# HG changeset patch
# User Scott Morrison <scott@tqft.net>
# Date 1301497407 25200
# Node ID db9d3a27647a77fa18cf2f573bb29cdde6dfefb1
# Parent  6fd9b377be3be5b8c1c2743ef1542879bbc1c23d# Parent  bd0d5b2155a72867a862766b18e8be5b4d08974b
Automated merge with https://tqft.net/hg/blob/

diff -r bd0d5b2155a7 -r db9d3a27647a blob1.tex
--- a/blob1.tex	Wed Mar 30 08:03:22 2011 -0700
+++ b/blob1.tex	Wed Mar 30 08:03:27 2011 -0700
@@ -17,11 +17,6 @@
 
 \maketitle
 
-%[revision $\ge$ 527;  $\ge$ 30 August 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.
-
 \begin{abstract}
 Given an $n$-manifold $M$ and an $n$-category $\cC$, we define a chain complex
 (the ``blob complex") $\bc_*(M; \cC)$.
@@ -46,8 +41,6 @@
 }
 
 
-%\let\stdsection\section
-%\renewcommand\section{\newpage\stdsection}
 
 \input{text/intro}
 
@@ -73,8 +66,6 @@
 
 \input{text/appendixes/famodiff}
 
-%\input{text/appendixes/smallblobs}
-
 \input{text/appendixes/comparing_defs}
 
 %\input{text/comm_alg}
diff -r bd0d5b2155a7 -r db9d3a27647a text/appendixes/comparing_defs.tex
--- a/text/appendixes/comparing_defs.tex	Wed Mar 30 08:03:22 2011 -0700
+++ b/text/appendixes/comparing_defs.tex	Wed Mar 30 08:03:27 2011 -0700
@@ -118,12 +118,12 @@
 Each approach has advantages and disadvantages.
 For better or worse, we choose bigons here.
 
-Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard
+Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k) \trans E$, where $B^k$ denotes the standard
 $k$-ball, which we also think of as the standard bihedron (a.k.a.\ globe).
 (For $k=1$ this is an interval, and for $k=2$ it is a bigon.)
 Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$
 into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$.
-Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$
+Recall that the subscript in $\cC(B^k) \trans E$ means that we consider the subset of $\cC(B^k)$
 whose boundary is splittable along $E$.
 This allows us to define the domain and range of morphisms of $C$ using
 boundary and restriction maps of $\cC$.
diff -r bd0d5b2155a7 -r db9d3a27647a text/evmap.tex
--- a/text/evmap.tex	Wed Mar 30 08:03:22 2011 -0700
+++ b/text/evmap.tex	Wed Mar 30 08:03:27 2011 -0700
@@ -125,10 +125,10 @@
 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
-$g_{j-1}(|f_j|)$ is also contained is an open set of $\cV_1$.
+$g_{j-1}(|f_j|)$ 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 $\cV_1$
-(more specifically, $|c_{ij}| = g_{j-1}(B)$)
+(more specifically, $|c_{ij}| = g_{j-1}(|f_j|)$)
 and $\bd c_{ij} = g_{j-1}(a_i) - g_{j}(a_i)$.
 Define
 \[
@@ -346,8 +346,9 @@
 
 It suffices to show that we can deform a finite subcomplex $C_*$ of $\btc_*(X)$ into $\sbtc_*(X)$
 (relative to any designated subcomplex of $C_*$ already in $\sbtc_*(X)$).
-The first step is to replace families of general blob diagrams with families that are 
-small with respect to $\cU$.
+The first step is to replace families of general blob diagrams with families 
+of blob diagrams that are small with respect to $\cU$.
+(If $f:P \to \BD_k$ is the family then for all $p\in P$ we have that $f(p)$ is a diagram in which the blobs are small.)
 This is done as in the proof of Lemma \ref{small-blobs-b}; the technique of the proof works in families.
 Each such family is homotopic to a sum families which can be a ``lifted" to $\Homeo(X)$.
 That is, $f:P \to \BD_k$ has the form $f(p) = g(p)(b)$ for some $g:P\to \Homeo(X)$ and $b\in \BD_k$.
diff -r bd0d5b2155a7 -r db9d3a27647a text/intro.tex
--- a/text/intro.tex	Wed Mar 30 08:03:22 2011 -0700
+++ b/text/intro.tex	Wed Mar 30 08:03:27 2011 -0700
@@ -555,7 +555,17 @@
 and
 Alexander Kirillov
 for many interesting and useful conversations. 
-\nn{should add thanks to people from Teichner's reading course; Aaron Mazel-Gee, $\ldots$}
+Peter Teichner ran a reading course based on an earlier draft of this paper, and the detailed feedback
+we got from the student lecturers lead to very many improvements in later drafts.
+So big thanks to
+Aaron Mazel-Gee,
+Nate Watson,
+Alan Wilder,
+Dmitri Pavlov,
+Ansgar Schneider,
+and
+Dan Berwick-Evans.
+\nn{need to double-check this list once the reading course is over}
 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. We'd like to thank the Aspen Center for Physics for the pleasant and productive 
 environment provided there during the final preparation of this manuscript.
 
diff -r bd0d5b2155a7 -r db9d3a27647a text/kw_macros.tex
--- a/text/kw_macros.tex	Wed Mar 30 08:03:22 2011 -0700
+++ b/text/kw_macros.tex	Wed Mar 30 08:03:27 2011 -0700
@@ -32,6 +32,7 @@
 \def\BD{BD}
 
 \def\spl{_\pitchfork}
+\def\trans#1{_{\pitchfork #1}}
 
 %\def\nn#1{{{\it \small [#1]}}}
 \def\nn#1{{{\color[rgb]{.2,.5,.6} \small [[#1]]}}}
diff -r bd0d5b2155a7 -r db9d3a27647a text/ncat.tex
--- a/text/ncat.tex	Wed Mar 30 08:03:22 2011 -0700
+++ b/text/ncat.tex	Wed Mar 30 08:03:27 2011 -0700
@@ -14,13 +14,15 @@
 (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.
+Compared to other definitions in the literature,
+the definitions presented below tie the categories more closely to the topology
+and avoid combinatorial questions about, for example, finding a minimal sufficient
+collection of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets.
 It is easy to show that examples of topological origin
-(e.g.\ categories whose morphisms are maps into spaces or decorated balls), 
+(e.g.\ categories whose morphisms are maps into spaces or decorated balls, or bordism categories), 
 satisfy our axioms.
-For examples of a more purely algebraic origin, one would typically need the combinatorial
+To show that examples of a more purely algebraic origin satisfy our axioms, 
+one would typically need the combinatorial
 results that we have avoided here.
 
 See \S\ref{n-cat-names} for a discussion of $n$-category terminology.
@@ -30,6 +32,15 @@
 
 \medskip
 
+The axioms for an $n$-category are spread throughout this section.
+Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, \ref{nca-boundary}, \ref{axiom:composition},  \ref{nca-assoc}, \ref{axiom:product} and \ref{axiom:extended-isotopies}; for an $A_\infty$ $n$-category, we replace Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}.
+
+Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms 
+for $k{-}1$-morphisms.
+So readers who prefer things to be presented in a strictly logical order should read this subsection $n$ times, first imagining that $k=0$, then that $k=1$, and so on until they reach $k=n$.
+
+\medskip
+
 There are many existing definitions of $n$-categories, with various intended uses.
 In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$.
 Generally, these sets are indexed by instances of a certain typical shape. 
@@ -49,13 +60,9 @@
 We {\it do not} assume that it is equipped with a 
 preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below.
 
-The axioms for an $n$-category are spread throughout this section.
-Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, \ref{nca-boundary}, \ref{axiom:composition},  \ref{nca-assoc}, \ref{axiom:product} and \ref{axiom:extended-isotopies}; for an $A_\infty$ $n$-category, we replace Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}.
-
-
 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on 
 the boundary), we want a corresponding
-bijection of sets $f:\cC(X)\to \cC(Y)$.
+bijection of sets $f:\cC_k(X)\to \cC_k(Y)$.
 (This will imply ``strong duality", among other things.) Putting these together, we have
 
 \begin{axiom}[Morphisms]
@@ -103,7 +110,8 @@
 Morphisms are modeled on balls, so their boundaries are modeled on spheres.
 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
+At first it might seem that we need another axiom 
+(more specifically, additional data) for this, but in fact once we have
 all the axioms in this subsection for $0$ through $k-1$ we can use a colimit
 construction, as described in \S\ref{ss:ncat-coend} below, to extend $\cC_{k-1}$
 to spheres (and any other manifolds):
@@ -197,20 +205,30 @@
 The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}.
 %\nn{we might want a more official looking proof...}
 
-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 do not insist on surjectivity of the gluing map, since this is not satisfied by all of the examples
+we are trying to axiomatize.
+If our $k$-morphisms $\cC(X)$ are labeled cell complexes embedded in $X$ (c.f. Example \ref{ex:traditional-n-categories} below), then a $k$-morphism is
+in the image of the gluing map precisely which the cell complex is in general position
+with respect to $E$. On the other hand, in categories based on maps to a target space (c.f. Example \ref{ex:maps-to-a-space} below) the gluing map is always surjective
+
+If $S$ is a 0-sphere (the case $k=1$ above), then $S$ can be identified with the {\it disjoint} union
+of two 0-balls $B_1$ and $B_2$ and the colimit construction $\cl{\cC}(S)$ can be identified
+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.
 
 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 $\cC(X)_E = \bd^{-1}(\cl{\cC}(\bd X)_E)$.
+as above, then we define $\cC(X)\trans E = \bd^{-1}(\cl{\cC}(\bd X)\trans E)$.
 
-We will call the projection $\cl{\cC}(S)_E \to \cC(B_i)$
+We will call the projection $\cl{\cC}(S)\trans E \to \cC(B_i)$
 a {\it restriction} map and write $\res_{B_i}(a)$
-(or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)_E$.
+(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"
 the boundary maps of Axiom \ref{nca-boundary} above,
 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 $\cC(X)_E \to \cC(B_i)$
+In particular, we have restriction maps $\cC(X)\trans E \to \cC(B_i)$
 ($i = 1, 2$, notation from previous paragraph).
 These restriction maps can be thought of as 
 domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$.
@@ -229,11 +247,11 @@
 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}).
 Let $E = \bd Y$, which is a $k{-}2$-sphere.
 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$.
-We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$.
-Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. 
+We have restriction (domain or range) maps $\cC(B_i)\trans E \to \cC(Y)$.
+Let $\cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E$ denote the fibered product of these two maps. 
 We have a map
 \[
-	\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E
+	\gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B)\trans E
 \]
 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions
 to the intersection of the boundaries of $B$ and $B_i$.
@@ -269,16 +287,16 @@
 \caption{An example of strict associativity.}\label{blah6}\end{figure}
 
 We'll use the notation  $a\bullet b$ for the glued together field $\gl_Y(a, b)$.
-In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ 
-a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$.
+In the other direction, we will call the projection from $\cC(B)\trans E$ to $\cC(B_i)\trans E$ 
+a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)\trans E$.
 %Compositions of boundary and restriction maps will also be called restriction maps.
 %For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a
 %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 
+We will write $\cC(B)\trans Y$ for the image of $\gl_Y$ in $\cC(B)$.
+We will call elements of $\cC(B)\trans Y$ morphisms which are 
 ``splittable along $Y$'' or ``transverse to $Y$''.
-We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$.
+We have $\cC(B)\trans Y \sub \cC(B)\trans E \sub \cC(B)$.
 
 More generally, let $\alpha$ be a splitting of $X$ into smaller balls.
 Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from 
@@ -537,8 +555,9 @@
 This axiom needs to be strengthened to force product morphisms to act as the identity.
 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball.
 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 $Y\times J$ with boundary entirely pinched.)
+Let $s_{Y,J}: X\cup_Y (Y\times J) \to X$ be a collaring homeomorphism
+(see the end of \S\ref{ss:syst-o-fields}).
+Here we use $Y\times J$ with boundary entirely pinched.
 We define a map
 \begin{eqnarray*}
 	\psi_{Y,J}: \cC(X) &\to& \cC(X) \\
@@ -680,7 +699,7 @@
 \begin{itemize}
 \item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms});
 \item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary});
-\item ``composition'' or ``gluing'' maps $\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B_1\cup_Y B_2)_E$ (Axiom \ref{axiom:composition});
+\item ``composition'' or ``gluing'' maps $\gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B_1\cup_Y B_2)\trans E$ (Axiom \ref{axiom:composition});
 \item ``product'' or ``identity'' maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product});
 \item if enriching in an auxiliary category, additional structure on $\cC_n(X; c)$;
 \item in the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families}).
@@ -802,16 +821,18 @@
 }
 
 
-\begin{example}[The bordism $n$-category, ordinary version]
+\begin{example}[The bordism $n$-category of $d$-manifolds, ordinary version]
 \label{ex:bord-cat}
 \rm
 \label{ex:bordism-category}
-For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional
-submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse
-to $\bd X$.
-For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes (rel boundary) of such $n$-dimensional submanifolds;
+For a $k$-ball $X$, $k<n$, define $\Bord^{n,d}(X)$ to be the set of all $(d{-}n{+}k)$-dimensional PL
+submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W = W \cap \bd X \times \Real^\infty$.
+For an $n$-ball $X$ define $\Bord^{n,d}(X)$ to be homeomorphism classes (rel boundary) of such $d$-dimensional submanifolds;
 we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism
 $W \to W'$ which restricts to the identity on the boundary.
+For $n=1$ we have the familiar bordism 1-category of $d$-manifolds.
+The case $n=d$ captures the $n$-categorical nature of bordisms.
+The case $n > 2d$ captures the full symmetric monoidal $n$-category structure.
 \end{example}
 
 %\nn{the next example might be an unnecessary distraction.  consider deleting it.}
@@ -872,15 +893,14 @@
 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 $\pi^\infty_{\leq 0}(T) \iso C_* T$.
 
-\begin{example}[The bordism $n$-category, $A_\infty$ version]
+\begin{example}[The bordism $n$-category of $d$-manifolds, $A_\infty$ version]
 \rm
 \label{ex:bordism-category-ainf}
-As in Example \ref{ex:bord-cat}, for $X$ a $k$-ball, $k<n$, we define $\Bord^{n,\infty}(X)$
-to be the set of all $k$-dimensional
-submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse
-to $\bd X$.
+As in Example \ref{ex:bord-cat}, for $X$ a $k$-ball, $k<n$, we define $\Bord^{n,d}_\infty(X)$
+to be the set of all $(d{-}n{+}k)$-dimensional
+submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W = W \cap \bd X \times \Real^\infty$.
 For an $n$-ball $X$ with boundary condition $c$ 
-define $\Bord^{n,\infty}(X; c)$ to be the space of all $k$-dimensional
+define $\Bord^{n,d}_\infty(X; c)$ to be the space of all $d$-dimensional
 submanifolds $W$ of $X\times \Real^\infty$ such that 
 $W$ coincides with $c$ at $\bd X \times \Real^\infty$.
 (The topology on this space is induced by ambient isotopy rel boundary.
@@ -967,8 +987,11 @@
 and we  will define $\cl{\cC}(W)$ as a suitable colimit 
 (or homotopy colimit in the $A_\infty$ case) of this functor. 
 We'll later give a more explicit description of this colimit.
-In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-balls with boundary data), 
-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).
+In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain 
+complexes to $n$-balls with boundary data), 
+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).
 
 Recall (Definition \ref{defn:gluing-decomposition}) that a {\it ball decomposition} of $W$ is a 
 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls
@@ -985,7 +1008,8 @@
 
 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement
 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$
-with $\du_b Y_b = M_i$ for some $i$.
+with $\du_b Y_b = M_i$ for some $i$,
+and with $M_0,\ldots, M_i$ each being a disjoint union of balls.
 
 \begin{defn}
 The poset $\cell(W)$ has objects the permissible decompositions of $W$, 
@@ -1016,7 +1040,7 @@
 	\psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl
 \end{equation}
 where the restrictions to the various pieces of shared boundaries amongst the cells
-$X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells).
+$X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n{-}1$-cells).
 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$.
 \end{defn}
 
@@ -1266,8 +1290,8 @@
 \end{lem}
 Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}.
 
-Let $\cl\cM(H)_E$ denote the image of $\gl_E$.
-We will refer to elements of $\cl\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". 
+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 $E$". 
 
 \begin{lem}[Module to category restrictions]
 {For each marked $k$-hemisphere $H$ there is a restriction map
@@ -1331,10 +1355,10 @@
 and $Y = X\cap M'$ is a $k{-}1$-ball.
 Let $E = \bd Y$, which is a $k{-}2$-sphere.
 We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$.
-Let $\cC(X)_E \times_{\cC(Y)} \cM(M')_E$ denote the fibered product of these two maps. 
+Let $\cC(X)\trans E \times_{\cC(Y)} \cM(M')_E$ denote the fibered product of these two maps. 
 Then (axiom) we have a map
 \[
-	\gl_Y :\cC(X)_E \times_{\cC(Y)} \cM(M')_E \to \cM(M)_E
+	\gl_Y :\cC(X)\trans E \times_{\cC(Y)} \cM(M')_E \to \cM(M)_E
 \]
 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions
 to the intersection of the boundaries of $X$ and $M'$.
diff -r bd0d5b2155a7 -r db9d3a27647a text/tqftreview.tex
--- a/text/tqftreview.tex	Wed Mar 30 08:03:22 2011 -0700
+++ b/text/tqftreview.tex	Wed Mar 30 08:03:27 2011 -0700
@@ -100,6 +100,8 @@
 maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two
 copies of $Y$ in $\bd X$.
 Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps.
+(When $X$ is a disjoint union $X_1\du X_2$ the equalizer is the same as the fibered product
+$\cC_k(X_1)\times_{\cC(Y)} \cC_k(X_2)$.)
 Then (here's the axiom/definition part) there is an injective ``gluing" map
 \[
 	\Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) ,
@@ -213,17 +215,29 @@
 
 \medskip
 
-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 $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$.
+Extend the product structure on $Y\times I$ to a bicollar neighborhood of 
+$Y$ inside $M \cup (Y\times I)$.
+We call a homeomorphism
+\[
+	f: M \cup (Y\times I) \to M
+\]
+a {\it collaring homeomorphism} if $f$ is the identity outside of the bicollar
+and $f$ preserves the fibers of the bicollar.
+
+Using the functoriality and product field properties above, together
+with collaring homeomorphisms, we can define 
+{\it collar maps} $\cC(M)\to \cC(M)$.
+Let $M$ and $Y \sub \bd M$ be as above.
 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is splittable along $\bd Y$.
 Let $c$ be $x$ restricted to $Y$.
-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 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
@@ -231,54 +245,6 @@
 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.}
-
-\nn{remark that if top dimensional fields are not already linear
-then we will soon linearize them(?)}
-
-For top dimensional ($n$-dimensional) manifolds, we're actually interested
-in the linearized space of fields.
-By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is
-the vector space of finite
-linear combinations of fields on $X$.
-If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$.
-Thus the restriction (to boundary) maps are well defined because we never
-take linear combinations of fields with differing boundary conditions.
-
-In some cases we don't linearize the default way; instead we take the
-spaces $\lf(X; a)$ to be part of the data for the system of fields.
-In particular, for fields based on linear $n$-category pictures we linearize as follows.
-Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by
-obvious relations on 0-cell labels.
-More specifically, let $L$ be a cell decomposition of $X$
-and let $p$ be a 0-cell of $L$.
-Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that
-$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$.
-Then the subspace $K$ is generated by things of the form
-$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader
-to infer the meaning of $\alpha_{\lambda c + d}$.
-Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms.
-
-\nn{Maybe comment further: if there's a natural basis of morphisms, then no need;
-will do something similar below; in general, whenever a label lives in a linear
-space we do something like this; ? say something about tensor
-product of all the linear label spaces?  Yes:}
-
-For top dimensional ($n$-dimensional) manifolds, we linearize as follows.
-Define an ``almost-field" to be a field without labels on the 0-cells.
-(Recall that 0-cells are labeled by $n$-morphisms.)
-To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism
-space determined by the labeling of the link of the 0-cell.
-(If the 0-cell were labeled, the label would live in this space.)
-We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell).
-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 \texorpdfstring{$n$}{n}-categories}
@@ -425,7 +391,7 @@
 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:
-if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$
+if $B = B' \cup B''$, $x\in U(B')$, and $r\in \cC(B'')$, then $x\bullet r \in U(B)$
 \end{enumerate}
 \end{defn}
 See \cite{kw:tqft} for further details.