--- a/blob1.tex Sun Jul 04 13:15:03 2010 -0600
+++ b/blob1.tex Sun Jul 04 23:32:48 2010 -0600
@@ -16,7 +16,7 @@
\maketitle
-[revision $\ge$ 414; $\ge$ 3 July 2010]
+[revision $\ge$ 417; $\ge$ 4 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-\item[2] (S) needs explanation that this will be superseded by the n-cat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/text/a_inf_blob.tex Sun Jul 04 13:15:03 2010 -0600
+++ b/text/a_inf_blob.tex Sun Jul 04 23:32:48 2010 -0600
@@ -44,7 +44,7 @@
\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'
+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 \cl{\bc_*(F; C)}(Y)
--- a/text/appendixes/comparing_defs.tex Sun Jul 04 13:15:03 2010 -0600
+++ b/text/appendixes/comparing_defs.tex Sun Jul 04 23:32:48 2010 -0600
@@ -294,4 +294,4 @@
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 Sun Jul 04 13:15:03 2010 -0600
+++ b/text/appendixes/smallblobs.tex Sun Jul 04 23:32:48 2010 -0600
@@ -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}
--- a/text/basic_properties.tex Sun Jul 04 13:15:03 2010 -0600
+++ b/text/basic_properties.tex Sun Jul 04 23:32:48 2010 -0600
@@ -89,7 +89,7 @@
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 $T(B') \sub T(b)$.
Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models),
-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}
+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
@@ -111,7 +111,7 @@
}
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/evmap.tex Sun Jul 04 13:15:03 2010 -0600
+++ b/text/evmap.tex Sun Jul 04 23:32:48 2010 -0600
@@ -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 \\
--- a/text/hochschild.tex Sun Jul 04 13:15:03 2010 -0600
+++ b/text/hochschild.tex Sun Jul 04 23:32:48 2010 -0600
@@ -107,7 +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
@@ -130,8 +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 $\cP_i$ 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
$$\cP_*(M) \quismto \coinv(F_*).$$
--- a/text/intro.tex Sun Jul 04 13:15:03 2010 -0600
+++ b/text/intro.tex Sun Jul 04 23:32:48 2010 -0600
@@ -38,7 +38,7 @@
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
@@ -61,10 +61,10 @@
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 Theorem \ref{thm: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}.
@@ -115,7 +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.
@@ -436,7 +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.
--- a/text/ncat.tex Sun Jul 04 13:15:03 2010 -0600
+++ b/text/ncat.tex Sun Jul 04 23:32:48 2010 -0600
@@ -271,7 +271,7 @@
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.
@@ -667,7 +667,7 @@
\begin{example}[Maps to a space]
\rm
\label{ex:maps-to-a-space}%
-Fix a `target space' $T$, any topological space.
+Fix a ``target space" $T$, any 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 $T$.
@@ -704,12 +704,12 @@
\nn{need to say something about fundamental classes, or choose $\alpha$ carefully}
\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
@@ -725,7 +725,7 @@
to be the dual Hilbert space $A(X\times F; c)$.
\nn{refer elsewhere for details?}
-Recall we described a system of fields and local relations based on a `traditional $n$-category'
+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.
@@ -794,15 +794,15 @@
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"
+\nn{``cofibrant replacement"?}
of the topological $n$-category.
\todo{Say more here!}
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 $\pi^\infty_{\leq 0}(T) \iso C_* T$.
@@ -895,12 +895,12 @@
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.
@@ -909,7 +909,7 @@
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
\[
W = \bigcup_a X_a ,
\]
@@ -938,7 +938,7 @@
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.
@@ -1740,7 +1740,7 @@
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