Binary file RefereeReport.pdf has changed
--- a/blob to-do Sun Sep 25 14:33:30 2011 -0600
+++ b/blob to-do Sun Sep 25 14:44:38 2011 -0600
@@ -3,8 +3,10 @@
* better discussion of systems of fields from disk-like n-cats
(Is this done by now?)
+
+* ?? say clearly that certain lemmas don't work for TOP; we're only claiming DIFF and PL (requires small changes in many places)
-* need to fix fam-o-homeo argument per discussion with Rob
+* need to fix fam-o-homeo argument per discussion with Rob (or just remove it)
* need to change module axioms to follow changes in n-cat axioms; search for and destroy all the "Homeo_\bd"'s, add a v-cone axiom
Binary file diagrams/ncat/boundary-collar.pdf has changed
--- a/preamble.tex Sun Sep 25 14:33:30 2011 -0600
+++ b/preamble.tex Sun Sep 25 14:44:38 2011 -0600
@@ -50,6 +50,9 @@
\theoremstyle{plain}
%\newtheorem*{fact}{Fact}
\newtheorem{prop}{Proposition}[subsection]
+\makeatletter
+\@addtoreset{prop}{section}
+\makeatother
\newtheorem{conj}[prop]{Conjecture}
\newtheorem{thm}[prop]{Theorem}
\newtheorem{lem}[prop]{Lemma}
--- a/sandbox.tex Sun Sep 25 14:33:30 2011 -0600
+++ b/sandbox.tex Sun Sep 25 14:44:38 2011 -0600
@@ -1,15 +1,12 @@
-\documentclass[11pt,leqno]{amsart}
-
-%\usepackage{amsthm}
+\documentclass[11pt,leqno]{article}
\newcommand{\pathtotrunk}{./}
+\input{preamble}
\input{text/article_preamble}
-\input{text/top_matter}
\input{text/kw_macros}
%\title{Blob Homology}
\title{Sandbox}
-
\begin{document}
--- a/text/a_inf_blob.tex Sun Sep 25 14:33:30 2011 -0600
+++ b/text/a_inf_blob.tex Sun Sep 25 14:44:38 2011 -0600
@@ -418,14 +418,31 @@
\begin{rem}
Lurie has shown in \cite[Theorem 3.8.6]{0911.0018} that the topological chiral homology
of an $n$-manifold $M$ with coefficients in a certain $E_n$ algebra constructed from $T$ recovers
-the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected.
+the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n{-}1$-connected.
This extra hypothesis is not surprising, in view of the idea described in Example \ref{ex:e-n-alg}
that an $E_n$ algebra is roughly equivalent data to an $A_\infty$ $n$-category which
is trivial at levels 0 through $n-1$.
Ricardo Andrade also told us about a similar result.
+
+Specializing still further, Theorem \ref{thm:map-recon} is related to the classical result that for connected spaces $T$
+we have $HH_*(C_*(\Omega T)) \cong H_*(LT)$, that is, the Hochschild homology of based loops in $T$ is isomorphic
+to the homology of the free loop space of $T$ (see \cite{MR793184} and \cite{MR842427}).
+Theorem \ref{thm:map-recon} says that for any space $T$ (connected or not) we have
+$\bc_*(S^1; C_*(\pi^\infty_{\le 1}(T))) \simeq C_*(LT)$.
+Here $C_*(\pi^\infty_{\le 1}(T))$ denotes the singular chain version of the fundamental infinity-groupoid of $T$,
+whose objects are points in $T$ and morphism chain complexes are $C_*(\paths(t_1 \to t_2))$ for $t_1, t_2 \in T$.
+If $T$ is connected then the $A_\infty$ 1-category $C_*(\pi^\infty_{\le 1}(T))$ is Morita equivalent to the
+$A_\infty$ algebra $C_*(\Omega T)$;
+the bimodule for the equivalence is the singular chains of the space of paths which start at the base point of $T$.
+Theorem \ref{thm:hochschild} holds for $A_\infty$ 1-categories (though we do not prove that in this paper),
+which then implies that
+\[
+ Hoch_*(C_*(\Omega T)) \simeq Hoch_*(C_*(\pi^\infty_{\le 1}(T)))
+ \simeq \bc_*(S^1; C_*(\pi^\infty_{\le 1}(T))) \simeq C_*(LT) .
+\]
\end{rem}
-\begin{proof}
+\begin{proof}[Proof of Theorem \ref{thm:map-recon}]
The proof is again similar to that of Theorem \ref{thm:product}.
We begin by constructing a chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$.
--- a/text/appendixes/comparing_defs.tex Sun Sep 25 14:33:30 2011 -0600
+++ b/text/appendixes/comparing_defs.tex Sun Sep 25 14:44:38 2011 -0600
@@ -4,8 +4,8 @@
\label{sec:comparing-defs}
In \S\ref{sec:example:traditional-n-categories(fields)} we showed how to construct
-a topological $n$-category from a traditional $n$-category; the morphisms of the
-topological $n$-category are string diagrams labeled by the traditional $n$-category.
+a disk-like $n$-category from a traditional $n$-category; the morphisms of the
+disk-like $n$-category are string diagrams labeled by the traditional $n$-category.
In this appendix we sketch how to go the other direction, for $n=1$ and 2.
The basic recipe, given a disk-like $n$-category $\cC$, is to define the $k$-morphisms
of the corresponding traditional $n$-category to be $\cC(B^k)$, where
@@ -585,6 +585,7 @@
The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$
($\cC$ applied to the standard interval with boundary labeled by $x$ and $y$).
For simplicity we will now assume there is only one object and suppress it from the notation.
+Henceforth $A$ will also denote its unique morphism space.
A choice of homeomorphism $I\cup I \to I$ induces a chain map $m_2: A\otimes A\to A$.
We now have two different homeomorphisms $I\cup I\cup I \to I$, but they are isotopic.
@@ -610,7 +611,7 @@
We define a $\Homeo(J)$ action on $\cC(J)$ via $g_*(f, a) = (g\circ f, a)$.
The $C_*(\Homeo(J))$ action is defined similarly.
-Let $J_1$ and $J_2$ be intervals.
+Let $J_1$ and $J_2$ be intervals, and let $J_1\cup J_2$ denote their union along a single boundary point.
We must define a map $\cC(J_1)\ot\cC(J_2)\to\cC(J_1\cup J_2)$.
Choose a homeomorphism $g:I\to J_1\cup J_2$.
Let $(f_i, a_i)\in \cC(J_i)$.
--- a/text/appendixes/famodiff.tex Sun Sep 25 14:33:30 2011 -0600
+++ b/text/appendixes/famodiff.tex Sun Sep 25 14:44:38 2011 -0600
@@ -236,14 +236,13 @@
Since $X$ is compact, we may assume without loss of generality that the cover $\cU$ is finite.
Let $\cU = \{U_\alpha\}$, $1\le \alpha\le N$.
-We will need some wiggle room, so for each $\alpha$ choose a large finite number of open sets
+We will need some wiggle room, so for each $\alpha$ choose $2N$ additional open sets
\[
- U_\alpha = U_\alpha^0 \supset U_\alpha^1 \supset U_\alpha^2 \supset \cdots
+ U_\alpha = U_\alpha^0 \supset U_\alpha^\frac12 \supset U_\alpha^1 \supset U_\alpha^\frac32 \supset \cdots \supset U_\alpha^N
\]
-so that for each fixed $i$ the sets $\{U_\alpha^i\}$ are an open cover of $X$, and also so that
-the closure $\ol{U_\alpha^i}$ is compact and $U_\alpha^{i-1} \supset \ol{U_\alpha^i}$.
-\nn{say specifically how many we need?}
-
+so that for each fixed $i$ the set $\cU^i = \{U_\alpha^i\}$ is an open cover of $X$, and also so that
+the closure $\ol{U_\alpha^i}$ is compact and $U_\alpha^{i-\frac12} \supset \ol{U_\alpha^i}$.
+%\nn{say specifically how many we need?}
Let $P$ be some $k$-dimensional polyhedron and $f:P\to \Homeo(X)$.
After subdividing $P$, we may assume that there exists $g\in \Homeo(X)$
@@ -252,34 +251,97 @@
The sense of ``small" we mean will be explained below.
It depends only on $\cU$ and the choice of $U_\alpha^i$'s.
-We may assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU$.
+Our goal is to homotope $P$, rel boundary, so that it is adapted to $\cU$.
+By the local contractibility of $\Homeo(X)$ (Corollary 1.1 of \cite{MR0283802}),
+it suffices to find $f':P\to \Homeo(X)$ such that $f' = f$ on $\bd P$ and $f'$ is adapted to $\cU$.
+
+We may assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU^N$.
+So $\bd P = \sum Q_\beta$, and the support of $f$ restricted to $Q_\beta$ is $V_\beta^N$, the union of $k-1$ of
+the $U_\alpha^N$'s. Define $V_\beta^i \sup V_\beta^N$ to be the corresponding union of $k-1$
+of the $U_\alpha^i$'s.
+
+Define
+\[
+ W_j^i = U_1^i \cup U_2^i \cup \cdots \cup U_j^i .
+\]
+
+By the local contractibility of $\Homeo(X)$ (Corollary 1.1 of \cite{MR0283802}),
+
+We will construct a sequence of maps $f_i : \bd P\to \Homeo(X)$, for $i = 0, 1, \ldots, N$, with the following properties:
+\begin{itemize}
+\item[(A)] $f_0 = f|_{\bd P}$;
+\item[(B)] $f_i = g$ on $W_i^i$;
+\item[(C)] $f_i$ restricted to $Q_\beta$ has support contained in $V_\beta^{N-i}$; and
+\item[(D)] there is a homotopy $F_i : \bd P\times I \to \Homeo(X)$ from $f_{i-1}$ to $f_i$ such that the
+support of $F_i$ restricted to $Q_\beta\times I$ is contained in $U_i^i\cup V_\beta^{N-i}$.
+\nn{check this when done writing}
+\end{itemize}
+
+Once we have the $F_i$'s as in (D), we can finish the argument as follows.
+Assemble the $F_i$'s into a map $F: \bd P\times [0,N] \to \Homeo(X)$.
+$F$ is adapted to $\cU$ by (D).
+$F$ restricted to $\bd P\times\{N\}$ is constant on $W_N^N = X$ by (B).
+We can therefore view $F$ as a map $f'$ from $\Cone(\bd P) \cong P$ to $\Homeo(X)$
+which is adapted to $\cU$.
+
+The homotopies $F_i$ will be composed of three types of pieces, $A_\beta$, $B_\beta$ and $C$, % NOT C_\beta
+as illustrated in Figure \nn{xxxx}.
+($A_\beta$, $B_\beta$ and $C$ also depend on $i$, but we are suppressing that from the notation.)
+The homotopy $A_\beta : Q_\beta \times I \to \Homeo(X)$ will arrange that $f_i$ agrees with $g$
+on $U_i^i \setmin V_\beta^{N-i+1}$.
+The homotopy $B_\beta : Q_\beta \times I \to \Homeo(X)$ will extend the agreement with $g$ to all of $U_i^i$.
+The homotopies $C$ match things up between $\bd Q_\beta \times I$ and $\bd Q_{\beta'} \times I$ when
+$Q_\beta$ and $Q_{\beta'}$ are adjacent.
+
+Assume inductively that we have defined $f_{i-1}$.
+
+Now we define $A_\beta$.
+Choose $q_0\in Q_\beta$.
+Theorem 5.1 of \cite{MR0283802} implies that we can choose a homotopy $h:I \to \Homeo(X)$, with $h(0)$ the identity, such that
+\begin{itemize}
+\item[(E)] the support of $h$ is contained in $U_i^{i-1} \setmin W_{i-1}^{i-\frac12}$; and
+\item[(F)] $h(1) \circ f_{i-1}(q_0) = g$ on $U_i^i$.
+\end{itemize}
+Define $A_\beta$ by
+\[
+ A_\beta(q, t) = h(t) \circ f_{i-1}(q) .
+\]
+It follows that
+\begin{itemize}
+\item[(G)] $A_\beta(q,1) = A(q',1)$, for all $q,q' \in Q_\beta$, on $X \setmin V_\beta^{N-i+1}$;
+\item[(H)] $A_\beta(q, 1) = g$ on $(U_i^i \cup W_{i-1}^{i-\frac12})\setmin V_\beta^{N-i+1}$; and
+\item[(I)] the support of $A_\beta$ is contained in $U_i^{i-1} \cup V_\beta^{N-i+1}$.
+\end{itemize}
+
+Next we define $B_\beta$.
+Theorem 5.1 of \cite{MR0283802} implies that we can choose a homotopy $B_\beta:Q_\beta\times I\to \Homeo(X)$
+such that
+\begin{itemize}
+\item[(J)] $B_\beta(\cdot, 0) = A_\beta(\cdot, 1)$;
+\item[(K)] $B_\beta(q,1) = g$ on $W_i^i$;
+\item[(L)] the support of $B_\beta(\cdot,1)$ is contained in $V_\beta^{N-i}$; and
+\item[(M)] the support of $B_\beta$ is contained in $U_i^i \cup V_\beta^{N-i}$.
+\end{itemize}
+
+All that remains is to define the ``glue" $C$ which interpolates between adjacent $Q_\beta$ and $Q_{\beta'}$.
+First consider the $k=2$ case.
+(In this case Figure \nn{xxxx} is literal rather than merely schematic.)
+Let $q = Q_\beta \cap Q_{\beta'}$ be a point on the boundaries of both $Q_\beta$ and $Q_{\beta'}$.
+We have an arc of Homeomorphisms, composed of $B_\beta(q, \cdot)$, $A_\beta(q, \cdot)$,
+$A_{\beta'}(q, \cdot)$ and $B_{\beta'}(q, \cdot)$, which connects $B_\beta(q, 1)$ to $B_{\beta'}(q, 1)$.
+
+\nn{Hmmmm..... I think there's a problem here}
-Theorem 5.1 of \cite{MR0283802}, applied $N$ times, allows us
-to choose a homotopy $h:P\times [0,N] \to \Homeo(X)$ with the following properties:
-\begin{itemize}
-\item $h(p, 0) = f(p)$ for all $p\in P$.
-\item The restriction of $h(p, i)$ to $U_0^i \cup \cdots \cup U_i^i$ is equal to the homeomorphism $g$,
-for all $p\in P$.
-\item For each fixed $p\in P$, the family of homeomorphisms $h(p, [i-1, i])$ is supported on
-$U_i^{i-1} \setmin (U_0^i \cup \cdots \cup U_{i-1}^i)$
-(and hence supported on $U_i$).
-\end{itemize}
+
+\nn{resume revising here}
+
+
+\nn{scraps:}
+
To apply Theorem 5.1 of \cite{MR0283802}, the family $f(P)$ must be sufficiently small,
and the subdivision mentioned above is chosen fine enough to insure this.
-By reparametrizing, we can view $h$ as a homotopy (rel boundary) from $h(\cdot,0) = f: P\to\Homeo(X)$
-to the family
-\[
- h(\bd(P\times [0,N]) \setmin P\times \{0\}) = h(\bd P \times [0,N]) \cup h(P \times \{N\}) .
-\]
-We claim that the latter family of homeomorphisms is adapted to $\cU$.
-By the second bullet above, $h(P\times \{N\})$ is the constant family $g$, and is therefore supported on the empty set.
-Via an inductive assumption and a preliminary homotopy, we have arranged above that $f(\bd P)$ is
-adapted to $\cU$, which means that $\bd P = \cup_j Q_j$ with $f(Q_j)$ supported on the union of $k-1$
-of the $U_\alpha$'s for each $j$.
-It follows (using the third bullet above) that $h(Q_j \times [i-1,i])$ is supported on the union of $k$
-of the $U_\alpha$'s, specifically, the support of $f(Q_j)$ union $U_i$.
\end{proof}
--- a/text/article_preamble.tex Sun Sep 25 14:33:30 2011 -0600
+++ b/text/article_preamble.tex Sun Sep 25 14:44:38 2011 -0600
@@ -18,7 +18,7 @@
\usetikzlibrary{decorations,decorations.pathreplacing}
\usetikzlibrary{fit,calc,through}
-\pgfrealjobname{blob1}
+%\pgfrealjobname{blob1}
\makeatletter
\@ifclassloaded{beamer}{}{%
--- a/text/basic_properties.tex Sun Sep 25 14:33:30 2011 -0600
+++ b/text/basic_properties.tex Sun Sep 25 14:44:38 2011 -0600
@@ -90,7 +90,7 @@
$r$ 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, \S\ref{sec:moam}),
so $f$ and the identity map are homotopic.
\end{proof}
--- a/text/blobdef.tex Sun Sep 25 14:33:30 2011 -0600
+++ b/text/blobdef.tex Sun Sep 25 14:44:38 2011 -0600
@@ -43,7 +43,7 @@
``the space of all local relations that can be imposed on $\bc_0(X)$".
Thus we say a $1$-blob diagram consists of:
\begin{itemize}
-\item An closed ball in $X$ (``blob") $B \sub X$.
+\item A closed ball in $X$ (``blob") $B \sub X$.
\item A boundary condition $c \in \cF(\bdy B) = \cF(\bd(X \setmin B))$.
\item A field $r \in \cF(X \setmin B; c)$.
\item A local relation field $u \in U(B; c)$.
@@ -156,7 +156,7 @@
\end{itemize}
Combining these two operations can give rise to configurations of blobs whose complement in $X$ is not
a manifold.
-Thus will need to be more careful when speaking of a field $r$ on the complement of the blobs.
+Thus we will need to be more careful when speaking of a field $r$ on the complement of the blobs.
\begin{example} \label{sin1x-example}
Consider the four subsets of $\Real^3$,
@@ -208,7 +208,7 @@
%and the entire configuration should be compatible with some gluing decomposition of $X$.
\begin{defn}
\label{defn:configuration}
-A configuration of $k$ blobs in $X$ is an ordered collection of $k$ subsets $\{B_1, \ldots B_k\}$
+A configuration of $k$ blobs in $X$ is an ordered collection of $k$ subsets $\{B_1, \ldots, B_k\}$
of $X$ such that there exists a gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$ and
for each subset $B_i$ there is some $0 \leq l \leq m$ and some connected component $M_l'$ of
$M_l$ which is a ball, so $B_i$ is the image of $M_l'$ in $X$.
@@ -238,7 +238,7 @@
\label{defn:blob-diagram}
A $k$-blob diagram on $X$ consists of
\begin{itemize}
-\item a configuration $\{B_1, \ldots B_k\}$ of $k$ blobs in $X$,
+\item a configuration $\{B_1, \ldots, B_k\}$ of $k$ blobs in $X$,
\item and a field $r \in \cF(X)$ which is splittable along some gluing decomposition compatible with that configuration,
\end{itemize}
such that
--- a/text/deligne.tex Sun Sep 25 14:33:30 2011 -0600
+++ b/text/deligne.tex Sun Sep 25 14:44:38 2011 -0600
@@ -160,7 +160,7 @@
We have now defined a map from the little $n{+}1$-balls operad to the $n$-SC operad,
with contractible fibers.
(The fibers correspond to moving the $D_i$'s in the $x_{n+1}$
-direction without changing their ordering.)
+direction while keeping them disjoint.)
%\nn{issue: we've described this by varying the $R_i$'s, but above we emphasize varying the $f_i$'s.
%does this need more explanation?}
@@ -178,7 +178,8 @@
p(\ol{f}): \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k))
\to \hom(\bc_*(M_0), \bc_*(N_0)) .
\]
-Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define $p(\ol{f}$) to be the composition
+Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define
+$p(\ol{f})(\alpha_1\ot\cdots\ot\alpha_k)$ to be the composition
\[
\bc_*(M_0) \stackrel{f_0}{\to} \bc_*(R_1\cup M_1)
\stackrel{\id\ot\alpha_1}{\to} \bc_*(R_1\cup N_1)
@@ -201,7 +202,7 @@
\label{thm:deligne}
There is a collection of chain maps
\[
- C_*(SC^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes
+ C_*(SC^n_{\ol{M}\ol{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes
\hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0))
\]
which satisfy the operad compatibility conditions.
@@ -216,7 +217,7 @@
a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disks operad.
\begin{proof}
-As described above, $SC^n_{\overline{M}, \overline{N}}$ is equal to the disjoint
+As described above, $SC^n_{\ol{M}\ol{N}}$ is equal to the disjoint
union of products of homeomorphism spaces, modulo some relations.
By Theorem \ref{thm:CH} and the Eilenberg-Zilber theorem, we have for each such product $P$
a chain map
@@ -225,7 +226,7 @@
\hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) .
\]
It suffices to show that the above maps are compatible with the relations whereby
-$SC^n_{\overline{M}, \overline{N}}$ is constructed from the various $P$'s.
+$SC^n_{\ol{M}\ol{N}}$ is constructed from the various $P$'s.
This in turn follows easily from the fact that
the actions of $C_*(\Homeo(\cdot\to\cdot))$ are local (compatible with gluing) and associative.
%\nn{should add some detail to above}
--- a/text/evmap.tex Sun Sep 25 14:33:30 2011 -0600
+++ b/text/evmap.tex Sun Sep 25 14:44:38 2011 -0600
@@ -123,7 +123,7 @@
Let $g_j = f_1\circ f_2\circ\cdots\circ f_j$.
Let $g$ be the last of the $g_j$'s.
Choose the sequence $\bar{f}_j$ so that
-$g(B)$ is contained is an open set of $\cV_1$ and
+$g(B)$ is contained in an open set of $\cV_1$ and
$g_{j-1}(|f_j|)$ is also contained in an open set of $\cV_1$.
There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$
@@ -325,7 +325,7 @@
\end{proof}
For $S\sub X$, we say that $a\in \btc_k(X)$ is {\it supported on $S$}
-if there exists $a'\in \btc_k(S)$
+if there exist $a'\in \btc_k(S)$
and $r\in \btc_0(X\setmin S)$ such that $a = a'\bullet r$.
\newcommand\sbtc{\btc^{\cU}}
@@ -385,7 +385,7 @@
Now let $b$ be a generator of $C_2$.
If $\cU$ is fine enough, there is a disjoint union of balls $V$
on which $b + h_1(\bd b)$ is supported.
-Since $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_2(X)$, we can find
+Since $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_1(X)$, we can find
$s(b)\in \bc_2(X)$ with $\bd(s(b)) = \bd(b + h_1(\bd b))$ (by Corollary \ref{disj-union-contract}).
By Lemmas \ref{bt-contract} and \ref{btc-prod}, we can now find
$h_2(b) \in \btc_3(X)$, also supported on $V$, such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$
--- a/text/hochschild.tex Sun Sep 25 14:33:30 2011 -0600
+++ b/text/hochschild.tex Sun Sep 25 14:44:38 2011 -0600
@@ -12,7 +12,7 @@
Hochschild complex of the 1-category $\cC$.
(Recall from \S \ref{sec:example:traditional-n-categories(fields)} that a
$1$-category gives rise to a $1$-dimensional system of fields; as usual,
-talking about the blob complex with coefficients in a $n$-category means
+talking about the blob complex with coefficients in an $n$-category means
first passing to the corresponding $n$ dimensional system of fields.)
Thus the blob complex is a natural generalization of something already
known to be interesting in higher homological degrees.
@@ -293,7 +293,7 @@
$\widetilde{q} \in \ker(C \tensor E \tensor C \to E)$.
Further,
\begin{align*}
-\hat{g}(\widetilde{q}) & = \sum_i \left(a_i \tensor g(\widetilde{q_i}) \tensor b_i\right) - 1 \tensor \left(\sum_i a_i g(\widetilde{q_i}\right) b_i) \tensor 1 \\
+\hat{g}(\widetilde{q}) & = \sum_i \left(a_i \tensor g(\widetilde{q_i}) \tensor b_i\right) - 1 \tensor \left(\sum_i a_i g(\widetilde{q_i}) b_i\right) \tensor 1 \\
& = q - 0
\end{align*}
(here we used that $g$ is a map of $C$-$C$ bimodules, and that $\sum_i a_i q_i b_i = 0$).
@@ -341,11 +341,11 @@
$j$-th term have labelled points $d_{j,i}$ to the left of $*$, $m_j$ at $*$, and $d_{j,i}'$ to the right of $*$,
and there are labels $c_i$ at the labeled points outside the blob.
We know that
-$$\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \tensor m_j \tensor d_{j,1}' \tensor \cdots \tensor d_{j,k'_j}' \in \ker(\DirectSum_{k,k'} C^{\tensor k} \tensor M \tensor C^{\tensor k'} \tensor \to M),$$
+$$\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \tensor m_j \tensor d_{j,1}' \tensor \cdots \tensor d_{j,k'_j}' \in \ker(\DirectSum_{k,k'} C^{\tensor k} \tensor M \tensor C^{\tensor k'} \to M),$$
and so
\begin{align*}
-\ev(\bdy y) & = \sum_j m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k d_{j,1} \cdots d_{j,k_j} \\
- & = \sum_j d_{j,1} \cdots d_{j,k_j} m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k \\
+\pi\left(\ev(\bdy y)\right) & = \pi\left(\sum_j m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k d_{j,1} \cdots d_{j,k_j}\right) \\
+ & = \pi\left(\sum_j d_{j,1} \cdots d_{j,k_j} m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k\right) \\
& = 0
\end{align*}
where this time we use the fact that we're mapping to $\coinv{M}$, not just $M$.
--- a/text/intro.tex Sun Sep 25 14:33:30 2011 -0600
+++ b/text/intro.tex Sun Sep 25 14:44:38 2011 -0600
@@ -260,8 +260,7 @@
Note that this includes the case of gluing two disjoint manifolds together.
\begin{property}[Gluing map]
\label{property:gluing-map}%
-Given a gluing $X \to X_\mathrm{gl}$, there is
-a natural map
+Given a gluing $X \to X_\mathrm{gl}$, there is an injective natural map
\[
\bc_*(X) \to \bc_*(X_\mathrm{gl})
\]
--- a/text/kw_macros.tex Sun Sep 25 14:33:30 2011 -0600
+++ b/text/kw_macros.tex Sun Sep 25 14:44:38 2011 -0600
@@ -64,7 +64,7 @@
% \DeclareMathOperator{\pr}{pr} etc.
\def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}}
-\applytolist{declaremathop}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}{Compat}{Coll}{Cone};
+\applytolist{declaremathop}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}{Compat}{Coll}{Cone}{pr}{paths};
\DeclareMathOperator*{\colim}{colim}
\DeclareMathOperator*{\hocolim}{hocolim}
--- a/text/ncat.tex Sun Sep 25 14:33:30 2011 -0600
+++ b/text/ncat.tex Sun Sep 25 14:44:38 2011 -0600
@@ -214,12 +214,14 @@
with the (ordinary, not fibered) product $\cC(B_1) \times \cC(B_2)$.
Let $\cl{\cC}(S)\trans E$ denote the image of $\gl_E$.
-We will refer to elements of $\cl{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$". When the gluing map is surjective every such element is splittable.
+We will refer to elements of $\cl{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$".
+When the gluing map is surjective every such element is splittable.
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 $\cC(X)\trans E = \bd^{-1}(\cl{\cC}(\bd X)\trans E)$.
-We will call the projection $\cl{\cC}(S)\trans E \to \cC(B_i)$
+We will call the projection $\cl{\cC}(S)\trans E \to \cC(B_i)$ given by the composition
+$$\cl{\cC}(S)\trans E \xrightarrow{\gl^{-1}} \cC(B_1) \times \cC(B_2) \xrightarrow{\pr_i} \cC(B_i)$$
a {\it restriction} map and write $\res_{B_i}(a)$
(or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)\trans E$.
More generally, we also include under the rubric ``restriction map"
@@ -227,9 +229,14 @@
another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition
of restriction maps.
In particular, we have restriction maps $\cC(X)\trans E \to \cC(B_i)$
-($i = 1, 2$, notation from previous paragraph).
+defined as the composition of the boundary with the first restriction map described above:
+$$
+\cC(X) \trans E \xrightarrow{\bdy} \cl{\cC}(\bdy X)\trans E \xrightarrow{\res} \cC(B_i)
+.$$
These restriction maps can be thought of as
domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$.
+These restriction maps in fact have their image in the subset $\cC(B_i)\trans E$,
+and so to emphasize this we will sometimes write the restriction map as $\cC(X)\trans E \to \cC(B_i)\trans E$.
Next we consider composition of morphisms.
@@ -977,7 +984,7 @@
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
+Second, in the category definition we directly impose isotopy
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.)
@@ -994,6 +1001,8 @@
In the $n$-category axioms above we have intermingled data and properties for expository reasons.
Here's a summary of the definition which segregates the data from the properties.
+We also remind the reader of the inductive nature of the definition: All the data for $k{-}1$-morphisms must be in place
+before we can describe the data for $k$-morphisms.
An $n$-category consists of the following data:
\begin{itemize}
@@ -1164,8 +1173,8 @@
\label{ex:blob-complexes-of-balls}
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,
+When $X$ is an $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$.
+When $X$ is a $k$-ball,
define $\cC(X; c) = \bc^\cE_*(X\times F; c)$
where $\bc^\cE_*$ denotes the blob complex based on $\cE$.
\end{example}
@@ -1226,12 +1235,13 @@
Note that this implies a $\Diff(B^n)$ action on $A$,
since $\cE\cB_n$ contains a copy of $\Diff(B^n)$.
We will define a strict $A_\infty$ $n$-category $\cC^A$.
+(We enrich in topological spaces, though this could easily be adapted to, say, chain complexes.)
If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point.
In other words, the $k$-morphisms are trivial for $k<n$.
If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction.
(Plain colimit, not homotopy colimit.)
Let $J$ be the category whose objects are embeddings of a disjoint union of copies of
-the standard ball $B^n$ into $X$, and who morphisms are given by engulfing some of the
+the standard ball $B^n$ into $X$, and whose morphisms are given by engulfing some of the
embedded balls into a single larger embedded ball.
To each object of $J$ we associate $A^{\times m}$ (where $m$ is the number of balls), and
to each morphism of $J$ we associate a morphism coming from the $\cE\cB_n$ action on $A$.
@@ -1248,7 +1258,17 @@
%\nn{The paper is already long; is it worth giving details here?}
% According to the referee, yes it is...
Let $A = \cC(B^n)$, where $B^n$ is the standard $n$-ball.
-\nn{need to finish this}
+We must define maps
+\[
+ \cE\cB_n^k \times A \times \cdots \times A \to A ,
+\]
+where $\cE\cB_n^k$ is the $k$-th space of the $\cE\cB_n$ operad.
+Let $(b, a_1,\ldots,a_k)$ be a point of $\cE\cB_n^k \times A \times \cdots \times A \to A$.
+The $i$-th embedding of $b$ together with $a_i$ determine an element of $\cC(B_i)$,
+where $B_i$ denotes the $i$-th little ball.
+Using composition of $n$-morphsims in $\cC$, and padding the spaces between the little balls with the
+(essentially unique) identity $n$-morphism of $\cC$, we can construct a well-defined element
+of $\cC(B^n) = A$.
If we apply the homotopy colimit construction of the next subsection to this example,
we get an instance of Lurie's topological chiral homology construction.
@@ -1265,7 +1285,7 @@
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 ordinary $n$-category $\cC$ and pass to the ``free resolution",
+Recall that we can take an ordinary $n$-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
@@ -1470,7 +1490,7 @@
\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 $g: \psi_{\cC;W,c}(x)
-\to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$.
+\to \psi_{\cC;W,c}(y)$ is the value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$.
In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit
is more involved.
@@ -1558,7 +1578,7 @@
Let $z$ be a decomposition of $W$ which is in general position with respect to all of the
$x_i$'s and $v_i$'s.
-There there decompositions $x'_i$ and $v'_i$ (for all $i$) such that
+There exist decompositions $x'_i$ and $v'_i$ (for all $i$) such that
\begin{itemize}
\item $x'_i$ antirefines to $x_i$ and $z$;
\item $v'_i$ antirefines to $x'_i$, $x'_{i-1}$ and $v_i$;
@@ -1584,7 +1604,7 @@
%define $k$-cat $\cC(\cdot\times W)$}
\subsection{Modules}
-
+\label{sec:modules}
Next we define ordinary and $A_\infty$ $n$-category modules.
The definition will be very similar to that of $n$-categories,
but with $k$-balls replaced by {\it marked $k$-balls,} defined below.
@@ -1654,6 +1674,7 @@
then for each marked $n$-ball $M=(B,N)$ and $c\in \cC(\bd B \setminus N)$, the set $\cM(M; c)$ should be an object in that category.
\begin{lem}[Boundary from domain and range]
+\label{lem:module-boundary}
{Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$),
$M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere.
Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the
@@ -1664,7 +1685,32 @@
\]
which is natural with respect to the actions of homeomorphisms.}
\end{lem}
-Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}.
+Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}, and illustrated in Figure \ref{fig:module-boundary}.
+\begin{figure}[t]
+\tikzset{marked/.style={line width=3pt}}
+
+\begin{equation*}
+\begin{tikzpicture}[baseline=0]
+\coordinate (a) at (0,1);
+\coordinate (b) at (4,1);
+\draw[marked] (a) arc (180:0:2);
+\draw (b) -- (a);
+\node at (2,2) {$M_1$};
+
+\draw (0,0) node[fill, circle] {} -- (4,0) node[fill,circle] {};
+\node at (-0.6,0) {$E$};
+
+\draw[marked] (0,-1) arc(-180:0:2);
+\draw (4,-1) -- (0,-1);
+\node at (2,-2) {$M_2$};
+\end{tikzpicture}
+\qquad \qquad \qquad
+\begin{tikzpicture}[baseline=0]
+\draw[marked] (0,0) node {$H$} circle (2);
+\end{tikzpicture}
+\end{equation*}\caption{The marked hemispheres and marked balls from Lemma \ref{lem:module-boundary}.}
+\label{fig:module-boundary}
+\end{figure}
Let $\cl\cM(H)\trans E$ denote the image of $\gl_E$.
We will refer to elements of $\cl\cM(H)\trans E$ as ``splittable along $E$" or ``transverse to $E$".
@@ -2073,7 +2119,7 @@
associated to $L$ by $\cX$ and $\cC$.
(See \S \ref{ss:ncat_fields} and \S \ref{moddecss}.)
Define $\cl{\cY}(L)$ similarly.
-For $K$ an unmarked 1-ball let $\cl{\cC(K)}$ denote the local homotopy colimit
+For $K$ an unmarked 1-ball let $\cl{\cC}(K)$ denote the local homotopy colimit
construction associated to $K$ by $\cC$.
Then we have an injective gluing map
\[
@@ -2130,7 +2176,7 @@
The sphere module $n{+}1$-category is a natural generalization of the
algebra-bimodule-intertwiner 2-category to higher dimensions.
-Another possible name for this $n{+}1$-category is $n{+}1$-category of defects.
+Another possible name for this $n{+}1$-category is the $n{+}1$-category of defects.
The $n$-categories are thought of as representing field theories, and the
$0$-sphere modules are codimension 1 defects between adjacent theories.
In general, $m$-sphere modules are codimension $m{+}1$ defects;
@@ -2181,7 +2227,7 @@
We only consider those decompositions in which the smaller balls are either
$0$-marked (i.e. intersect the $0$-marking of the large ball in a disc)
or plain (don't intersect the $0$-marking of the large ball).
-We can also take the boundary of a $0$-marked ball, which is $0$-marked sphere.
+We can also take the boundary of a $0$-marked ball, which is a $0$-marked sphere.
Fix $n$-categories $\cA$ and $\cB$.
These will label the two halves of a $0$-marked $k$-ball.
@@ -2574,7 +2620,6 @@
\caption{Moving $B$ from bottom to top}
\label{jun23c}
\end{figure}
-Let $D' = B\cap C$.
It is not hard too show that the above two maps are mutually inverse.
\begin{lem} \label{equator-lemma}
@@ -2738,19 +2783,19 @@
\medskip
-We end this subsection with some remarks about Morita equivalence of disklike $n$-categories.
+We end this subsection with some remarks about Morita equivalence of disk-like $n$-categories.
Recall that two 1-categories $\cC$ and $\cD$ are Morita equivalent if and only if they are equivalent
objects in the 2-category of (linear) 1-categories, bimodules, and intertwiners.
-Similarly, we define two disklike $n$-categories to be Morita equivalent if they are equivalent objects in the
+Similarly, we define two disk-like $n$-categories to be Morita equivalent if they are equivalent objects in the
$n{+}1$-category of sphere modules.
-Because of the strong duality enjoyed by disklike $n$-categories, the data for such an equivalence lives only in
+Because of the strong duality enjoyed by disk-like $n$-categories, the data for such an equivalence lives only in
dimensions 1 and $n+1$ (the middle dimensions come along for free).
The $n{+}1$-dimensional part of the data must be invertible and satisfy
identities corresponding to Morse cancellations in $n$-manifolds.
We will treat this in detail for the $n=2$ case; the case for general $n$ is very similar.
-Let $\cC$ and $\cD$ be (unoriented) disklike 2-categories.
+Let $\cC$ and $\cD$ be (unoriented) disk-like 2-categories.
Let $\cS$ denote the 3-category of 2-category sphere modules.
The 1-dimensional part of the data for a Morita equivalence between $\cC$ and $\cD$ is a 0-sphere module $\cM = {}_\cC\cM_\cD$
(categorified bimodule) connecting $\cC$ and $\cD$.
--- a/text/tqftreview.tex Sun Sep 25 14:33:30 2011 -0600
+++ b/text/tqftreview.tex Sun Sep 25 14:44:38 2011 -0600
@@ -51,7 +51,7 @@
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 $\cS$
+An $n$-dimensional {\it system of fields} in $\cS$
is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$
together with some additional data and satisfying some additional conditions, all specified below.
@@ -85,13 +85,15 @@
\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 $\Kom$),
+If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$ (chain complexes)),
then this extra structure is considered part of the definition of $\cC_n$.
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 $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$,
+structures on $\cM_k$, $\Set$ and $\cS$.
+For $k<n$ we have $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$,
compatibly with homeomorphisms and restriction to boundary.
-We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$
+For $k=n$ we require $\cC_n(X \du W; c\du d) \cong \cC_k(X, c)\ot \cC_k(W, d)$.
+We will call the projections $\cC_k(X_1 \du X_2) \to \cC_k(X_i)$
restriction maps.
\item Gluing without corners.
Let $\bd X = Y \du Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds.
@@ -299,7 +301,7 @@
domain and range determined by the transverse orientation and the labelings of the 1-cells.
\end{itemize}
-We want fields on 1-manifolds to be enriched over Vect, so we also allow formal linear combinations
+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
@@ -371,7 +373,7 @@
\subsection{Local relations}
\label{sec:local-relations}
-For convenience we assume that fields are enriched over Vect.
+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.
@@ -393,14 +395,14 @@
These motivate the following definition.
\begin{defn}
-A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$,
+A {\it local relation} is a collection of subspaces $U(B; c) \sub \lf(B; c)$,
for all $n$-manifolds $B$ which are
homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$,
satisfying the following properties.
\begin{enumerate}
\item Functoriality:
$f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$
-\item Local relations imply extended isotopy:
+\item Local relations imply extended isotopy invariance:
if $x, y \in \cC(B; c)$ and $x$ is extended isotopic
to $y$, then $x-y \in U(B; c)$.
\item Ideal with respect to gluing:
@@ -449,7 +451,7 @@
The above construction can be extended to higher codimensions, assigning
a $k$-category $A(Y)$ to an $n{-}k$-manifold $Y$, for $0 \le k \le n$.
These invariants fit together via actions and gluing formulas.
-We describe only the case $k=1$ below.
+We describe only the case $k=1$ below. We describe these extensions in the more general setting of the blob complex later, in particular in Examples \ref{ex:ncats-from-tqfts} and \ref{ex:blob-complexes-of-balls} and in \S \ref{sec:modules}.
The construction of the $n{+}1$-dimensional part of the theory (the path integral)
requires that the starting data (fields and local relations) satisfy additional
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