hochschild cohomology.
The first several paragraphs lead up to a precise statement of the result
-(Proposition \ref{prop:deligne} below).
+(Theorem \ref{thm:deligne} below).
Then we sketch the proof.
\nn{Does this generalization encompass Kontsevich's proposed generalization from \cite[\S2.5]{MR1718044},
@@ -206,8 +206,8 @@
The main result of this section is that this chain map extends to the full singular
chain complex $C_*(FG^n_{\ol{M}\ol{N}})$.
-\begin{prop}
-\label{prop:deligne}
+\begin{thm}
+\label{thm:deligne}
There is a collection of chain maps
\[
C_*(FG^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes
@@ -216,7 +216,7 @@
which satisfy the operad compatibility conditions.
On $C_0(FG^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above.
When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of Section \ref{sec:evaluation}.
-\end{prop}
+\end{thm}
If, in analogy to Hochschild cochains, we define elements of $\hom(M, N)$
to be ``blob cochains", we can summarize the above proposition by saying that the $n$-FG operad acts on
--- a/text/evmap.tex Mon Jul 05 10:27:45 2010 -0700
+++ b/text/evmap.tex Mon Jul 05 10:27:51 2010 -0700
@@ -21,7 +21,7 @@
such that
\begin{enumerate}
\item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of
-$\Homeo(X, Y)$ on $\bc_*(X)$ (Proposition (\ref{diff0prop})), and
+$\Homeo(X, Y)$ on $\bc_*(X)$ (Property (\ref{property:functoriality})), and
\item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$,
the following diagram commutes up to homotopy
\eq{ \xymatrix{
@@ -46,7 +46,7 @@
and let $S \sub X$.
We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all
$x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if
-there is a family of homeomorphisms $f' : P \times S \to S$ and a `background'
+there is a family of homeomorphisms $f' : P \times S \to S$ and a ``background"
homeomorphism $f_0 : X \to X$ so that
\begin{align*}
f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\
@@ -60,7 +60,6 @@
Let $\cU = \{U_\alpha\}$ be an open cover of $X$.
A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is
{\it adapted to $\cU$}
-\nn{or `weakly adapted'; need to decide on terminology}
if the support of $f$ is contained in the union
of at most $k$ of the $U_\alpha$'s.
@@ -228,8 +227,8 @@
We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$.
%We also have that $\deg(b'') = 0 = \deg(p'')$.
Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$.
-This is possible by \ref{bcontract}, \ref{disjunion} and the fact that isotopic fields
-differ by a local relation \nn{give reference?}.
+This is possible by Properties \ref{property:disjoint-union} and \ref{property:contractibility} and the fact that isotopic fields
+differ by a local relation.
Finally, define
\[
e(p\ot b) \deq x' \bullet p''(b'') .
@@ -618,9 +617,6 @@
\end{proof}
-
-\nn{this should perhaps be a numbered remark, so we can cite it more easily}
-
\begin{rem*}
\label{rem:for-small-blobs}
For the proof of Lemma \ref{lem:CH-small-blobs} below we will need the following observation on the action constructed above.
@@ -633,7 +629,6 @@
\end{rem*}
-
\begin{prop}
The $CH_*(X, Y)$ actions defined above are associative.
That is, the following diagram commutes up to homotopy:
--- a/text/hochschild.tex Mon Jul 05 10:27:45 2010 -0700
+++ b/text/hochschild.tex Mon Jul 05 10:27:51 2010 -0700
@@ -66,7 +66,8 @@
so it suffices to show that they are quasi-isomorphic.
We claim that
-\begin{thm} \label{hochthm}
+\begin{thm}
+\label{thm:hochschild}
The blob complex $\bc_*(S^1; C)$ on the circle is homotopy equivalent to the
usual Hochschild complex for $C$.
\end{thm}
@@ -106,7 +107,7 @@
quasi-isomorphic to its $0$-th homology (which in turn, by \ref{item:hochschild-coinvariants}
above, is just $C$) via the quotient map $\HC_0 \onto \HH_0$.
\end{enumerate}
-(Together, these just say that Hochschild homology is `the derived functor of coinvariants'.)
+(Together, these just say that Hochschild homology is ``the derived functor of coinvariants".)
We'll first recall why these properties are characteristic.
Take some $C$-$C$ bimodule $M$, and choose a free resolution
@@ -129,8 +130,8 @@
\cP_*(F_j) & \xrightarrow{\cP_0(F_j) \onto H_0(\cP_*(F_j))} \coinv(F_j).
\end{align*}
The cone of each chain map is acyclic.
-In the first case, this is because the `rows' indexed by $i$ are acyclic since $\HC_i$ is exact.
-In the second case, this is because the `columns' indexed by $j$ are acyclic, since $F_j$ is free.
+In the first case, this is because the ``rows" indexed by $i$ are acyclic since $\cP_i$ is exact.
+In the second case, this is because the ``columns" indexed by $j$ are acyclic, since $F_j$ is free.
Because the cones are acyclic, the chain maps are quasi-isomorphisms.
Composing one with the inverse of the other, we obtain the desired quasi-isomorphism
$$\cP_*(M) \quismto \coinv(F_*).$$
@@ -203,7 +204,8 @@
We claim that $J_*$ is homotopy equivalent to $\bc_*(S^1)$.
Let $F_*^\ep \sub \bc_*(S^1)$ be the subcomplex where either
(a) the point * is not on the boundary of any blob or
-(b) there are no labeled points or blob boundaries within distance $\ep$ of *.
+(b) there are no labeled points or blob boundaries within distance $\ep$ of *,
+other than blob boundaries at * itself.
Note that all blob diagrams are in $F_*^\ep$ for $\ep$ sufficiently small.
Let $b$ be a blob diagram in $F_*^\ep$.
Define $f(b)$ to be the result of moving any blob boundary points which lie on *
@@ -235,7 +237,9 @@
If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding
$N_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction
of $x$ to $N_\ep$.
-If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$,
+If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$,
+\nn{SM: I don't think we need to consider sums here}
+\nn{KW: It depends on whether we allow linear combinations of fields outside of twig blobs}
write $y_i$ for the restriction of $z_i$ to $N_\ep$, and let
$x_i$ be equal to $x$ on $S^1 \setmin B$, equal to $z_i$ on $B \setmin N_\ep$,
and have an additional blob $N_\ep$ with label $y_i - s(y_i)$.
--- a/text/intro.tex Mon Jul 05 10:27:45 2010 -0700
+++ b/text/intro.tex Mon Jul 05 10:27:51 2010 -0700
@@ -3,13 +3,13 @@
\section{Introduction}
We construct the ``blob complex'' $\bc_*(M; \cC)$ associated to an $n$-manifold $M$ and a ``linear $n$-category with strong duality'' $\cC$.
-This blob complex provides a simultaneous generalisation of several well-understood constructions:
+This blob complex provides a simultaneous generalization of several well-understood constructions:
\begin{itemize}
\item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$.
-(See Property \ref{property:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.)
+(See Theorem \ref{thm:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.)
\item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra),
the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$.
-(See Property \ref{property:hochschild} and \S \ref{sec:hochschild}.)
+(See Theorem \ref{thm:hochschild} and \S \ref{sec:hochschild}.)
\item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have
that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains
on the configuration space of unlabeled points in $M$.
@@ -23,22 +23,22 @@
The blob complex has good formal properties, summarized in \S \ref{sec:properties}.
These include an action of $\CH{M}$,
-extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Property \ref{property:evaluation}) and a gluing
-formula allowing calculations by cutting manifolds into smaller parts (see Property \ref{property:gluing}).
+extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Theorem \ref{thm:evaluation}) and a gluing
+formula allowing calculations by cutting manifolds into smaller parts (see Theorem \ref{thm:gluing}).
We expect applications of the blob complex to contact topology and Khovanov homology but do not address these in this paper.
See \S \ref{sec:future} for slightly more detail.
-\subsubsection{Structure of the paper}
-The three subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}),
-summarise the formal properties of the blob complex (see \S \ref{sec:properties})
-and outline anticipated future directions and applications (see \S \ref{sec:future}).
+\subsection{Structure of the paper}
+The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}),
+summarise the formal properties of the blob complex (see \S \ref{sec:properties}), describe known specializations (see \S \ref{sec:specializations}), outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications})
+and outline anticipated future directions (see \S \ref{sec:future}).
The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex,
and establishes some of its properties.
There are many alternative definitions of $n$-categories, and part of our difficulty defining the blob complex is
simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs.
-At first we entirely avoid this problem by introducing the notion of a `system of fields', and define the blob complex
+At first we entirely avoid this problem by introducing the notion of a ``system of fields", and define the blob complex
associated to an $n$-manifold and an $n$-dimensional system of fields.
We sketch the construction of a system of fields from a $1$-category or from a pivotal $2$-category.
@@ -50,7 +50,7 @@
We call these ``topological $n$-categories'', to differentiate them from previous versions.
Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms.
-The basic idea is that each potential definition of an $n$-category makes a choice about the `shape' of morphisms.
+The basic idea is that each potential definition of an $n$-category makes a choice about the ``shape" of morphisms.
We try to be as lax as possible: a topological $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball.
These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid.
For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to each such $B$ and ask that the action of
@@ -61,15 +61,13 @@
In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category
(using a colimit along cellulations of a manifold), and in \S \ref{sec:ainfblob} give an alternative definition
of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit).
-Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an
+Using these definitions, we show how to use the blob complex to ``resolve" any topological $n$-category as an
$A_\infty$ $n$-category, and relate the first and second definitions of the blob complex.
We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants),
-in particular the `gluing formula' of Property \ref{property:gluing} below.
+in particular the ``gluing formula" of Theorem \ref{thm:gluing} below.
The relationship between all these ideas is sketched in Figure \ref{fig:outline}.
-\nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.}
-
\tikzstyle{box} = [rectangle, rounded corners, draw,outer sep = 5pt, inner sep = 5pt, line width=0.5pt]
\begin{figure}[!ht]
@@ -101,7 +99,7 @@
Example \ref{ex:traditional-n-categories(fields)} \\ and \S \ref{ss:ncat_fields}
%$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle \cU(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$
} (FU);
-\draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Property \ref{property:skein-modules}} (A);
+\draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Theorem \ref{thm:skein-modules}} (A);
\draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs);
\draw (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs);
@@ -117,7 +115,7 @@
thought of as a topological $n$-category, in terms of the topology of $M$.
Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves)
a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex.
-The appendixes prove technical results about $\CH{M}$ and the `small blob complex',
+The appendixes prove technical results about $\CH{M}$ and the ``small blob complex",
and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$,
as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras.
@@ -217,7 +215,7 @@
complexes and isomorphisms between them.
\end{property}
As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*^\cC(X)$;
-this action is extended to all of $C_*(\Homeo(X))$ in Property \ref{property:evaluation} below.
+this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:evaluation} below.
The blob complex is also functorial (indeed, exact) with respect to $\cC$,
although we will not address this in detail here.
@@ -256,8 +254,17 @@
\end{equation}
\end{property}
-\begin{property}[Skein modules]
-\label{property:skein-modules}%
+Properties \ref{property:functoriality} will be immediate from the definition given in
+\S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there.
+Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.
+
+\subsection{Specializations}
+\label{sec:specializations}
+
+The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology.
+
+\begin{thm}[Skein modules]
+\label{thm:skein-modules}%
The $0$-th blob homology of $X$ is the usual
(dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$
by $\cC$.
@@ -265,23 +272,30 @@
\begin{equation*}
H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X)
\end{equation*}
-\end{property}
+\end{thm}
-\todo{Somehow, the Hochschild homology thing isn't a "property".
-Let's move it and call it a theorem? -S}
-\begin{property}[Hochschild homology when $X=S^1$]
-\label{property:hochschild}%
+\newtheorem*{thm:hochschild}{Theorem \ref{thm:hochschild}}
+
+\begin{thm:hochschild}[Hochschild homology when $X=S^1$]
The blob complex for a $1$-category $\cC$ on the circle is
quasi-isomorphic to the Hochschild complex.
\begin{equation*}
\xymatrix{\bc_*^{\cC}(S^1) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).}
\end{equation*}
-\end{property}
+\end{thm:hochschild}
+
+Theorem \ref{thm:skein-modules} is immediate from the definition, and
+Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}.
+We also note Appendix \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of certain commutative algebras thought of as an $n$-category.
+
+
+\subsection{Structure of the blob complex}
+\label{sec:structure}
In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$.
-\begin{property}[$C_*(\Homeo(-))$ action]\mbox{}\\
+\begin{thm}[$C_*(\Homeo(-))$ action]\mbox{}\\
\vspace{-0.5cm}
-\label{property:evaluation}%
+\label{thm:evaluation}%
\begin{enumerate}
\item There is a chain map
\begin{equation*}
@@ -311,7 +325,7 @@
}
\end{equation*}
\end{enumerate}
-\end{property}
+\end{thm}
Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps
$$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$
@@ -322,8 +336,8 @@
Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields.
Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category.
-\begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category]
-\label{property:blobs-ainfty}
+\begin{thm}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category]
+\label{thm:blobs-ainfty}
Let $\cC$ be a topological $n$-category.
Let $Y$ be an $n{-}k$-manifold.
There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$,
@@ -331,8 +345,8 @@
$$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$
(When $m=k$ the subsets with fixed boundary conditions form a chain complex.)
These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in
-Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Property \ref{property:evaluation}.
-\end{property}
+Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}.
+\end{thm}
\begin{rem}
Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from a topological $n$-category.
We think of this $A_\infty$ $n$-category as a free resolution.
@@ -342,24 +356,26 @@
instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}.
The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$.
-\begin{property}[Product formula]
-\label{property:product}
+\newtheorem*{thm:product}{Theorem \ref{thm:product}}
+
+\begin{thm:product}[Product formula]
Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold.
Let $\cC$ be an $n$-category.
-Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}).
+Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Theorem \ref{thm:blobs-ainfty}).
Then
\[
\bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W).
\]
-\end{property}
+\end{thm:product}
We also give a generalization of this statement for arbitrary fibre bundles, in \S \ref{moddecss}, and a sketch of a statement for arbitrary maps.
Fix a topological $n$-category $\cC$, which we'll omit from the notation.
Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category.
(See Appendix \ref{sec:comparing-A-infty} for the translation between topological $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.)
-\begin{property}[Gluing formula]
-\label{property:gluing}%
+\newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}}
+
+\begin{thm:gluing}[Gluing formula]
\mbox{}% <-- gets the indenting right
\begin{itemize}
\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an
@@ -371,32 +387,37 @@
\bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow
\end{equation*}
\end{itemize}
-\end{property}
+\end{thm:gluing}
+
+Theorem \ref{thm:evaluation} is proved in
+in \S \ref{sec:evaluation}, Theorem \ref{thm:blobs-ainfty} appears as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats},
+and Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, with Theorem \ref{thm:gluing} then a relatively straightforward consequence of the proof, explained in \S \ref{sec:gluing}.
-Finally, we prove two theorems which we consider as applications.
+\subsection{Applications}
+\label{sec:applications}
+Finally, we give two theorems which we consider as applications.
-\begin{thm}[Mapping spaces]
+\newtheorem*{thm:map-recon}{Theorem \ref{thm:map-recon}}
+
+\begin{thm:map-recon}[Mapping spaces]
Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps
$B^n \to T$.
(The case $n=1$ is the usual $A_\infty$-category of paths in $T$.)
Then
$$\bc_*(X, \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$
-\end{thm}
+\end{thm:map-recon}
-This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data.
+This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data. The proof appears in \S \ref{sec:map-recon}.
-\begin{thm}[Higher dimensional Deligne conjecture]
-\label{thm:deligne}
+\newtheorem*{thm:deligne}{Theorem \ref{thm:deligne}}
+
+\begin{thm:deligne}[Higher dimensional Deligne conjecture]
The singular chains of the $n$-dimensional fat graph operad act on blob cochains.
-\end{thm}
+\end{thm:deligne}
See \S \ref{sec:deligne} for a full explanation of the statement, and an outline of the proof.
-Properties \ref{property:functoriality} and \ref{property:skein-modules} will be immediate from the definition given in
-\S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there.
-Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.
-Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation}
-in \S \ref{sec:evaluation}, Property \ref{property:blobs-ainfty} as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats},
-and Properties \ref{property:product} and \ref{property:gluing} in \S \ref{sec:ainfblob} as consequences of Theorem \ref{product_thm}.
+
+
\subsection{Future directions}
\label{sec:future}
@@ -415,7 +436,7 @@
The paper ``Skein homology'' \cite{MR1624157} has similar motivations, and it may be
interesting to investigate if there is a connection with the material here.
-Many results in Hochschild homology can be understood `topologically' via the blob complex.
+Many results in Hochschild homology can be understood ``topologically" via the blob complex.
For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$
(see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$,
but haven't investigated the details.
@@ -424,7 +445,19 @@
\subsection{Thanks and acknowledgements}
-We'd like to thank David Ben-Zvi, Kevin Costello, Chris Douglas,
-Michael Freedman, Vaughan Jones, Justin Roberts, Chris Schommer-Pries, Peter Teichner \nn{and who else?} for many interesting and useful conversations.
+% attempting to make this chronological rather than alphabetical
+We'd like to thank
+Justin Roberts,
+Michael Freedman,
+Peter Teichner,
+David Ben-Zvi,
+Vaughan Jones,
+Chris Schommer-Pries,
+Thomas Tradler,
+Kevin Costello,
+Chris Douglas,
+and
+Alexander Kirillov
+for many interesting and useful conversations.
During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley.
--- a/text/kw_macros.tex Mon Jul 05 10:27:45 2010 -0700
+++ b/text/kw_macros.tex Mon Jul 05 10:27:51 2010 -0700
@@ -33,7 +33,7 @@
\def\spl{_\pitchfork}
%\def\nn#1{{{\it \small [#1]}}}
-\def\nn#1{{{\color[rgb]{.2,.5,.6} \small [#1]}}}
+\def\nn#1{{{\color[rgb]{.2,.5,.6} \small [[#1]]}}}
\long\def\noop#1{}
% equations
--- a/text/ncat.tex Mon Jul 05 10:27:45 2010 -0700
+++ b/text/ncat.tex Mon Jul 05 10:27:51 2010 -0700
@@ -11,13 +11,15 @@
Before proceeding, we need more appropriate definitions of $n$-categories,
$A_\infty$ $n$-categories, modules for these, and tensor products of these modules.
-(As is the case throughout this paper, by ``$n$-category" we implicitly intend some notion of
-a `weak' $n$-category with `strong duality'.)
+(As is the case throughout this paper, by ``$n$-category" we mean some notion of
+a ``weak" $n$-category with ``strong duality".)
The definitions presented below tie the categories more closely to the topology
and avoid combinatorial questions about, for example, the minimal sufficient
collections of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets.
-For examples of topological origin, it is typically easy to show that they
+For examples of topological origin
+(e.g.\ categories whose morphisms are maps into spaces or decorated balls),
+it is easy to show that they
satisfy our axioms.
For examples of a more purely algebraic origin, one would typically need the combinatorial
results that we have avoided here.
@@ -36,7 +38,7 @@
model the $k$-morphisms on more complicated combinatorial polyhedra.
For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball.
-Thus we expect to associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic
+Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic
to the standard $k$-ball.
By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the
standard $k$-ball.
@@ -58,7 +60,7 @@
(Note: We usually omit the subscript $k$.)
-We are so far being deliberately vague about what flavor of $k$-balls
+We are being deliberately vague about what flavor of $k$-balls
we are considering.
They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$.
They could be topological or PL or smooth.
@@ -66,13 +68,13 @@
(If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need
to be fussier about corners and boundaries.)
For each flavor of manifold there is a corresponding flavor of $n$-category.
-We will concentrate on the case of PL unoriented manifolds.
+For simplicity, we will concentrate on the case of PL unoriented manifolds.
(The ambitious reader may want to keep in mind two other classes of balls.
The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}).
This will be used below to describe the blob complex of a fiber bundle with
base space $Y$.
-The second is balls equipped with a section of the the tangent bundle, or the frame
+The second is balls equipped with a section of the tangent bundle, or the frame
bundle (i.e.\ framed balls), or more generally some flag bundle associated to the tangent bundle.
These can be used to define categories with less than the ``strong" duality we assume here,
though we will not develop that idea fully in this paper.)
@@ -94,7 +96,7 @@
In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for
$1\le k \le n$.
At first it might seem that we need another axiom for this, but in fact once we have
-all the axioms in the subsection for $0$ through $k-1$ we can use a colimit
+all the axioms in this subsection for $0$ through $k-1$ we can use a colimit
construction, as described in Subsection \ref{ss:ncat-coend} below, to extend $\cC_{k-1}$
to spheres (and any other manifolds):
@@ -105,9 +107,9 @@
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.
+along with the data described in the other axioms at lower levels.
%In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point.
@@ -131,28 +133,28 @@
$\cC(Y; c)$ is just a plain set if $\dim(Y) < n$.
\medskip
-\nn{
-%At the moment I'm a little confused about orientations, and more specifically
-%about the role of orientation-reversing maps of boundaries when gluing oriented manifolds.
-Maybe need a discussion about what the boundary of a manifold with a
-structure (e.g. orientation) means.
-Tentatively, I think we need to redefine the oriented boundary of an oriented $n$-manifold.
-Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal
-first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold
-equipped with an orientation of its once-stabilized tangent bundle.
-Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of
-their $k$ times stabilized tangent bundles.
-(cf. \cite{MR2079378}.)
-Probably should also have a framing of the stabilized dimensions in order to indicate which
-side the bounded manifold is on.
-For the moment just stick with unoriented manifolds.}
+
+(In order to simplify the exposition we have concentrated on the case of
+unoriented PL manifolds and avoided the question of what exactly we mean by
+the boundary a manifold with extra structure, such as an oriented manifold.
+In general, all manifolds of dimension less than $n$ should be equipped with the germ
+of a thickening to dimension $n$, and this germ should carry whatever structure we have
+on $n$-manifolds.
+In addition, lower dimensional manifolds should be equipped with a framing
+of their normal bundle in the thickening; the framing keeps track of which
+side (iterated) bounded manifolds lie on.
+For example, the boundary of an oriented $n$-ball
+should be an $n{-}1$-sphere equipped with an orientation of its once stabilized tangent
+bundle and a choice of direction in this bundle indicating
+which side the $n$-ball lies on.)
+
\medskip
We have just argued that the boundary of a morphism has no preferred splitting into
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 +165,7 @@
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
@@ -174,8 +176,8 @@
$$
\begin{tikzpicture}[%every label/.style={green}
]
-\node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {};
-\node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {};
+\node[fill=black, circle, label=below:$E$, inner sep=1.5pt](S) at (0,0) {};
+\node[fill=black, circle, label=above:$E$, inner sep=1.5pt](N) at (0,2) {};
\draw (S) arc (-90:90:1);
\draw (N) arc (90:270:1);
\node[left] at (-1,1) {$B_1$};
@@ -184,10 +186,11 @@
$$
\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.
+The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}.
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 $\cC(X)_E = \bd^{-1}(\cl{\cC}(\bd X)_E)$.
@@ -261,13 +264,14 @@
%restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$.
We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$.
-We will call elements of $\cC(B)_Y$ morphisms which are `splittable along $Y$' or `transverse to $Y$'.
+We will call elements of $\cC(B)_Y$ morphisms which are
+``splittable along $Y$'' or ``transverse to $Y$''.
We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$.
More generally, let $\alpha$ be a subdivision of a ball $X$ into smaller balls.
Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from
the smaller balls to $X$.
-We say that elements of $\cC(X)_\alpha$ are morphisms which are `splittable along $\alpha$'.
+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.
@@ -298,7 +302,7 @@
These maps must satisfy the following conditions.
\begin{enumerate}
\item
-If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram
+If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are homeomorphisms such that the diagram
\[ \xymatrix{
X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\
X \ar[r]^{f} & X'
@@ -363,7 +367,7 @@
$$
\caption{Examples of pinched products}\label{pinched_prods}
\end{figure}
-(The need for a strengthened version will become apparent in appendix \ref{sec:comparing-defs}
+(The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs}
where we construct a traditional category from a topological category.)
Define a {\it pinched product} to be a map
\[
@@ -525,7 +529,7 @@
Let $J$ be a 1-ball (interval).
We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$.
(Here we use the ``pinched" version of $Y\times J$.
-\nn{need notation for this})
+\nn{do we need notation for this?})
We define a map
\begin{eqnarray*}
\psi_{Y,J}: \cC(X) &\to& \cC(X) \\
@@ -577,42 +581,40 @@
\end{equation*}
\caption{Extended homeomorphism.}\label{glue-collar}\end{figure}
-We say that $\psi_{Y,J}$ is {\it extended isotopic} to the identity map.
-\nn{bad terminology; fix it later}
-\nn{also need to make clear that plain old isotopic to the identity implies
-extended isotopic}
-\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}
+We call a map of this form a {\it collar map}.
It can be thought of as the action of the inverse of
-a map which projects a collar neighborhood of $Y$ onto $Y$.
+a map which projects a collar neighborhood of $Y$ onto $Y$,
+or as the limit of homeomorphisms $X\to X$ which expand a very thin collar of $Y$
+to a larger collar.
+We call the equivalence relation generated by collar maps and homeomorphisms
+isotopic (rel boundary) to the identity {\it extended isotopy}.
The revised axiom is
\addtocounter{axiom}{-1}
-\begin{axiom}{\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$}
+\begin{axiom}{\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$.}
\label{axiom:extended-isotopies}
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
-to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity.
+to the identity on $\bd X$ and isotopic (rel boundary) to the identity.
Then $f$ acts trivially on $\cC(X)$.
+In addition, collar maps act trivially on $\cC(X)$.
\end{axiom}
-\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.}
-
\smallskip
For $A_\infty$ $n$-categories, we replace
isotopy invariance with the requirement that families of homeomorphisms act.
For the moment, assume that our $n$-morphisms are enriched over chain complexes.
+Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and
+$C_*(\Homeo_\bd(X))$ denote the singular chains on this space.
+
\addtocounter{axiom}{-1}
-\begin{axiom}{\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$}
+\begin{axiom}{\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.}
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 $\bd X$.
These action maps are required to be associative up to homotopy
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that
a diagram like the one in Proposition \ref{CHprop} commutes.
@@ -620,16 +622,16 @@
\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?}
\end{axiom}
-We should strengthen the above axiom to apply to families of extended homeomorphisms.
-To do this we need to explain how extended homeomorphisms form a topological space.
-Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology,
+We should strengthen the above axiom to apply to families of collar maps.
+To do this we need to explain how collar maps form a topological space.
+Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology,
and we can replace the class of all intervals $J$ with intervals contained in $\r$.
-\nn{need to also say something about collaring homeomorphisms.}
-\nn{this paragraph needs work.}
+Having chains on the space of collar maps act gives rise to coherence maps involving
+weak identities.
+We will not pursue this in this draft of the paper.
Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category
into a plain $n$-category (enriched over graded groups).
-\nn{say more here?}
In a different direction, if we enrich over topological spaces instead of chain complexes,
we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting
instead of $C_*(\Homeo_\bd(X))$.
@@ -639,13 +641,13 @@
\medskip
The alert reader will have already noticed that our definition of a (plain) $n$-category
-is extremely similar to our definition of a topological system of fields.
-There are two essential differences.
+is extremely similar to our definition of a system of fields.
+There are two differences.
First, for the $n$-category definition we restrict our attention to balls
(and their boundaries), while for fields we consider all manifolds.
Second, in category definition we directly impose isotopy
-invariance in dimension $n$, while in the fields definition we have do not expect isotopy invariance on fields
-but instead remember a subspace of local relations which contain differences of isotopic fields.
+invariance in dimension $n$, while in the fields definition we
+instead remember a subspace of local relations which contain differences of isotopic fields.
(Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.)
Thus a system of fields and local relations $(\cF,\cU)$ determines an $n$-category $\cC_ {\cF,\cU}$ simply by restricting our attention to
balls and, at level $n$, quotienting out by the local relations:
@@ -661,11 +663,13 @@
We now describe several classes of examples of $n$-categories satisfying our axioms.
+We typically specify only the morphisms; the rest of the data for the category
+(restriction maps, gluing, product morphisms, action of homeomorphisms) is usually obvious.
\begin{example}[Maps to a space]
\rm
\label{ex:maps-to-a-space}%
-Fix a `target space' $T$, any topological space.
+Let $T$be a topological space.
We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows.
For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of
all continuous maps from $X$ to $T$.
@@ -674,10 +678,14 @@
(Note that homotopy invariance implies isotopy invariance.)
For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to
be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection.
+\end{example}
+\noop{
Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above.
Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example.
-\end{example}
+\nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from
+an n-cat}
+}
\begin{example}[Maps to a space, with a fiber]
\rm
@@ -699,15 +707,16 @@
the $R$-module of finite linear combinations of continuous maps from $X\times F$ to $T$,
modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy
$h: X\times F\times I \to T$, then $a = \alpha(h)b$.
-\nn{need to say something about fundamental classes, or choose $\alpha$ carefully}
+(In order for this to be well-defined we must choose $\alpha$ to be zero on degenerate simplices.
+Alternatively, we could equip the balls with fundamental classes.)
\end{example}
-The next example is only intended to be illustrative, as we don't specify which definition of a `traditional $n$-category' we intend.
-Further, most of these definitions don't even have an agreed-upon notion of `strong duality', which we assume here.
+The next example is only intended to be illustrative, as we don't specify which definition of a ``traditional $n$-category" we intend.
+Further, most of these definitions don't even have an agreed-upon notion of ``strong duality", which we assume here.
\begin{example}[Traditional $n$-categories]
\rm
\label{ex:traditional-n-categories}
-Given a `traditional $n$-category with strong duality' $C$
+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 embedded cell complexes of $X$ (c.f. \S \ref{sec:fields}).
For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X; c)$ to be finite linear
@@ -721,9 +730,13 @@
to be the set of all $C$-labeled embedded cell complexes of $X\times F$.
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?}
+(See Subsection \ref{sec:constructing-a-tqft}.)
+\end{example}
-Recall we described a system of fields and local relations based on a `traditional $n$-category'
+\noop{
+\nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from
+an n-cat}
+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.
\nn{KW: We already refer to \S \ref{sec:fields} above}
Constructing a system of fields from $\cC$ recovers that example.
@@ -732,11 +745,8 @@
where the quotient is built in.
but (string diagrams)/(relations) is isomorphic to
(pasting diagrams composed of smaller string diagrams)/(relations)}
-\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}
\newcommand{\Bord}{\operatorname{Bord}}
\begin{example}[The bordism $n$-category, plain version]
@@ -764,15 +774,19 @@
%We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex.
-\begin{example}[Chains of maps to a space]
+\begin{example}[Chains (or space) of maps to a space]
\rm
\label{ex:chains-of-maps-to-a-space}
We can modify Example \ref{ex:maps-to-a-space} above to define the fundamental $A_\infty$ $n$-category $\pi^\infty_{\le n}(T)$ of a topological space $T$.
For a $k$-ball $X$, with $k < n$, the set $\pi^\infty_{\leq n}(T)(X)$ is just $\Maps(X \to T)$.
Define $\pi^\infty_{\leq n}(T)(X; c)$ for an $n$-ball $X$ and $c \in \pi^\infty_{\leq n}(T)(\bdy X)$ to be the chain complex
-$$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary,
+\[
+ C_*(\Maps_c(X\times F \to T)),
+\]
+where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary,
and $C_*$ denotes singular chains.
-\nn{maybe should also mention version where we enrich over spaces rather than chain complexes}
+Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X\times F \to T)$,
+we get an $A_\infty$ $n$-category enriched over spaces.
\end{example}
See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to
@@ -781,7 +795,7 @@
\begin{example}[Blob complexes of balls (with a fiber)]
\rm
\label{ex:blob-complexes-of-balls}
-Fix an $n-k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$.
+Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$.
We will define an $A_\infty$ $k$-category $\cC$.
When $X$ is a $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$.
When $X$ is an $k$-ball,
@@ -789,18 +803,17 @@
where $\bc^\cE_*$ denotes the blob complex based on $\cE$.
\end{example}
-This example will be essential for Theorem \ref{product_thm} below, which allows us to compute the blob complex of a product.
+This example will be essential for Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product.
Notice that with $F$ a point, the above example is a construction turning a topological
$n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$.
-We think of this as providing a `free resolution'
-\nn{`cofibrant replacement'?}
+We think of this as providing a ``free resolution"
of the topological $n$-category.
-\todo{Say more here!}
+\nn{say something about cofibrant replacements?}
In fact, there is also a trivial, but mostly uninteresting, way to do this:
we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$,
and take $\CD{B}$ to act trivially.
-Be careful that the `free resolution' of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$.
+Be careful that the ``free resolution" of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$.
It's easy to see that with $n=0$, the corresponding system of fields is just
linear combinations of connected components of $T$, and the local relations are trivial.
There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$.
@@ -832,10 +845,10 @@
the embeddings of a ``little" ball with image all of the big ball $B^n$.
\nn{should we warn that the inclusion of this copy of $\Diff(B^n)$ is not a homotopy equivalence?})
The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad.
-(By shrinking the little balls (precomposing them with dilations),
+By shrinking the little balls (precomposing them with dilations),
we see that both operads are homotopic to the space of $k$ framed points
-in $B^n$.)
-It is easy to see that $n$-fold loop spaces $\Omega^n(T)$ have the structure have
+in $B^n$.
+It is easy to see that $n$-fold loop spaces $\Omega^n(T)$ have
an action of $\cE\cB_n$.
\nn{add citation for this operad if we can find one}
@@ -884,7 +897,7 @@
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,
@@ -893,12 +906,12 @@
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',
+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 $1 \leq k \leq n$.
+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.
@@ -907,12 +920,11 @@
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).
\begin{defn}
-Say that a `permissible decomposition' of $W$ is a cell decomposition
+Say that a ``permissible decomposition" of $W$ is a cell decomposition
\[
W = \bigcup_a X_a ,
\]
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 $x$.
@@ -937,7 +949,7 @@
Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls,
and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries
are splittable along this decomposition.
-%For a $k$-cell $X$ in a cell composition of $W$, we can consider the `splittable fields' $\cC(X)_{\bdy X}$, the subset of $\cC(X)$ consisting of fields which are splittable with respect to each boundary $k-1$-cell.
+%For a $k$-cell $X$ in a cell composition of $W$, we can consider the ``splittable fields" $\cC(X)_{\bdy X}$, the subset of $\cC(X)$ consisting of fields which are splittable with respect to each boundary $k-1$-cell.
\begin{defn}
Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows.
@@ -962,26 +974,27 @@
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 $\psi_{\cC;W}$.
+\label{def:colim-fields}
+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 +1005,17 @@
%\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 +1034,21 @@
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.
+\begin{lem}
+\label{lem:colim-injective}
+Let $W$ be a manifold of dimension less than $n$. Then for each
+decomposition $x$ of $W$ the natural map $\psi_{\cC;W}(x)\to \cl{\cC}(W)$ is injective.
+\end{lem}
+\begin{proof}
+\nn{...}
+\end{proof}
+
\nn{need to finish explaining why we have a system of fields;
need to say more about ``homological" fields?
(actions of homeomorphisms);
@@ -1398,14 +1420,6 @@
We will define a more general self tensor product (categorified coend) below.
-%\nn{what about self tensor products /coends ?}
-
-\nn{maybe ``tensor product" is not the best name?}
-
-%\nn{start with (less general) tensor products; maybe change this later}
-
-
-
\subsection{Morphisms of $A_\infty$ $1$-category modules}
\label{ss:module-morphisms}
@@ -1607,8 +1621,7 @@
$g((\bd \olD)\ot -)$ (where the $g((\bd_0 \olD)\ot -)$ part of $g((\bd \olD)\ot -)$
should be interpreted as above).
-Define a {\it naive morphism}
-\nn{should consider other names for this}
+Define a {\it strong morphism}
of modules to be a collection of {\it chain} maps
\[
h_K : \cX(K)\to \cY(K)
@@ -1622,7 +1635,7 @@
\ar[d]^{\gl} \\
\cX(K) \ar[r]^{h_{K}} & \cY(K)
} \]
-Given such an $h$ we can construct a non-naive morphism $g$, with $\bd g = 0$, as follows.
+Given such an $h$ we can construct a morphism $g$, with $\bd g = 0$, as follows.
Define $g(\olD\ot - ) = 0$ if the length/degree of $\olD$ is greater than 0.
If $\olD$ consists of the single subdivision $K = I_0\cup\cdots\cup I_q$ then define
\[
@@ -1738,7 +1751,7 @@
morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side)
or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side)
or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball).
-Corresponding to this decomposition we have a composition (or `gluing') map
+Corresponding to this decomposition we have a composition (or ``gluing") map
from the product (fibered over the boundary data) of these various sets into $\cM_k(X)$.
\medskip
@@ -1832,8 +1845,9 @@
For the time being, let's say they are.}
A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$,
where $B^j$ is the standard $j$-ball.
-A 1-marked $k$-balls can be decomposed in various ways into smaller balls, which are either
-smaller 1-marked $k$-balls or the product of an unmarked ball with a marked interval. \todo{I'm confused by this last sentence. By `the product of an unmarked ball with a marked internal', you mean a 0-marked $k$-ball, right? If so, we should say it that way. Further, there are also just some entirely unmarked balls. -S}
+A 1-marked $k$-ball can be decomposed in various ways into smaller balls, which are either
+(a) smaller 1-marked $k$-balls, (b) 0-marked $k$-balls, or (c) plain $k$-balls.
+(See Figure xxxx.)
We now proceed as in the above module definitions.
\begin{figure}[!ht]
@@ -2080,6 +2094,7 @@
\end{figure}
Invariance under this movie move follows from the compatibility of the inner
product for $B_1\cup B_2$ with the inner products for $B_1$ and $B_2$.
+\nn{should also say something about locality/distant-commutativity}
If $n\ge 2$, these two movie move suffice:
--- a/text/tqftreview.tex Mon Jul 05 10:27:45 2010 -0700
+++ b/text/tqftreview.tex Mon Jul 05 10:27:51 2010 -0700
@@ -4,16 +4,31 @@
\label{sec:fields}
\label{sec:tqftsviafields}
-In this section we review the notion of a ``system of fields and local relations".
+In this section we review the construction of TQFTs from fields and local relations.
For more details see \cite{kw:tqft}.
-From a system of fields and local relations we can readily construct TQFT invariants of manifolds.
-This is described in \S \ref{sec:constructing-a-tqft}.
+For our purposes, a TQFT is {\it defined} to be something which arises
+from this construction.
+This is an alternative to the more common definition of a TQFT
+as a functor on cobordism categories satisfying various conditions.
+A fully local (``down to points") version of the cobordism-functor TQFT definition
+should be equivalent to the fields-and-local-relations definition.
+
A system of fields is very closely related to an $n$-category.
-In Example \ref{ex:traditional-n-categories(fields)}, which runs throughout this section,
-we sketch the construction of a system of fields from an $n$-category.
-We make this more precise for $n=1$ or $2$ in \S \ref{sec:example:traditional-n-categories(fields)},
-and much later, after we've have given our own definition of a `topological $n$-category' in \S \ref{sec:ncats},
-we explain precisely how to go back and forth between a topological $n$-category and a system of fields and local relations.
+In one direction, Example \ref{ex:traditional-n-categories(fields)}
+shows how to construct a system of fields from a (traditional) $n$-category.
+We do this in detail for $n=1,2$ (Subsection \ref{sec:example:traditional-n-categories(fields)})
+and more informally for general $n$.
+In the other direction,
+our preferred definition of an $n$-category in Section \ref{sec:ncats} is essentially
+just a system of fields restricted to balls of dimensions 0 through $n$;
+one could call this the ``local" part of a system of fields.
+
+Since this section is intended primarily to motivate
+the blob complex construction of Section \ref{sec:blob-definition},
+we suppress some technical details.
+In Section \ref{sec:ncats} the analogous details are treated more carefully.
+
+\medskip
We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0
submanifold of $X$, then $X \setmin Y$ implicitly means the closure
@@ -26,11 +41,12 @@
unoriented PL manifolds of dimension
$k$ and morphisms homeomorphisms.
(We could equally well work with a different category of manifolds ---
-oriented, topological, smooth, spin, etc. --- but for definiteness we
+oriented, topological, smooth, spin, etc. --- but for simplicity we
will stick with unoriented PL.)
Fix a symmetric monoidal category $\cS$.
-While reading the definition, you should just think about the cases $\cS = \Set$ or $\cS = \Vect$.
+Fields on $n$-manifolds will be enriched over $\cS$.
+Good examples to keep in mind are $\cS = \Set$ or $\cS = \Vect$.
The presentation here requires that the objects of $\cS$ have an underlying set, but this could probably be avoided if desired.
A $n$-dimensional {\it system of fields} in $\cS$
@@ -64,10 +80,12 @@
For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of
$\cC(X)$ which restricts to $c$.
In this context, we will call $c$ a boundary condition.
-\item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$.
-(This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$),
+\item The subset $\cC_n(X;c)$ of top-dimensional fields
+with a given boundary condition is an object in our symmetric monoidal category $\cS$.
+(This condition is of course trivial when $\cS = \Set$.)
+If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$),
then this extra structure is considered part of the definition of $\cC_n$.
-Any maps mentioned below between top level fields must be morphisms in $\cS$.
+Any maps mentioned below between fields on $n$-manifolds must be morphisms in $\cS$.
\item $\cC_k$ is compatible with the symmetric monoidal
structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$,
compatibly with homeomorphisms and restriction to boundary.
@@ -86,22 +104,24 @@
\]
and this gluing map is compatible with all of the above structure (actions
of homeomorphisms, boundary restrictions, disjoint union).
-Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity,
+Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity
+and collaring maps,
the gluing map is surjective.
We say that fields on $X\sgl$ in the image of the gluing map
are transverse to $Y$ or splittable along $Y$.
\item Gluing with corners.
Let $\bd X = Y \cup Y \cup W$, where the two copies of $Y$ and
$W$ might intersect along their boundaries.
-Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$.
+Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$
+(Figure xxxx).
Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself
(without corners) along two copies of $\bd Y$.
Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a splittable field on $W\sgl$ and let
$c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$.
Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$.
(This restriction map uses the gluing without corners map above.)
-Using the boundary restriction, gluing without corners, and (in one case) orientation reversal
-maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two
+Using the boundary restriction and gluing without corners maps,
+we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two
copies of $Y$ in $\bd X$.
Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps.
Then (here's the axiom/definition part) there is an injective ``gluing" map
@@ -109,12 +129,14 @@
\Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) ,
\]
and this gluing map is compatible with all of the above structure (actions
-of homeomorphisms, boundary restrictions, orientation reversal, disjoint union).
-Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity,
+of homeomorphisms, boundary restrictions, disjoint union).
+Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity
+and collaring maps,
the gluing map is surjective.
We say that fields in the image of the gluing map
are transverse to $Y$ or splittable along $Y$.
-\item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted
+\item Product fields.
+There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted
$c \mapsto c\times I$.
These maps comprise a natural transformation of functors, and commute appropriately
with all the structure maps above (disjoint union, boundary restriction, etc.).
@@ -136,9 +158,9 @@
\medskip
-Using the functoriality and $\cdot\times I$ properties above, together
-with boundary collar homeomorphisms of manifolds, we can define the notion of
-{\it extended isotopy}.
+Using the functoriality and product field properties above, together
+with boundary collar homeomorphisms of manifolds, we can define
+{\it collar maps} $\cC(M)\to \cC(M)$.
Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold
of $\bd M$.
Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is splittable along $\bd Y$.
@@ -146,10 +168,16 @@
Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$.
Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$.
Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism.
-Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$.
-More generally, we define extended isotopy to be the equivalence relation on fields
-on $M$ generated by isotopy plus all instance of the above construction
-(for all appropriate $Y$ and $x$).
+Then we call the map $x \mapsto f(x \bullet (c\times I))$ a {\it collar map}.
+We call the equivalence relation generated by collar maps and
+homeomorphisms isotopic to the identity {\it extended isotopy}, since the collar maps
+can be thought of (informally) as the limit of homeomorphisms
+which expand an infinitesimally thin collar neighborhood of $Y$ to a thicker
+collar neighborhood.
+
+
+% all this linearizing stuff is unnecessary, I think
+\noop{
\nn{the following discussion of linearizing fields is kind of lame.
maybe just assume things are already linearized.}
@@ -195,6 +223,8 @@
We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the
above tensor products.
+} % end \noop
+
\subsection{Systems of fields from $n$-categories}
\label{sec:example:traditional-n-categories(fields)}
@@ -209,6 +239,15 @@
with codimension $i$ cells labeled by $i$-morphisms of $C$.
We'll spell this out for $n=1,2$ and then describe the general case.
+This way of decorating an $n$-manifold with an $n$-category is sometimes referred to
+as a ``string diagram".
+It can be thought of as (geometrically) dual to a pasting diagram.
+One of the advantages of string diagrams over pasting diagrams is that one has more
+flexibility in slicing them up in various ways.
+In addition, string diagrams are traditional in quantum topology.
+The diagrams predate by many years the terms ``string diagram" and ``quantum topology".
+\nn{?? cite penrose, kauffman, jones(?)}
+
If $X$ has boundary, we require that the cell decompositions are in general
position with respect to the boundary --- the boundary intersects each cell
transversely, so cells meeting the boundary are mere half-cells.
@@ -227,11 +266,17 @@
by an object (0-morphism) of $C$;
\item a transverse orientation of each 0-cell, thought of as a choice of
``domain" and ``range" for the two adjacent 1-cells; and
- \item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with
+ \item a labeling of each 0-cell by a 1-morphism of $C$, with
domain and range determined by the transverse orientation and the labelings of the 1-cells.
\end{itemize}
-If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels
+We want fields on 1-manifolds to be enriched over Vect, so we also allow formal linear combinations
+of the above fields on a 1-manifold $X$ so long as these fields restrict to the same field on $\bd X$.
+
+In addition, we mod out by the relation which replaces
+a 1-morphism label $a$ of a 0-cell $p$ with $a^*$ and reverse the transverse orientation of $p$.
+
+If $C$ is a *-algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels
of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the
interior of $S$, each transversely oriented and each labeled by an element (1-morphism)
of the algebra.
@@ -258,12 +303,23 @@
and the labelings of the 2-cells;
\item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood
of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped
-to $\pm 1 \in S^1$; and
+to $\pm 1 \in S^1$
+(this amounts to splitting of the link of the 0-cell into domain and range); and
\item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range
determined by the labelings of the 1-cells and the parameterizations of the previous
bullet.
\end{itemize}
-\nn{need to say this better; don't try to fit everything into the bulleted list}
+
+As in the $n=1$ case, we allow formal linear combinations of fields on 2-manifolds,
+so long as their restrictions to the boundary coincide.
+
+In addition, we regard the labelings as being equivariant with respect to the * structure
+on 1-morphisms and pivotal structure on 2-morphisms.
+That is, we mod out my the relation which flips the transverse orientation of a 1-cell
+and replaces its label $a$ by $a^*$, as well as the relation which changes the parameterization of the link
+of a 0-cell and replaces its label by the appropriate pivotal conjugate.
+
+\medskip
For general $n$, a field on a $k$-manifold $X^k$ consists of
\begin{itemize}
@@ -274,18 +330,14 @@
domain and range determined by the labelings of the link of $j$-cell.
\end{itemize}
-%\nn{next definition might need some work; I think linearity relations should
-%be treated differently (segregated) from other local relations, but I'm not sure
-%the next definition is the best way to do it}
-
-\medskip
-
-
\subsection{Local relations}
\label{sec:local-relations}
-Local relations are certain subspaces of the fields on balls, which form an ideal under gluing.
+
+For convenience we assume that fields are enriched over Vect.
+
+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}
@@ -324,13 +376,14 @@
\label{sec:constructing-a-tqft}
In this subsection we briefly review the construction of a TQFT from a system of fields and local relations.
-(For more details, see \cite{kw:tqft}.)
+As usual, see \cite{kw:tqft} for more details.
Let $W$ be an $n{+}1$-manifold.
We can think of the path integral $Z(W)$ as assigning to each
boundary condition $x\in \cC(\bd W)$ a complex number $Z(W)(x)$.
In other words, $Z(W)$ lies in $\c^{\lf(\bd W)}$, the vector space of linear
maps $\lf(\bd W)\to \c$.
+(We haven't defined a path integral in this context; this is just for motivation.)
The locality of the TQFT implies that $Z(W)$ in fact lies in a subspace
$Z(\bd W) \sub \c^{\lf(\bd W)}$ defined by local projections.
@@ -349,7 +402,7 @@
$$A(X) \deq \lf(X) / U(X),$$
where $\cU(X) \sub \lf(X)$ to be the space of local relations in $\lf(X)$;
$\cU(X)$ is generated by things of the form $u\bullet r$, where
-$u\in \cU(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$.
+$u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$.
\end{defn}
(The blob complex, defined in the next section,
is in some sense the derived version of $A(X)$.)