--- a/blob1.tex Sun Jun 06 09:49:06 2010 -0700
+++ b/blob1.tex Sun Jun 06 09:49:57 2010 -0700
@@ -20,8 +20,6 @@
\textbf{Draft version, read with caution.}
-\nn{maybe to do: add appendix on various versions of acyclic models}
-
\paragraph{To do list}
\begin{itemize}
\item[1] (K) tweak intro
@@ -43,10 +41,22 @@
\item[A] may need to weaken statement to get boundaries working (K) finish
\item[B] (S) look at this, decide what to keep
-\item Make clear exactly what counts as a "blob diagram", and search for
-"blob diagram"
+\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.
+I suppose our work could be interpreted as a alternative proof of cobordism hypothesis, but we
+don't emphasize that at the moment.
+On the other hand, I'm happy to do Chris a favor by citing this stuff.}
+
+\item Make clear exactly what counts as a ``blob diagram", and search for
+``blob diagram"
\item Say something about stabilizing an $n$-category (centre), taking the top $k$ levels of a category, and the stabilization hypothesis?
+
+\item say something about starting with semisimple n-cat (trivial?? not trivial?)
+
+\item maybe to do: add appendix on various versions of acyclic models
+
+
\end{itemize}
\tableofcontents
--- a/preamble.tex Sun Jun 06 09:49:06 2010 -0700
+++ b/preamble.tex Sun Jun 06 09:49:57 2010 -0700
@@ -70,6 +70,7 @@
\newtheorem{question}{Question}
\newtheorem{property}{Property}
\newtheorem{axiom}{Axiom}
+\newtheorem{module-axiom}{Module Axiom}
%\newenvironment{axiom-numbered}[2]{\textbf{Axiom #1(#2)}\it}{}
%\newenvironment{preliminary-axiom}[2]{\textbf{Axiom #1 [preliminary] (#2)}\it}{}
\newtheorem{example}[prop]{Example}
--- a/text/a_inf_blob.tex Sun Jun 06 09:49:06 2010 -0700
+++ b/text/a_inf_blob.tex Sun Jun 06 09:49:57 2010 -0700
@@ -16,7 +16,8 @@
\medskip
An important technical tool in the proofs of this section is provided by the idea of `small blobs'.
-Fix $\cU$, an open cover of $M$. Define the `small blob complex' $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$.
+Fix $\cU$, an open cover of $M$.
+Define the `small blob complex' $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$.
\nn{KW: We need something a little stronger: Every blob diagram (even a 0-blob diagram) is splittable into pieces which are small w.r.t.\ $\cU$.
If field have potentially large coupons/boxes, then this is a non-trivial constraint.
On the other hand, we could probably get away with ignoring this point.
@@ -46,11 +47,14 @@
\nn{need to settle on notation; proof and statement are inconsistent}
\begin{thm} \label{product_thm}
-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
+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
\begin{equation*}
C^{\times F}(B) = \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 $C^{\times F}$:
+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 $C^{\times F}$:
\begin{align*}
\cB_*(Y \times F, C) & \htpy \cB_*(Y, C^{\times F})
\end{align*}
@@ -305,7 +309,8 @@
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 $J$.
-We define a map $\psi:\cT\to \bc_*(X)$. On filtration degree zero summands it is given
+We define a map $\psi:\cT\to \bc_*(X)$.
+On filtration degree zero summands it is given
by gluing the pieces together to get a blob diagram on $X$.
On filtration degree 1 and greater $\psi$ is zero.
@@ -353,13 +358,18 @@
To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$.
\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 $T$.
+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 $T$.
$$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$
\end{thm}
\begin{rem}
-\nn{This just isn't true, Lurie doesn't do this! I just heard this from Ricardo...}
-\nn{KW: Are you sure about that?}
-Lurie has shown in \cite{0911.0018} that the topological chiral homology of an $n$-manifold $M$ with coefficients in \nn{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. This extra hypothesis is not surprising, in view of the idea that an $E_n$ algebra is roughly equivalent data as an $A_\infty$ $n$-category which is trivial at all but the topmost level.
+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.
+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 all but the topmost level.
+Ricardo Andrade also told us about a similar result.
\end{rem}
\nn{proof is again similar to that of Theorem \ref{product_thm}. should probably say that explicitly}
--- a/text/appendixes/comparing_defs.tex Sun Jun 06 09:49:06 2010 -0700
+++ b/text/appendixes/comparing_defs.tex Sun Jun 06 09:49:57 2010 -0700
@@ -12,14 +12,18 @@
\subsection{$1$-categories over $\Set$ or $\Vect$}
\label{ssec:1-cats}
Given a topological $1$-category $\cX$ we construct a $1$-category in the conventional sense, $c(\cX)$.
-This construction is quite straightforward, but we include the details for the sake of completeness, because it illustrates the role of structures (e.g. orientations, spin structures, etc) on the underlying manifolds, and
+This construction is quite straightforward, but we include the details for the sake of completeness,
+because it illustrates the role of structures (e.g. orientations, spin structures, etc)
+on the underlying manifolds, and
to shed some light on the $n=2$ case, which we describe in \S \ref{ssec:2-cats}.
Let $B^k$ denote the \emph{standard} $k$-ball.
-Let the objects of $c(\cX)$ be $c(\cX)^0 = \cX(B^0)$ and the morphisms of $c(\cX)$ be $c(\cX)^1 = \cX(B^1)$. The boundary and restriction maps of $\cX$ give domain and range maps from $c(\cX)^1$ to $c(\cX)^0$.
+Let the objects of $c(\cX)$ be $c(\cX)^0 = \cX(B^0)$ and the morphisms of $c(\cX)$ be $c(\cX)^1 = \cX(B^1)$.
+The boundary and restriction maps of $\cX$ give domain and range maps from $c(\cX)^1$ to $c(\cX)^0$.
Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$.
-Define composition in $c(\cX)$ to be the induced map $c(\cX)^1\times c(\cX)^1 \to c(\cX)^1$ (defined only when range and domain agree).
+Define composition in $c(\cX)$ to be the induced map $c(\cX)^1\times c(\cX)^1 \to c(\cX)^1$
+(defined only when range and domain agree).
By isotopy invariance in $\cX$, any other choice of homeomorphism gives the same composition rule.
Also by isotopy invariance, composition is strictly associative.
@@ -27,9 +31,12 @@
By extended isotopy invariance in $\cX$, this has the expected properties of an identity morphism.
-If the underlying manifolds for $\cX$ have further geometric structure, then we obtain certain functors. The base case is for oriented manifolds, where we obtain no extra algebraic data.
+If the underlying manifolds for $\cX$ have further geometric structure, then we obtain certain functors.
+The base case is for oriented manifolds, where we obtain no extra algebraic data.
-For 1-categories based on unoriented manifolds (somewhat confusingly, we're thinking of being unoriented as requiring extra data beyond being oriented, namely the identification between the orientations), there is a map $*:c(\cX)^1\to c(\cX)^1$
+For 1-categories based on unoriented manifolds (somewhat confusingly, we're thinking of being
+unoriented as requiring extra data beyond being oriented, namely the identification between the orientations),
+there is a map $*:c(\cX)^1\to c(\cX)^1$
coming from $\cX$ applied to an orientation-reversing homeomorphism (unique up to isotopy)
from $B^1$ to itself.
Topological properties of this homeomorphism imply that
@@ -71,7 +78,8 @@
\medskip
-The compositions of the constructions above, $$\cX\to c(\cX)\to t(c(\cX))$$ and $$C\to t(C)\to c(t(C)),$$ give back
+The compositions of the constructions above, $$\cX\to c(\cX)\to t(c(\cX))$$
+and $$C\to t(C)\to c(t(C)),$$ give back
more or less exactly the same thing we started with.
As we will see below, for $n>1$ the compositions yield a weaker sort of equivalence.
@@ -169,7 +177,8 @@
We first collapse the red region, then remove a product morphism from the boundary,
We define horizontal composition of 2-morphisms as shown in Figure \ref{fzo5}.
-It is not hard to show that this is independent of the arbitrary (left/right) choice made in the definition, and that it is associative.
+It is not hard to show that this is independent of the arbitrary (left/right)
+choice made in the definition, and that it is associative.
\begin{figure}[t]
\begin{equation*}
\mathfig{.83}{tempkw/zo5}
@@ -191,22 +200,48 @@
\subsection{$A_\infty$ $1$-categories}
\label{sec:comparing-A-infty}
-In this section, we make contact between the usual definition of an $A_\infty$ algebra and our definition of a topological $A_\infty$ algebra, from Definition \ref{defn:topological-Ainfty-category}.
+In this section, we make contact between the usual definition of an $A_\infty$ algebra
+and our definition of a topological $A_\infty$ algebra, from Definition \ref{defn:topological-Ainfty-category}.
-We begin be restricting the data of a topological $A_\infty$ algebra to the standard interval $[0,1]$, which we can alternatively characterise as:
+We begin be restricting the data of a topological $A_\infty$ algebra to the standard interval $[0,1]$,
+which we can alternatively characterise as:
\begin{defn}
-A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$, and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with
+A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$,
+and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with
\begin{itemize}
\item an action of the operad of $\Obj(\cC)$-labeled cell decompositions
\item and a compatible action of $\CD{[0,1]}$.
\end{itemize}
\end{defn}
-Here the operad of cell decompositions of $[0,1]$ has operations indexed by a finite set of points $0 < x_1< \cdots < x_k < 1$, cutting $[0,1]$ into subintervals. An $X$-labeled cell decomposition labels $\{0, x_1, \ldots, x_k, 1\}$ by $X$. Given two cell decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$, we can compose them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$. In the $X$-labeled case, we insist that the appropriate labels match up. Saying we have an action of this operad means that for each labeled cell decomposition $0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain map $$\cC_{a_0,a_1} \tensor \cdots \tensor \cC_{a_k,a_{k+1}} \to \cC(a_0,a_{k+1})$$ and these chain maps compose exactly as the cell decompositions.
-An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which is supported on the subintervals determined by $\pi$, then the two possible operations (glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms separately to the subintervals, then glue) commute (as usual, up to a weakly unique homotopy).
+Here the operad of cell decompositions of $[0,1]$ has operations indexed by a finite set of
+points $0 < x_1< \cdots < x_k < 1$, cutting $[0,1]$ into subintervals.
+An $X$-labeled cell decomposition labels $\{0, x_1, \ldots, x_k, 1\}$ by $X$.
+Given two cell decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$, we can compose
+them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points
+of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$.
+In the $X$-labeled case, we insist that the appropriate labels match up.
+Saying we have an action of this operad means that for each labeled cell decomposition
+$0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain
+map $$\cC_{a_0,a_1} \tensor \cdots \tensor \cC_{a_k,a_{k+1}} \to \cC(a_0,a_{k+1})$$ and these
+chain maps compose exactly as the cell decompositions.
+An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad
+if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which
+is supported on the subintervals determined by $\pi$, then the two possible operations
+(glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms
+separately to the subintervals, then glue) commute (as usual, up to a weakly unique homotopy).
-Translating between this notion and the usual definition of an $A_\infty$ category is now straightforward. To restrict to the standard interval, define $\cC_{a,b} = \cC([0,1];a,b)$. Given a cell decomposition $0 < x_1< \cdots < x_k < 1$, we use the map (suppressing labels)
+Translating between this notion and the usual definition of an $A_\infty$ category is now straightforward.
+To restrict to the standard interval, define $\cC_{a,b} = \cC([0,1];a,b)$.
+Given a cell decomposition $0 < x_1< \cdots < x_k < 1$, we use the map (suppressing labels)
$$\cC([0,1])^{\tensor k+1} \to \cC([0,x_1]) \tensor \cdots \tensor \cC[x_k,1] \to \cC([0,1])$$
-where the factors of the first map are induced by the linear isometries $[0,1] \to [x_i, x_{i+1}]$, and the second map is just gluing. The action of $\CD{[0,1]}$ carries across, and is automatically compatible. Going the other way, we just declare $\cC(J;a,b) = \cC_{a,b}$, pick a diffeomorphism $\phi_J : J \isoto [0,1]$ for every interval $J$, define the gluing map $\cC(J_1) \tensor \cC(J_2) \to \cC(J_1 \cup J_2)$ by the first applying the cell decomposition map for $0 < \frac{1}{2} < 1$, then the self-diffeomorphism of $[0,1]$ given by $\frac{1}{2} (\phi_{J_1} \cup (1+ \phi_{J_2})) \circ \phi_{J_1 \cup J_2}^{-1}$. You can readily check that this gluing map is associative on the nose. \todo{really?}
+where the factors of the first map are induced by the linear isometries $[0,1] \to [x_i, x_{i+1}]$, and the second map is just gluing.
+The action of $\CD{[0,1]}$ carries across, and is automatically compatible.
+Going the other way, we just declare $\cC(J;a,b) = \cC_{a,b}$, pick a diffeomorphism
+$\phi_J : J \isoto [0,1]$ for every interval $J$, define the gluing map
+$\cC(J_1) \tensor \cC(J_2) \to \cC(J_1 \cup J_2)$ by the first applying
+the cell decomposition map for $0 < \frac{1}{2} < 1$, then the self-diffeomorphism of $[0,1]$
+given by $\frac{1}{2} (\phi_{J_1} \cup (1+ \phi_{J_2})) \circ \phi_{J_1 \cup J_2}^{-1}$.
+You can readily check that this gluing map is associative on the nose. \todo{really?}
%First recall the \emph{coloured little intervals operad}. Given a set of labels $\cL$, the operations are indexed by \emph{decompositions of the interval}, each of which is a collection of disjoint subintervals $\{(a_i,b_i)\}_{i=1}^k$ of $[0,1]$, along with a labeling of the complementary regions by $\cL$, $\{l_0, \ldots, l_k\}$. Given two decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$ such that $l^{(1)}_{m-1} = l^{(2)}_0$ and $l^{(1)}_{m} = l^{(2)}_{k^{(2)}}$, we can form a new decomposition by inserting the intervals of $\cJ^{(2)}$ linearly inside the $m$-th interval of $\cJ^{(1)}$. We call the resulting decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$.
@@ -235,7 +270,15 @@
%\end{enumerate}
%\end{defn}
-From a topological $A_\infty$ category on $[0,1]$ $\cC$ we can produce a `conventional' $A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}. We'll just describe the algebra case (that is, a category with only one object), as the modifications required to deal with multiple objects are trivial. Define $A = \cC$ as a chain complex (so $m_1 = d$). Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$. To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms of $[0,1]$ that interpolates linearly between the identity and the piecewise linear diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define
+From a topological $A_\infty$ category on $[0,1]$ $\cC$ we can produce a `conventional'
+$A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}.
+We'll just describe the algebra case (that is, a category with only one object),
+as the modifications required to deal with multiple objects are trivial.
+Define $A = \cC$ as a chain complex (so $m_1 = d$).
+Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$.
+To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms
+of $[0,1]$ that interpolates linearly between the identity and the piecewise linear
+diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define
\begin{equation*}
m_3(a,b,c) = ev(\phi_3, m_2(m_2(a,b), c)).
\end{equation*}
@@ -250,4 +293,5 @@
\end{align*}
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
+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
--- a/text/appendixes/famodiff.tex Sun Jun 06 09:49:06 2010 -0700
+++ b/text/appendixes/famodiff.tex Sun Jun 06 09:49:57 2010 -0700
@@ -39,7 +39,9 @@
Furthermore, if $Q$ is a convex linear subpolyhedron of $\bd P$ and $f$ restricted to $Q$
has support $S' \subset X$, then
$F: (I\times Q)\times X\to T$ also has support $S'$.
-\item Suppose both $X$ and $T$ are smooth manifolds, metric spaces, or PL manifolds, and let $\cX$ denote the subspace of $\Maps(X \to T)$ consisting of immersions or of diffeomorphisms (in the smooth case), bi-Lipschitz homeomorphisms (in the metric case), or PL homeomorphisms (in the PL case).
+\item Suppose both $X$ and $T$ are smooth manifolds, metric spaces, or PL manifolds, and
+let $\cX$ denote the subspace of $\Maps(X \to T)$ consisting of immersions or of diffeomorphisms (in the smooth case),
+bi-Lipschitz homeomorphisms (in the metric case), or PL homeomorphisms (in the PL case).
If $f$ is smooth, Lipschitz or PL, as appropriate, and $f(p, \cdot):X\to T$ is in $\cX$ for all $p \in P$
then $F(t, p, \cdot)$ is also in $\cX$ for all $t\in I$ and $p\in P$.
\end{enumerate}
@@ -128,7 +130,10 @@
\right) .
\end{equation}
-This completes the definition of $u: I \times P \times X \to P$. The formulas above are consistent: for $p$ at the boundary between a $k-j$-handle and a $k-(j+1)$-handle the corresponding expressions in Equation \eqref{eq:u} agree, since one of the normal coordinates becomes $0$ or $1$.
+This completes the definition of $u: I \times P \times X \to P$.
+The formulas above are consistent: for $p$ at the boundary between a $k-j$-handle and
+a $k-(j+1)$-handle the corresponding expressions in Equation \eqref{eq:u} agree,
+since one of the normal coordinates becomes $0$ or $1$.
Note that if $Q\sub \bd P$ is a convex linear subpolyhedron, then $u(I\times Q\times X) \sub Q$.
\medskip
@@ -208,7 +213,9 @@
\end{proof}
\begin{lemma} \label{extension_lemma_c}
-Let $\cX_*$ be any of $C_*(\Maps(X \to T))$ or singular chains on the subspace of $\Maps(X\to T)$ consisting of immersions, diffeomorphisms, bi-Lipschitz homeomorphisms or PL homeomorphisms.
+Let $\cX_*$ be any of $C_*(\Maps(X \to T))$ or singular chains on the
+subspace of $\Maps(X\to T)$ consisting of immersions, diffeomorphisms,
+bi-Lipschitz homeomorphisms or PL homeomorphisms.
Let $G_* \subset \cX_*$ denote the chains adapted to an open cover $\cU$
of $X$.
Then $G_*$ is a strong deformation retract of $\cX_*$.
--- a/text/basic_properties.tex Sun Jun 06 09:49:06 2010 -0700
+++ b/text/basic_properties.tex Sun Jun 06 09:49:57 2010 -0700
@@ -3,9 +3,15 @@
\section{Basic properties of the blob complex}
\label{sec:basic-properties}
-In this section we complete the proofs of Properties 1-5. Throughout the paper, where possible, we prove results using Properties 1-5, rather than the actual definition of blob homology. This allows the possibility of future improvements to or alternatives on our definition. In fact, we hope that there may be a characterisation of blob homology in terms of Properties 1-5, but at this point we are unaware of one.
+In this section we complete the proofs of Properties 2-4.
+Throughout the paper, where possible, we prove results using Properties 1-4,
+rather than the actual definition of blob homology.
+This allows the possibility of future improvements to or alternatives on our definition.
+In fact, we hope that there may be a characterisation of blob homology in
+terms of Properties 1-4, but at this point we are unaware of one.
-Recall Property \ref{property:disjoint-union}, that there is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$.
+Recall Property \ref{property:disjoint-union},
+that there is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$.
\begin{proof}[Proof of Property \ref{property:disjoint-union}]
Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them
@@ -15,7 +21,9 @@
In the other direction, any blob diagram on $X\du Y$ is equal (up to sign)
to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines
a pair of blob diagrams on $X$ and $Y$.
-These two maps are compatible with our sign conventions. (We follow the usual convention for tensors products of complexes, as in e.g. \cite{MR1438306}: $d(a \tensor b) = da \tensor b + (-1)^{\deg(a)} a \tensor db$.)
+These two maps are compatible with our sign conventions.
+(We follow the usual convention for tensors products of complexes,
+as in e.g. \cite{MR1438306}: $d(a \tensor b) = da \tensor b + (-1)^{\deg(a)} a \tensor db$.)
The two maps are inverses of each other.
\end{proof}
@@ -43,7 +51,8 @@
Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to
the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$.
\end{proof}
-This proves Property \ref{property:contractibility} (the second half of the statement of this Property was immediate from the definitions).
+This proves Property \ref{property:contractibility} (the second half of the
+statement of this Property was immediate from the definitions).
Note that even when there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy
equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$.
@@ -92,7 +101,8 @@
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
-$X$ to get blob diagrams on $X\sgl$. This proves Property \ref{property:gluing-map}, which we restate here in more detail.
+$X$ 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
@@ -106,5 +116,3 @@
This map is very far from being an isomorphism, even on homology.
We fix this deficit in Section \ref{sec:gluing} below.
-
-As we pointed out earlier, Property \ref{property:skein-modules} is immediate from the definitions.
--- a/text/blobdef.tex Sun Jun 06 09:49:06 2010 -0700
+++ b/text/blobdef.tex Sun Jun 06 09:49:57 2010 -0700
@@ -57,9 +57,12 @@
(but keeping the blob label $u$).
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 half of Property \ref{property:contractibility}.
+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
+half of Property \ref{property:contractibility}.
-Next, we want the vector space $\bc_2(X)$ to capture `the space of all relations (redundancies, syzygies) among the
+Next, we want the vector space $\bc_2(X)$ to capture `the space of all relations
+(redundancies, syzygies) among the
local relations encoded in $\bc_1(X)$'.
More specifically, a $2$-blob diagram, comes in one of two types, disjoint and nested.
A disjoint 2-blob diagram consists of
@@ -85,7 +88,8 @@
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 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 $r' \in \cC(B_2 \setminus B_1; c_1, c_2)$
+(for some $c_1 \in \cC(\bdy B_1)$ and $c_2 \in \cC(\bdy B_2)$).
\item A field $r \in \cC(X \setminus B_2; c_2)$.
\item A local relation field $u \in U(B_1; c_1)$.
\end{itemize}
@@ -114,7 +118,10 @@
\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). \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}
+(rather than a new, linearly independent 2-blob diagram).
+\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}
Now for the general case.
A $k$-blob diagram consists of
@@ -158,7 +165,8 @@
\left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) .
\]
Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above.
-The index $\overline{c}$ runs over all boundary conditions, again as described above and $j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$.
+The index $\overline{c}$ runs over all boundary conditions, again as described above and $j$ runs over all indices of twig blobs.
+The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$.
The boundary map
\[
@@ -180,6 +188,9 @@
The $(-1)^{j+1}$ factors imply that the terms of $\bd^2(b)$ all cancel.
Thus we have a chain complex.
+Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, is immediately obvious from the definition.
+A homeomorphism acts in an obvious on blobs and on fields.
+
We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$,
to be the union of the blobs of $b$.
For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram),
@@ -193,8 +204,10 @@
(equivalently, to each rooted tree) according to the following rules:
\begin{itemize}
\item $p(\emptyset) = pt$, where $\emptyset$ denotes a 0-blob diagram or empty tree;
-\item $p(a \du b) = p(a) \times p(b)$, where $a \du b$ denotes the distant (non-overlapping) union of two blob diagrams (equivalently, join two trees at the roots); and
-\item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root).
+\item $p(a \du b) = p(a) \times p(b)$, where $a \du b$ denotes the distant (non-overlapping) union
+of two blob diagrams (equivalently, join two trees at the roots); and
+\item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which
+encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root).
\end{itemize}
For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while
a diagram of $k$ disjoint blobs corresponds to a $k$-cube.
--- a/text/comm_alg.tex Sun Jun 06 09:49:06 2010 -0700
+++ b/text/comm_alg.tex Sun Jun 06 09:49:57 2010 -0700
@@ -13,7 +13,10 @@
The goal of this \nn{subsection?} is to compute
$\bc_*(M^n, C)$ for various commutative algebras $C$.
-Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative algebra is homotopy equivalent to the higher Hochschild complex for $M^n$ with coefficients in $C$ (see \cite{MR0339132, MR1755114, MR2383113}). This possibility was suggested to us by Thomas Tradler.
+Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative
+algebra is homotopy equivalent to the higher Hochschild complex for $M^n$ with
+coefficients in $C$ (see \cite{MR0339132, MR1755114, MR2383113}).
+This possibility was suggested to us by Thomas Tradler.
\medskip
@@ -108,8 +111,12 @@
\nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section}
Let us check this directly.
-The algebra $k[t]$ has Koszul resolution $k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$, which has coinvariants $k[t] \xrightarrow{0} k[t]$. This is equal to its homology, so we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$.
-(See also \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings: $HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one.
+The algebra $k[t]$ has Koszul resolution
+$k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$,
+which has coinvariants $k[t] \xrightarrow{0} k[t]$.
+This is equal to its homology, so we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$.
+(See also \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings:
+$HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one.
We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other.
The fixed points of this flow are the equally spaced configurations.
@@ -152,7 +159,8 @@
\]
Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$.
We will content ourselves with the case $k = \z$.
-One can define a flow on $\Sigma_m^\infty(S^1)$ where points of the same color repel each other and points of different colors do not interact.
+One can define a flow on $\Sigma_m^\infty(S^1)$ where points of the
+same color repel each other and points of different colors do not interact.
This shows that a component $X$ of $\Sigma_m^\infty(S^1)$ is homotopy equivalent
to the torus $(S^1)^l$, where $l$ is the number of non-zero entries in the $m$-tuple
corresponding to $X$.
--- a/text/deligne.tex Sun Jun 06 09:49:06 2010 -0700
+++ b/text/deligne.tex Sun Jun 06 09:49:57 2010 -0700
@@ -11,7 +11,8 @@
(Proposition \ref{prop:deligne} below).
Then we sketch the proof.
-\nn{Does this generalisation encompass Kontsevich's proposed generalisation from \cite[\S2.5]{MR1718044}, that (I think...) the Hochschild homology of an $E_n$ algebra is an $E_{n+1}$ algebra? -S}
+\nn{Does this generalisation encompass Kontsevich's proposed generalisation from \cite[\S2.5]{MR1718044},
+that (I think...) the Hochschild homology of an $E_n$ algebra is an $E_{n+1}$ algebra? -S}
%from http://www.ams.org/mathscinet-getitem?mr=1805894
%Different versions of the geometric counterpart of Deligne's conjecture have been proven by Tamarkin [``Formality of chain operad of small squares'', preprint, http://arXiv.org/abs/math.QA/9809164], the reviewer [in Confˇrence Moshˇ Flato 1999, Vol. II (Dijon), 307--331, Kluwer Acad. Publ., Dordrecht, 2000; MR1805923 (2002d:55009)], and J. E. McClure and J. H. Smith [``A solution of Deligne's conjecture'', preprint, http://arXiv.org/abs/math.QA/9910126] (see also a later simplified version [J. E. McClure and J. H. Smith, ``Multivariable cochain operations and little $n$-cubes'', preprint, http://arXiv.org/abs/math.QA/0106024]). The paper under review gives another proof of Deligne's conjecture, which, as the authors indicate, may be generalized to a proof of a higher-dimensional generalization of Deligne's conjecture, suggested in [M. Kontsevich, Lett. Math. Phys. 48 (1999), no. 1, 35--72; MR1718044 (2000j:53119)].
@@ -151,15 +152,15 @@
\medskip
%The little $n{+}1$-ball operad injects into the $n$-FG operad.
-The $n$-FG operad contains the little $n{+}1$-ball operad.
+The $n$-FG operad contains the little $n{+}1$-balls operad.
Roughly speaking, given a configuration of $k$ little $n{+}1$-balls in the standard
$n{+}1$-ball, we fiber the complement of the balls by vertical intervals
and let $M_i$ [$N_i$] be the southern [northern] hemisphere of the $i$-th ball.
-More precisely, let $x_0,\ldots,x_n$ be the coordinates of $\r^{n+1}$.
+More precisely, let $x_1,\ldots,x_{n+1}$ be the coordinates of $\r^{n+1}$.
Let $z$ be a point of the $k$-th space of the little $n{+}1$-ball operad, with
little balls $D_1,\ldots,D_k$ inside the standard $n{+}1$-ball.
-We assume the $D_i$'s are ordered according to the $x_n$ coordinate of their centers.
-Let $\pi:\r^{n+1}\to \r^n$ be the projection corresponding to $x_n$.
+We assume the $D_i$'s are ordered according to the $x_{n+1}$ coordinate of their centers.
+Let $\pi:\r^{n+1}\to \r^n$ be the projection corresponding to $x_{n+1}$.
Let $B\sub\r^n$ be the standard $n$-ball.
Let $M_i$ and $N_i$ be $B$ for all $i$.
Identify $\pi(D_i)$ with $B$ (a.k.a.\ $M_i$ or $N_i$) via translations and dilations (no rotations).
@@ -167,7 +168,8 @@
Let $f_i = \rm{id}$ for all $i$.
We have now defined a map from the little $n{+}1$-ball operad to the $n$-FG operad,
with contractible fibers.
-(The fibers correspond to moving the $D_i$'s in the $x_n$ direction without changing their ordering.)
+(The fibers correspond to moving the $D_i$'s in the $x_{n+1}$
+direction without changing their ordering.)
\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?}
@@ -193,6 +195,7 @@
\stackrel{f_k}{\to} \bc_*(N_0)
\]
(Recall that the maps $\id\ot\alpha_i$ were defined in \nn{need ref}.)
+\nn{issue: haven't we only defined $\id\ot\alpha_i$ when $\alpha_i$ is closed?}
It is easy to check that the above definition is compatible with the equivalence relations
and also the operad structure.
We can reinterpret the above as a chain map
@@ -221,6 +224,10 @@
As noted above, the $n$-FG operad contains the little $n{+}1$-ball operad, so this constitutes
a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disk operad.
+\begin{proof}
+
+
\nn{...}
+\end{proof}
\nn{maybe point out that even for $n=1$ there's something new here.}
--- a/text/evmap.tex Sun Jun 06 09:49:06 2010 -0700
+++ b/text/evmap.tex Sun Jun 06 09:49:57 2010 -0700
@@ -41,7 +41,8 @@
I lean toward the latter.}
\medskip
-Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, and then give an outline of the method of proof.
+Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma},
+and then give an outline of the method of proof.
Without loss of generality, we will assume $X = Y$.
@@ -50,7 +51,8 @@
Let $f: P \times X \to X$ be a family of homeomorphisms (e.g. a generator of $CH_*(X)$)
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'
+$x \notin S$ 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'
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 \\
@@ -313,7 +315,9 @@
$G_*^{i,m}$.
Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero.
Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$.
-Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is spanned by families of homeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{extension_lemma}.
+Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is
+spanned by families of homeomorphisms with support compatible with $\cU_j$,
+as described in Lemma \ref{extension_lemma}.
Recall that $h_j$ and also the homotopy connecting it to the identity do not increase
supports.
Define
--- a/text/hochschild.tex Sun Jun 06 09:49:06 2010 -0700
+++ b/text/hochschild.tex Sun Jun 06 09:49:57 2010 -0700
@@ -7,7 +7,11 @@
greater than zero.
In this section we analyze the blob complex in dimension $n=1$.
We find that $\bc_*(S^1, \cC)$ is homotopy equivalent to the
-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 first passing to the corresponding $n$ dimensional system of fields.)
+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
+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.
@@ -67,12 +71,14 @@
usual Hochschild complex for $C$.
\end{thm}
-This follows from two results. First, we see that
+This follows from two results.
+First, we see that
\begin{lem}
\label{lem:module-blob}%
The complex $K_*(C)$ (here $C$ is being thought of as a
$C$-$C$-bimodule, not a category) is homotopy equivalent to the blob complex
-$\bc_*(S^1; C)$. (Proof later.)
+$\bc_*(S^1; C)$.
+(Proof later.)
\end{lem}
Next, we show that for any $C$-$C$-bimodule $M$,
@@ -114,17 +120,19 @@
$$\cP_*(M) \iso \coinv(F_*).$$
%
Observe that there's a quotient map $\pi: F_0 \onto M$, and by
-construction the cone of the chain map $\pi: F_* \to M$ is acyclic. Now
-construct the total complex $\cP_i(F_j)$, with $i,j \geq 0$, graded by
-$i+j$. We have two chain maps
+construction the cone of the chain map $\pi: F_* \to M$ is acyclic.
+Now construct the total complex $\cP_i(F_j)$, with $i,j \geq 0$, graded by $i+j$.
+We have two chain maps
\begin{align*}
\cP_i(F_*) & \xrightarrow{\cP_i(\pi)} \cP_i(M) \\
\intertext{and}
\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.
+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.
-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
+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
$$\cP_*(M) \quismto \coinv(F_*).$$
%If $M$ is free, that is, a direct sum of copies of
@@ -150,7 +158,8 @@
%and higher homology groups are determined by lower ones in $\HC_*(K)$, and
%hence recursively as coinvariants of some other bimodule.
-Proposition \ref{prop:hoch} then follows from the following lemmas, establishing that $K_*$ has precisely these required properties.
+Proposition \ref{prop:hoch} then follows from the following lemmas,
+establishing that $K_*$ has precisely these required properties.
\begin{lem}
\label{lem:hochschild-additive}%
Directly from the definition, $K_*(M_1 \oplus M_2) \cong K_*(M_1) \oplus K_*(M_2)$.
@@ -185,7 +194,8 @@
We want to define a homotopy inverse to the above inclusion, but before doing so
we must replace $\bc_*(S^1)$ with a homotopy equivalent subcomplex.
Let $J_* \sub \bc_*(S^1)$ be the subcomplex where * does not lie on the boundary
-of any blob. Note that the image of $i$ is contained in $J_*$.
+of any blob.
+Note that the image of $i$ is contained in $J_*$.
Note also that in $\bc_*(S^1)$ (away from $J_*$)
a blob diagram could have multiple (nested) blobs whose
boundaries contain *, on both the right and left of *.
@@ -219,10 +229,13 @@
every blob in the diagram.
Note that for any chain $x \in \bc_*(S^1)$, $x \in L_*^\ep$ for sufficiently small $\ep$.
-We define a degree $1$ map $j_\ep: L_*^\ep \to L_*^\ep$ as follows. Let $x \in L_*^\ep$ be a blob diagram.
+We define a degree $1$ map $j_\ep: L_*^\ep \to L_*^\ep$ as follows.
+Let $x \in L_*^\ep$ be a blob diagram.
\nn{maybe add figures illustrating $j_\ep$?}
-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 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 $u=\sum z_i$,
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 $y_i - s(y_i)$.
@@ -256,14 +269,24 @@
\]
and similarly for $\hat{g}$.
Most of what we need to check is easy.
-Suppose we have $\sum_i (a_i \tensor k_i \tensor b_i) \in \ker(C \tensor K \tensor C \to K)$, assuming without loss of generality that $\{a_i \tensor b_i\}_i$ is linearly independent in $C \tensor C$, and $\hat{f}(a \tensor k \tensor b) = 0 \in \ker(C \tensor E \tensor C \to E)$. We must then have $f(k_i) = 0 \in E$ for each $i$, which implies $k_i=0$ itself.
-If $\sum_i (a_i \tensor e_i \tensor b_i) \in \ker(C \tensor E \tensor C \to E)$ is in the image of $\ker(C \tensor K \tensor C \to K)$ under $\hat{f}$, again by assuming the set $\{a_i \tensor b_i\}_i$ is linearly independent we can deduce that each
-$e_i$ is in the image of the original $f$, and so is in the kernel of the original $g$, and so $\hat{g}(\sum_i a_i \tensor e_i \tensor b_i) = 0$.
-If $\hat{g}(\sum_i a_i \tensor e_i \tensor b_i) = 0$, then each $g(e_i) = 0$, so $e_i = f(\widetilde{e_i})$ for some $\widetilde{e_i} \in K$, and $\sum_i a_i \tensor e_i \tensor b_i = \hat{f}(\sum_i a_i \tensor \widetilde{e_i} \tensor b_i)$.
-Finally, the interesting step is in checking that any $q = \sum_i a_i \tensor q_i \tensor b_i$ such that $\sum_i a_i q_i b_i = 0$ is in the image of $\ker(C \tensor E \tensor C \to C)$ under $\hat{g}$.
-For each $i$, we can find $\widetilde{q_i}$ so $g(\widetilde{q_i}) = q_i$. However $\sum_i a_i \widetilde{q_i} b_i$ need not be zero.
+Suppose we have $\sum_i (a_i \tensor k_i \tensor b_i) \in \ker(C \tensor K \tensor C \to K)$,
+assuming without loss of generality that $\{a_i \tensor b_i\}_i$ is linearly independent in $C \tensor C$,
+and $\hat{f}(a \tensor k \tensor b) = 0 \in \ker(C \tensor E \tensor C \to E)$.
+We must then have $f(k_i) = 0 \in E$ for each $i$, which implies $k_i=0$ itself.
+If $\sum_i (a_i \tensor e_i \tensor b_i) \in \ker(C \tensor E \tensor C \to E)$
+is in the image of $\ker(C \tensor K \tensor C \to K)$ under $\hat{f}$,
+again by assuming the set $\{a_i \tensor b_i\}_i$ is linearly independent we can deduce that each
+$e_i$ is in the image of the original $f$, and so is in the kernel of the original $g$,
+and so $\hat{g}(\sum_i a_i \tensor e_i \tensor b_i) = 0$.
+If $\hat{g}(\sum_i a_i \tensor e_i \tensor b_i) = 0$, then each $g(e_i) = 0$, so $e_i = f(\widetilde{e_i})$
+for some $\widetilde{e_i} \in K$, and $\sum_i a_i \tensor e_i \tensor b_i = \hat{f}(\sum_i a_i \tensor \widetilde{e_i} \tensor b_i)$.
+Finally, the interesting step is in checking that any $q = \sum_i a_i \tensor q_i \tensor b_i$
+such that $\sum_i a_i q_i b_i = 0$ is in the image of $\ker(C \tensor E \tensor C \to C)$ under $\hat{g}$.
+For each $i$, we can find $\widetilde{q_i}$ so $g(\widetilde{q_i}) = q_i$.
+However $\sum_i a_i \widetilde{q_i} b_i$ need not be zero.
Consider then $$\widetilde{q} = \sum_i (a_i \tensor \widetilde{q_i} \tensor b_i) - 1 \tensor (\sum_i a_i \widetilde{q_i} b_i) \tensor 1.$$ Certainly
-$\widetilde{q} \in \ker(C \tensor E \tensor C \to E)$. Further,
+$\widetilde{q} \in \ker(C \tensor E \tensor C \to E)$.
+Further,
\begin{align*}
\hat{g}(\widetilde{q}) & = \sum_i (a_i \tensor g(\widetilde{q_i}) \tensor b_i) - 1 \tensor (\sum_i a_i g(\widetilde{q_i}) b_i) \tensor 1 \\
& = q - 0
@@ -275,32 +298,44 @@
\label{eq:ker-functor}%
M \mapsto \ker(C^{\tensor k} \tensor M \tensor C^{\tensor l} \to M)
\end{equation}
-are all exact too. Moreover, tensor products of such functors with each
+are all exact too.
+Moreover, tensor products of such functors with each
other and with $C$ or $\ker(C^{\tensor k} \to C)$ (e.g., producing the functor $M \mapsto \ker(M \tensor C \to M)
\tensor C \tensor \ker(C \tensor C \to M)$) are all still exact.
Finally, then we see that the functor $K_*$ is simply an (infinite)
-direct sum of copies of this sort of functor. The direct sum is indexed by
-configurations of nested blobs and of labels; for each such configuration, we have one of the above tensor product functors,
-with the labels of twig blobs corresponding to tensor factors as in \eqref{eq:ker-functor} or $\ker(C^{\tensor k} \to C)$ (depending on whether they contain a marked point $p_i$), and all other labelled points corresponding
+direct sum of copies of this sort of functor.
+The direct sum is indexed by
+configurations of nested blobs and of labels; for each such configuration, we have one of
+the above tensor product functors,
+with the labels of twig blobs corresponding to tensor factors as in \eqref{eq:ker-functor}
+or $\ker(C^{\tensor k} \to C)$ (depending on whether they contain a marked point $p_i$), and all other labelled points corresponding
to tensor factors of $C$ and $M$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:hochschild-coinvariants}]
We show that $H_0(K_*(M))$ is isomorphic to the coinvariants of $M$.
-We define a map $\ev: K_0(M) \to M$. If $x \in K_0(M)$ has the label $m \in M$ at $*$, and labels $c_i \in C$ at the other labeled points of $S^1$, reading clockwise from $*$,
-we set $\ev(x) = m c_1 \cdots c_k$. We can think of this as $\ev : M \tensor C^{\tensor k} \to M$, for each direct summand of $K_0(M)$ indexed by a configuration of labeled points.
+We define a map $\ev: K_0(M) \to M$.
+If $x \in K_0(M)$ has the label $m \in M$ at $*$, and labels $c_i \in C$ at the other
+labeled points of $S^1$, reading clockwise from $*$,
+we set $\ev(x) = m c_1 \cdots c_k$.
+We can think of this as $\ev : M \tensor C^{\tensor k} \to M$, for each direct summand of
+$K_0(M)$ indexed by a configuration of labeled points.
There is a quotient map $\pi: M \to \coinv{M}$.
We claim that the composition $\pi \compose \ev$ is well-defined on the quotient $H_0(K_*(M))$;
i.e.\ that $\pi(\ev(\bd y)) = 0$ for all $y \in K_1(M)$.
There are two cases, depending on whether the blob of $y$ contains the point *.
If it doesn't, then
-suppose $y$ has label $m$ at $*$, labels $c_i$ at other labeled points outside the blob, and the field inside the blob is a sum, with the $j$-th term having
-labeled points $d_{j,i}$. Then $\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \in \ker(\DirectSum_k C^{\tensor k} \to C)$, and so
+suppose $y$ has label $m$ at $*$, labels $c_i$ at other labeled points outside the blob,
+and the field inside the blob is a sum, with the $j$-th term having
+labeled points $d_{j,i}$.
+Then $\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \in \ker(\DirectSum_k C^{\tensor k} \to C)$, and so
$\ev(\bdy y) = 0$, because $$C^{\tensor \ell_1} \tensor \ker(\DirectSum_k C^{\tensor k} \to C) \tensor C^{\tensor \ell_2} \subset \ker(\DirectSum_k C^{\tensor k} \to C).$$
-Similarly, if $*$ is contained in the blob, then the blob label is a sum, with the $j$-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
+Similarly, if $*$ is contained in the blob, then the blob label is a sum, with the
+$j$-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
$$\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \tensor m_j \tensor d_{j,1}' \tensor \cdots \tensor d_{j,k'_j}' \in \ker(\DirectSum_{k,k'} C^{\tensor k} \tensor M \tensor C^{\tensor k'} \tensor \to M),$$
and so
\begin{align*}
@@ -310,7 +345,8 @@
\end{align*}
where this time we use the fact that we're mapping to $\coinv{M}$, not just $M$.
-The map $\pi \compose \ev: H_0(K_*(M)) \to \coinv{M}$ is clearly surjective ($\ev$ surjects onto $M$); we now show that it's injective.
+The map $\pi \compose \ev: H_0(K_*(M)) \to \coinv{M}$ is clearly
+surjective ($\ev$ surjects onto $M$); we now show that it's injective.
This is equivalent to showing that
\[
\ev\inv(\ker(\pi)) \sub \bd K_1(M) .
@@ -340,7 +376,8 @@
\begin{proof}[Proof of Lemma \ref{lem:hochschild-free}]
We show that $K_*(C\otimes C)$ is
-quasi-isomorphic to the 0-step complex $C$. We'll do this in steps, establishing quasi-isomorphisms and homotopy equivalences
+quasi-isomorphic to the 0-step complex $C$.
+We'll do this in steps, establishing quasi-isomorphisms and homotopy equivalences
$$K_*(C \tensor C) \quismto K'_* \htpyto K''_* \quismto C.$$
Let $K'_* \sub K_*(C\otimes C)$ be the subcomplex where the label of
@@ -355,7 +392,8 @@
%and the two boundary points of $N_\ep$ are not labeled points of $b$.
For a field $y$ on $N_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$
labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$.
-(See Figure \ref{fig:sy}.) Note that $y - s_\ep(y) \in U(N_\ep)$.
+(See Figure \ref{fig:sy}.)
+Note that $y - s_\ep(y) \in U(N_\ep)$.
Let $\sigma_\ep: K_*^\ep \to K_*^\ep$ be the chain map
given by replacing the restriction $y$ to $N_\ep$ of each field
appearing in an element of $K_*^\ep$ with $s_\ep(y)$.
@@ -512,7 +550,8 @@
\begin{equation*}
\mathfig{0.4}{hochschild/2-chains-1} \qquad \mathfig{0.4}{hochschild/2-chains-2}
\end{equation*}
-\caption{The 0-, 1- and 2-chains in the image of $m \tensor a \tensor b$. Only the supports of the 1- and 2-blobs are shown.}
+\caption{The 0-, 1- and 2-chains in the image of $m \tensor a \tensor b$.
+Only the supports of the 1- and 2-blobs are shown.}
\label{fig:hochschild-2-chains}
\end{figure}
@@ -529,7 +568,8 @@
\end{figure}
In degree 2, we send $m\ot a \ot b$ to the sum of 24 ($=6\cdot4$) 2-blob diagrams as shown in
-Figure \ref{fig:hochschild-2-chains}. In Figure \ref{fig:hochschild-2-chains} the 1- and 2-blob diagrams are indicated only by their support.
+Figure \ref{fig:hochschild-2-chains}.
+In Figure \ref{fig:hochschild-2-chains} the 1- and 2-blob diagrams are indicated only by their support.
We leave it to the reader to determine the labels of the 1-blob diagrams.
Each 2-cell in the figure is labeled by a ball $V$ in $S^1$ which contains the support of all
1-blob diagrams in its boundary.
--- a/text/intro.tex Sun Jun 06 09:49:06 2010 -0700
+++ b/text/intro.tex Sun Jun 06 09:49:57 2010 -0700
@@ -2,34 +2,69 @@
\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:
+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:
\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}.)
-\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}.)
+\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}.)
+\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}.)
\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 $M$.
%$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$
\end{itemize}
-The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space (replacing quotient of fields by local relations with some sort of resolution),
-and for a generalization of Hochschild homology to higher $n$-categories. We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. The blob complex gives us all of these! More detailed motivations are described in \S \ref{sec:motivations}.
+The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space
+(replacing quotient of fields by local relations with some sort of resolution),
+and for a generalization of Hochschild homology to higher $n$-categories.
+We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold.
+The blob complex gives us all of these! More detailed motivations are described in \S \ref{sec:motivations}.
-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}).
+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}).
-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.
+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}).
+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}).
-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 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.
+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
+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.
-Nevertheless, when we attempt to establish all of the observed properties of the blob complex, we find this situation unsatisfactory. Thus, in the second part of the paper (\S\S \ref{sec:ncats}-\ref{sec:ainfblob}) we give yet another definition of an $n$-category, or rather a definition of an $n$-category with strong duality. (It appears that removing the duality conditions from our definition would make it more complicated rather than less.) 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.
+Nevertheless, when we attempt to establish all of the observed properties of the blob complex,
+we find this situation unsatisfactory.
+Thus, in the second part of the paper (\S\S \ref{sec:ncats}-\ref{sec:ainfblob}) we give yet another
+definition of an $n$-category, or rather a definition of an $n$-category with strong duality.
+(It appears that removing the duality conditions from our definition would make it more complicated rather than less.)
+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. 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 homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$.
+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
+homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms.
+The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a
+topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$.
-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 $A_\infty$ $n$-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 \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
+$A_\infty$ $n$-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.
The relationship between all these ideas is sketched in Figure \ref{fig:outline}.
@@ -77,7 +112,14 @@
\label{fig:outline}
\end{figure}
-Finally, later sections address other topics. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, 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', 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.
+Finally, later sections address other topics.
+Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra,
+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',
+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.
\nn{some more things to cover in the intro}
@@ -165,16 +207,20 @@
\begin{property}[Functoriality]
\label{property:functoriality}%
-The blob complex is functorial with respect to homeomorphisms. That is,
+The blob complex is functorial with respect to homeomorphisms.
+That is,
for a fixed $n$-dimensional system of fields $\cC$, the association
\begin{equation*}
X \mapsto \bc_*^{\cC}(X)
\end{equation*}
-is a functor from $n$-manifolds and homeomorphisms between them to chain complexes and isomorphisms between them.
+is a functor from $n$-manifolds and homeomorphisms between them to chain
+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.
+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.
-The blob complex is also functorial (indeed, exact) with respect to $\cC$, although we will not address this in detail here.
+The blob complex is also functorial (indeed, exact) with respect to $\cC$,
+although we will not address this in detail here.
\begin{property}[Disjoint union]
\label{property:disjoint-union}
@@ -184,7 +230,9 @@
\end{equation*}
\end{property}
-If an $n$-manifold $X_\text{cut}$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, write $X_\text{glued} = X_\text{cut} \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. Note that this includes the case of gluing two disjoint manifolds together.
+If an $n$-manifold $X_\text{cut}$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary,
+write $X_\text{glued} = X_\text{cut} \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$.
+Note that this includes the case of gluing two disjoint manifolds together.
\begin{property}[Gluing map]
\label{property:gluing-map}%
%If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map
@@ -201,7 +249,8 @@
\begin{property}[Contractibility]
\label{property:contractibility}%
-With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology. Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cC$ to balls.
+With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology.
+Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cC$ to balls.
\begin{equation}
\xymatrix{\bc_*^{\cC}(B^n) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*^{\cC}(B^n)) \ar[r]^(0.6)\iso & \cC(B^n)}
\end{equation}
@@ -211,12 +260,15 @@
\label{property: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 $\cC$. (See \S \ref{sec:local-relations}.)
+by $\cC$.
+(See \S \ref{sec:local-relations}.)
\begin{equation*}
H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X)
\end{equation*}
\end{property}
+\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}%
The blob complex for a $1$-category $\cC$ on the circle is
@@ -249,7 +301,8 @@
\bc_*(X) \ar[u]_{\gl_Y}
}
\end{equation*}
-\item Any such chain map satisfying points 2. and 3. above is unique, up to an iterated homotopy. (That is, any pair of homotopies have a homotopy between them, and so on.)
+\item Any such chain map satisfying points 2. and 3. above is unique, up to an iterated homotopy.
+(That is, any pair of homotopies have a homotopy between them, and so on.)
\item This map is associative, in the sense that the following diagram commutes (up to homotopy).
\begin{equation*}
\xymatrix{
@@ -265,84 +318,107 @@
for any homeomorphic pair $X$ and $Y$,
satisfying corresponding conditions.
-In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. 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.
+In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields.
+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}
-Let $\cC$ be a topological $n$-category. Let $Y$ be an $n{-}k$-manifold.
-There is an $A_\infty$ $k$-category $A_*(Y, \cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, to be the set $$A_*(Y,\cC)(D) = A^\cC(Y \times D)$$ and on $k$-balls $D$ to be the set $$A_*(Y, \cC)(D) = \bc_*(Y \times D, \cC).$$ (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}.
+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 $0 \leq m < k$,
+to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set
+$$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$
+(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}
\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.
+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.
\end{rem}
There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category
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}
-Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. Let $\cC$ be an $n$-category.
-Let $A_*(Y,\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}).
+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}).
Then
\[
- \bc_*(Y\times W, \cC) \simeq \bc_*(W, A_*(Y,\cC)) .
+ \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W).
\]
-Note on the right hand side we have the version of the blob complex for $A_\infty$ $n$-categories.
\end{property}
-It seems reasonable to expect a generalization describing an arbitrary fibre bundle. See in particular \S \ref{moddecss} for the framework for such a statement.
+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}%
\mbox{}% <-- gets the indenting right
\begin{itemize}
-\item For any $(n-1)$-manifold $Y$, the blob complex of $Y \times I$ is
-naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below.
-
\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an
-$A_\infty$ module for $\bc_*(Y \times I)$.
+$A_\infty$ module for $\bc_*(Y)$.
\item For any $n$-manifold $X_\text{glued} = X_\text{cut} \bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{glued})$ is the $A_\infty$ self-tensor product of
-$\bc_*(X_\text{cut})$ as an $\bc_*(Y \times I)$-bimodule:
+$\bc_*(X_\text{cut})$ as an $\bc_*(Y)$-bimodule:
\begin{equation*}
-\bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \selfarrow
+\bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow
\end{equation*}
\end{itemize}
\end{property}
-Finally, we state two more properties, which we will not prove in this paper.
-\nn{revise this; expect that we will prove these in the paper}
+Finally, we prove two theorems which we consider as applications.
-\begin{property}[Mapping spaces]
+\begin{thm}[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{property}
+\end{thm}
This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data.
-\begin{property}[Higher dimensional Deligne conjecture]
-\label{property:deligne}
+\begin{thm}[Higher dimensional Deligne conjecture]
+\label{thm:deligne}
The singular chains of the $n$-dimensional fat graph operad act on blob cochains.
-\end{property}
-See \S \ref{sec:deligne} for an explanation of the terms appearing here. The proof will appear elsewhere.
+\end{thm}
+See \S \ref{sec:deligne} for a full explanation of the statement, and an outline of the proof.
-Properties \ref{property:functoriality}, \ref{property:gluing-map} 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. \todo{Make sure this gets done.}
-Properties \ref{property:disjoint-union} 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},
+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}
Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard).
-In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do. We could presumably also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories), and likely it will prove useful to think about the connections between what we do here and $(\infty,k)$-categories.
-More could be said about finite characteristic (there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$, for example). Much more could be said about other types of manifolds, in particular oriented, $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; there may be some differences for topological manifolds and smooth manifolds.
+In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do.
+We could presumably also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories),
+and likely it will prove useful to think about the connections between what we do here and $(\infty,k)$-categories.
+More could be said about finite characteristic
+(there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$, for example).
+Much more could be said about other types of manifolds, in particular oriented,
+$\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated.
+(We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.)
+We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds;
+there may be some differences for topological manifolds and smooth manifolds.
-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.
+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. 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.
+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.
Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh}
@@ -352,11 +428,3 @@
Michael Freedman, Vaughan Jones, Justin Roberts, Chris Schommer-Pries, Peter Teichner \nn{and who else?} 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.
-
-\medskip\hrule\medskip
-
-Still to do:
-\begin{itemize}
-\item say something about starting with semisimple n-cat (trivial?? not trivial?)
-\end{itemize}
-
--- a/text/ncat.tex Sun Jun 06 09:49:06 2010 -0700
+++ b/text/ncat.tex Sun Jun 06 09:49:57 2010 -0700
@@ -7,6 +7,7 @@
\label{sec:ncats}
\subsection{Definition of $n$-categories}
+\label{ss:n-cat-def}
Before proceeding, we need more appropriate definitions of $n$-categories,
$A_\infty$ $n$-categories, modules for these, and tensor products of these modules.
@@ -23,8 +24,10 @@
\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.
-Some $n$-category definitions model $k$-morphisms on the standard bihedrons (interval, bigon, and so on).
+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.
+Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, and so on).
Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$,
a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$,
and so on.
@@ -32,8 +35,10 @@
Still other definitions (see, for example, \cite{MR2094071})
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
-to the standard $k$-ball. By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the
+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
+to the standard $k$-ball.
+By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the
standard $k$-ball.
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.
@@ -78,7 +83,10 @@
Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range.
(Actually, this is only true in the oriented case, with 1-morphisms parameterized
by oriented 1-balls.)
-For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. (sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. We prefer to not make the distinction in the first place.
+For $k>1$ and in the presence of strong duality the division into domain and range makes less sense.
+For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc.
+(sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary.
+We prefer to not make the distinction in the first place.
Instead, we will combine the domain and range into a single entity which we call the
boundary of a morphism.
@@ -86,34 +94,36 @@
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 coend
+all the axioms in the 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):
-\begin{prop}
-\label{axiom:spheres}
-For each $1 \le k \le n$, we have a functor $\cC_{k-1}$ from
+\begin{lem}
+\label{lem:spheres}
+For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from
the category of $k{-}1$-spheres and
homeomorphisms to the category of sets and bijections.
-\end{prop}
+\end{lem}
-We postpone the proof \todo{} 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.
+We postpone the proof \todo{} 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.
%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.
\begin{axiom}[Boundaries]\label{nca-boundary}
-For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cC_{k-1}(\bd X)$.
+For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$.
These maps, for various $X$, comprise a natural transformation of functors.
\end{axiom}
(Note that the first ``$\bd$" above is part of the data for the category,
while the second is the ordinary boundary of manifolds.)
-Given $c\in\cC(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$.
+Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$.
Most of the examples of $n$-categories we are interested in are enriched in the following sense.
The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and
-all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category
+all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category
(e.g.\ vector spaces, or modules over some ring, or chain complexes),
and all the structure maps of the $n$-category should be compatible with the auxiliary
category structure.
@@ -142,27 +152,28 @@
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 proposition follows from the coend construction used to define $\cC_{k-1}$
+The following lemma follows from the colimit construction used to define $\cl{\cC}_{k-1}$
on spheres.
-\begin{prop}[Boundary from domain and range]
+\begin{lem}[Boundary from domain and range]
+\label{lem:domain-and-range}
Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$,
$B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}).
-Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the
-two maps $\bd: \cC(B_i)\to \cC(E)$.
+Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the
+two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$.
Then we have an injective map
\[
- \gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \into \cC(S)
+ \gl_E : \cC(B_1) \times_{\\cl{cC}(E)} \cC(B_2) \into \cl{\cC}(S)
\]
which is natural with respect to the actions of homeomorphisms.
-(When $k=1$ we stipulate that $\cC(E)$ is a point, so that the above fibered product
+(When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product
becomes a normal product.)
-\end{prop}
+\end{lem}
\begin{figure}[!ht]
$$
\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) {};
\draw (S) arc (-90:90:1);
@@ -175,15 +186,15 @@
Note that we insist on injectivity above.
-Let $\cC(S)_E$ denote the image of $\gl_E$.
-We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$".
+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 $E$".
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 $\cC(X)_E = \bd^{-1}(\cC(\bd X)_E)$.
+as above, then we define $\cC(X)_E = \bd^{-1}(\cl{\cC}(\bd X)_E)$.
-We will call the projection $\cC(S)_E \to \cC(B_i)$
+We will call the projection $\cl{\cC}(S)_E \to \cC(B_i)$
a {\it restriction} map and write $\res_{B_i}(a)$
-(or simply $\res(a)$ when there is no ambiguity), for $a\in \cC(S)_E$.
+(or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)_E$.
More generally, we also include under the rubric ``restriction map" the
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
@@ -260,7 +271,7 @@
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.
-If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cC(\bd X)_\beta)$;
+If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$;
this can also be denoted $\cC(X)\spl$ if the context contains an anonymous
subdivision of $\bd X$ and no competing subdivision of $X$.
@@ -281,8 +292,10 @@
The next axiom is related to identity morphisms, though that might not be immediately obvious.
-\begin{axiom}[Product (identity) morphisms]
-For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. These maps must satisfy the following conditions.
+\begin{axiom}[Product (identity) morphisms, preliminary version]
+For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$,
+usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$.
+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
@@ -303,7 +316,6 @@
\[
(a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') .
\]
-\nn{if pinched boundary, then remove first case above}
\item
Product morphisms are associative:
\[
@@ -320,24 +332,93 @@
\end{enumerate}
\end{axiom}
-\nn{need even more subaxioms for product morphisms?}
+We will need to strengthen the above preliminary version of the axiom to allow
+for products which are ``pinched" in various ways along their boundary.
+(See Figure xxxx.)
+(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
+\[
+ \pi: E\to X
+\]
+such that $E$ is a $k{+}m$-ball, $X$ is a $k$-ball ($m\ge 1$), and $\pi$ is locally modeled
+on a standard iterated degeneracy map
+\[
+ d: \Delta^{k+m}\to\Delta^k .
+\]
+In other words, \nn{each point has a neighborhood blah blah...}
+(We thank Kevin Costello for suggesting this approach.)
+
+Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball,
+and for for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension
+$l \le m$, with $l$ depending on $x$.
+
+It is easy to see that a composition of pinched products is again a pinched product.
+
+A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction
+$\pi:E'\to \pi(E')$ is again a pinched product.
+A {union} of pinched products is a decomposition $E = \cup_i E_i$
+such that each $E_i\sub E$ is a sub pinched product.
+(See Figure xxxx.)
+
+The product axiom will give a map $\pi^*:\cC(X)\to \cC(E)$ for each pinched product
+$\pi:E\to X$.
+Morphisms in the image of $\pi^*$ will be called product morphisms.
+Before stating the axiom, we illustrate it in our two motivating examples of $n$-categories.
+In the case where $\cC(X) = \{f: X\to T\}$, we define $\pi^*(f) = f\circ\pi$.
+In the case where $\cC(X)$ is the set of all labeled embedded cell complexes $K$ in $X$,
+define $\pi^*(K) = \pi\inv(K)$, with each codimension $i$ cell $\pi\inv(c)$ labeled by the
+same (traditional) $i$-morphism as the corresponding codimension $i$ cell $c$.
+
-\nn{Almost certainly we need a little more than the above axiom.
-More specifically, in order to bootstrap our way from the top dimension
-properties of identity morphisms to low dimensions, we need regular products,
-pinched products and even half-pinched products.
-I'm not sure what the best way to cleanly axiomatize the properties of these various
-products is.
-For the moment, I'll assume that all flavors of the product are at
-our disposal, and I'll plan on revising the axioms later.}
+\addtocounter{axiom}{-1}
+\begin{axiom}[Product (identity) morphisms]
+For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$),
+there is a map $\pi^*:\cC(X)\to \cC(E)$.
+These maps must satisfy the following conditions.
+\begin{enumerate}
+\item
+If $\pi:E\to X$ and $\pi':E'\to X'$ are pinched products, and
+if $f:X\to X'$ and $\tilde{f}:E \to E'$ are maps such that the diagram
+\[ \xymatrix{
+ E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\
+ X \ar[r]^{f} & X'
+} \]
+commutes, then we have
+\[
+ \pi'^*\circ f = \tilde{f}\circ \pi^*.
+\]
+\item
+Product morphisms are compatible with gluing (composition).
+Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$
+be pinched products with $E = E_1\cup E_2$.
+Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$.
+Then
+\[
+ \pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) .
+\]
+\item
+Product morphisms are associative.
+If $\pi:E\to X$ and $\rho:D\to E$ and pinched products then
+\[
+ \rho^*\circ\pi^* = (\pi\circ\rho)^* .
+\]
+\item
+Product morphisms are compatible with restriction.
+If we have a commutative diagram
+\[ \xymatrix{
+ D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\
+ Y \ar@{^(->}[r] & X
+} \]
+such that $\rho$ and $\pi$ are pinched products, then
+\[
+ \res_D\circ\pi^* = \rho^*\circ\res_Y .
+\]
+\end{enumerate}
+\end{axiom}
-\nn{current idea for fixing this: make the above axiom a ``preliminary version"
-(as we have already done with some of the other axioms), then state the official
-axiom for maps $\pi: E \to X$ which are almost fiber bundles.
-one option is to restrict E to be a (full/half/not)-pinched product (up to homeo).
-the alternative is to give some sort of local criterion for what's allowed.
-state a gluing axiom for decomps $E = E'\cup E''$ where all three are of the correct type.
-}
+
+\medskip
All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories.
The last axiom (below), concerning actions of
@@ -438,7 +519,7 @@
\addtocounter{axiom}{-1}
\begin{axiom}{\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$}
-For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes
+For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes
\[
C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) .
\]
@@ -476,7 +557,8 @@
(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. (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.)
+but 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:
\begin{align*}
@@ -495,7 +577,8 @@
\begin{example}[Maps to a space]
\rm
\label{ex:maps-to-a-space}%
-Fix a `target space' $T$, any topological space. We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows.
+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 $T$.
For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo
@@ -504,14 +587,17 @@
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.
-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.
+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}
\begin{example}[Maps to a space, with a fiber]
\rm
\label{ex:maps-to-a-space-with-a-fiber}%
We can modify the example above, by fixing a
-closed $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$, otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged. Taking $F$ to be a point recovers the previous case.
+closed $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$,
+otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged.
+Taking $F$ to be a point recovers the previous case.
\end{example}
\begin{example}[Linearized, twisted, maps to a space]
@@ -528,27 +614,36 @@
\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' $C$
define $\cC(X)$, for $X$ a $k$-ball with $k < n$,
-to be the set of all $C$-labeled sub cell complexes of $X$ (c.f. \S \ref{sec:fields}).
-For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear
-combinations of $C$-labeled sub cell complexes of $X$
+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
+combinations of $C$-labeled embedded cell complexes of $X$
modulo the kernel of the evaluation map.
Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$,
-with each cell labelled by the $m$-th iterated identity morphism of the corresponding cell for $a$.
+with each cell labelled according to the corresponding cell for $a$.
+(These two cells have the same codimension.)
More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$.
Define $\cC(X)$, for $\dim(X) < n$,
-to be the set of all $C$-labeled sub cell complexes of $X\times F$.
+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?}
-
-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. Constructing a system of fields from $\cC$ recovers that example.
+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.
+\todo{Except that it doesn't: pasting diagrams v.s. string diagrams.}
+\nn{KW: but the above example is all about string diagrams. the only difference is at the top level,
+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.
@@ -557,6 +652,7 @@
\newcommand{\Bord}{\operatorname{Bord}}
\begin{example}[The bordism $n$-category, plain 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
@@ -591,7 +687,8 @@
\nn{maybe should also mention version where we enrich over spaces rather than chain complexes}
\end{example}
-See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$.
+See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to
+homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$.
\begin{example}[Blob complexes of balls (with a fiber)]
\rm
@@ -604,29 +701,83 @@
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. 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' 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.
+This example will be essential for Theorem \ref{product_thm} 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'?}
+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 $\pi^\infty_{\leq n}(T)$. 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 $\pi^\infty_{\leq 0}(T) \iso C_* T$.
+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 $\pi^\infty_{\leq 0}(T) \iso C_* T$.
\begin{example}[The bordism $n$-category, $A_\infty$ version]
\rm
\label{ex:bordism-category-ainf}
-blah blah \nn{to do...}
+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 $\bd X$.
+For an $n$-ball $X$ with boundary condition $c$
+define $\Bord^{n,\infty}(X; c)$ to be the space of all $k$-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.
+This is homotopy equivalent to a disjoint union of copies $\mathrm{B}\!\Homeo(W')$, where
+$W'$ runs though representatives of homeomorphism types of such manifolds.)
+\nn{check this}
\end{example}
+
+Let $\cE\cB_n$ be the operad of smooth embeddings of $k$ (little)
+copies of the standard $n$-ball $B^n$ into another (big) copy of $B^n$.
+(We require that the interiors of the little balls be disjoint, but their
+boundaries are allowed to meet.
+Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely
+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),
+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
+an action of $\cE\cB_n$.
+\nn{add citation for this operad if we can find one}
+
\begin{example}[$E_n$ algebras]
\rm
\label{ex:e-n-alg}
-Let $\cE\cB_n$ be the operad of smooth embeddings of $k$ (little)
-copies of the standard $n$-ball $B^n$ into another (big) copy of $B^n$.
-The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad.
-(By peeling the little balls, we see that both are homotopic to the space of $k$ framed points
-in $B^n$.)
Let $A$ be an $\cE\cB_n$-algebra.
+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 an $A_\infty$ $n$-category $\cC^A$.
-\nn{...}
+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
+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 $A$.
+Alternatively and more simply, we could define $\cC^A(X)$ to be
+$\Diff(B^n\to X)\times A$ modulo the diagonal action of $\Diff(B^n)$.
+The remaining data for the $A_\infty$ $n$-category
+--- composition and $\Diff(X\to X')$ action ---
+also comes from the $\cE\cB_n$ action on $A$.
+\nn{should we spell this out?}
+
+\nn{Should remark that this is just Lurie's topological chiral homology construction
+applied to $n$-balls (check this).
+Hmmm... Does Lurie do both framed and unframed cases?}
\end{example}
@@ -637,15 +788,30 @@
%\subsection{From $n$-categories to systems of fields}
\subsection{From balls to manifolds}
\label{ss:ncat_fields} \label{ss:ncat-coend}
-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.
+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
from $k$-balls to arbitrary $k$-manifolds.
-In the case of plain $n$-categories, this construction factors into a construction of a system of fields and local relations, followed by the usual TQFT definition of a vector space invariant of manifolds of 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', 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 $\cC$.
+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,
+so that we can state the boundary axiom for $\cC$ on $k+1$-balls.
+In the case of plain $n$-categories, this construction factors into a construction of a
+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',
+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 $1 \leq k \leq n$.
-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.
-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 $n$-manifolds 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).
+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.
+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 $n$-manifolds 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).
\begin{defn}
Say that a `permissible decomposition' of $W$ is a cell decomposition
@@ -657,7 +823,8 @@
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 $x$.
-The category $\cell(W)$ has objects the permissible decompositions of $W$, and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.
+The category $\cell(W)$ has objects the permissible decompositions of $W$,
+and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.
See Figure \ref{partofJfig} for an example.
\end{defn}
@@ -693,7 +860,8 @@
When the $n$-category $\cC$ is enriched in some symmetric monoidal category $(A,\boxtimes)$, and $W$ is a
closed $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$ and
-we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$. (Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.)
+we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$.
+(Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.)
Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we
fix a field on $\bd W$
(i.e. fix an element of the colimit associated to $\bd W$).
@@ -708,12 +876,17 @@
\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 $k < n$ 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 $\cC$ is an $A_\infty$ $n$-category, $\cC(W)$ 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 $\psi_{\cC;W}$.
\end{defn}
-We can specify boundary data $c \in \cC(\bdy W)$, 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 can specify boundary data $c \in \cC(\bdy W)$, 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 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 $W$
\begin{equation*}
\cC(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K
\end{equation*}
@@ -730,7 +903,9 @@
\[
V = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] ,
\]
-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 $m$.)
+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 $\psi_{\cC;W}(x_0)$
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 $\bar{x}$
@@ -750,7 +925,8 @@
permissible decomposition (filtration degree 0).
Then we glue these together with mapping cylinders coming from gluing maps
(filtration degree 1).
-Then we kill the extra homology we just introduced with mapping cylinders between the mapping cylinders (filtration degree 2), and so on.
+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.
@@ -779,7 +955,9 @@
\nn{should also develop $\pi_{\le n}(T, S)$ as a module for $\pi_{\le n}(T)$, where $S\sub T$.}
-Throughout, we fix an $n$-category $\cC$. For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. We state the final axiom, on actions of homeomorphisms, differently in the two cases.
+Throughout, we fix an $n$-category $\cC$.
+For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category.
+We state the final axiom, on actions of homeomorphisms, differently in the two cases.
Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair
$$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$
@@ -787,10 +965,11 @@
A homeomorphism between marked $k$-balls is a homeomorphism of balls which
restricts to a homeomorphism of markings.
-\mmpar{Module axiom 1}{Module morphisms}
+\begin{module-axiom}[Module morphisms]
{For each $0 \le k \le n$, we have a functor $\cM_k$ from
the category of marked $k$-balls and
homeomorphisms to the category of sets and bijections.}
+\end{module-axiom}
(As with $n$-categories, we will usually omit the subscript $k$.)
@@ -810,16 +989,21 @@
Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$.
Call such a thing a {marked $k{-}1$-hemisphere}.
-\mmpar{Module axiom 2}{Module boundaries (hemispheres)}
+\begin{lem}
+\label{lem:hemispheres}
{For each $0 \le k \le n-1$, we have a functor $\cM_k$ from
the category of marked $k$-hemispheres and
homeomorphisms to the category of sets and bijections.}
+\end{lem}
+The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details.
+We use the same type of colimit construction.
In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$.
-\mmpar{Module axiom 3}{Module boundaries (maps)}
+\begin{module-axiom}[Module boundaries (maps)]
{For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$.
These maps, for various $M$, comprise a natural transformation of functors.}
+\end{module-axiom}
Given $c\in\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$.
@@ -827,7 +1011,7 @@
then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$
and $c\in \cC(\bd M)$.
-\mmpar{Module axiom 4}{Boundary from domain and range}
+\begin{lem}[Boundary from domain and range]
{Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k$-hemisphere ($0\le k\le n-1$),
$M_i$ is a marked $k$-ball, and $E = M_1\cap M_2$ is a marked $k{-}1$-hemisphere.
Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the
@@ -837,16 +1021,19 @@
\gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \hookrightarrow \cM(H)
\]
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}.
Let $\cM(H)_E$ denote the image of $\gl_E$.
We will refer to elements of $\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$".
-\mmpar{Module axiom 5}{Module to category restrictions}
+\begin{module-axiom}[Module to category restrictions]
{For each marked $k$-hemisphere $H$ there is a restriction map
$\cM(H)\to \cC(H)$.
($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.)
These maps comprise a natural transformation of functors.}
+\end{module-axiom}
Note that combining the various boundary and restriction maps above
(for both modules and $n$-categories)
@@ -873,7 +1060,7 @@
First, we can compose two module morphisms to get another module morphism.
-\mmpar{Module axiom 6}{Module composition}
+\begin{module-axiom}[Module composition]
{Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls (with $0\le k\le n$)
and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball.
Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere.
@@ -888,14 +1075,14 @@
to the intersection of the boundaries of $M$ and $M_i$.
If $k < n$ we require that $\gl_Y$ is injective.
(For $k=n$, see below.)}
-
+\end{module-axiom}
Second, we can compose an $n$-category morphism with a module morphism to get another
module morphism.
We'll call this the action map to distinguish it from the other kind of composition.
-\mmpar{Module axiom 7}{$n$-category action}
+\begin{module-axiom}[$n$-category action]
{Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$),
$X$ is a plain $k$-ball,
and $Y = X\cap M'$ is a $k{-}1$-ball.
@@ -910,9 +1097,11 @@
to the intersection of the boundaries of $X$ and $M'$.
If $k < n$ we require that $\gl_Y$ is injective.
(For $k=n$, see below.)}
+\end{module-axiom}
-\mmpar{Module axiom 8}{Strict associativity}
+\begin{module-axiom}[Strict associativity]
{The composition and action maps above are strictly associative.}
+\end{module-axiom}
Note that the above associativity axiom applies to mixtures of module composition,
action maps and $n$-category composition.
@@ -951,7 +1140,7 @@
\cite{MR1718089}.)
%\nn{need to double-check that this is true.}
-\mmpar{Module axiom 9}{Product/identity morphisms}
+\begin{module-axiom}[Product/identity morphisms]
{Let $M$ be a marked $k$-ball and $D$ be a plain $m$-ball, with $k+m \le n$.
Then we have a map $\cM(M)\to \cM(M\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cM(M)$.
If $f:M\to M'$ and $\tilde{f}:M\times D \to M'\times D'$ are maps such that the diagram
@@ -960,6 +1149,7 @@
M \ar[r]^{f} & M'
} \]
commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.}
+\end{module-axiom}
\nn{Need to add compatibility with various things, as in the n-cat version of this axiom above.}
@@ -969,10 +1159,11 @@
modules for plain $n$-categories or $A_\infty$ $n$-categories.
In the plain case we require
-\mmpar{Module axiom 10a}{Extended isotopy invariance in dimension $n$}
+\begin{module-axiom}[\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$]
{Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts
to the identity on $\bd M$ and is extended isotopic (rel boundary) to the identity.
Then $f$ acts trivially on $\cM(M)$.}
+\end{module-axiom}
\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.}
@@ -982,7 +1173,8 @@
For $A_\infty$ modules we require
-\mmpar{Module axiom 10b}{Families of homeomorphisms act}
+\addtocounter{module-axiom}{-1}
+\begin{module-axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act]
{For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes
\[
C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) .
@@ -994,13 +1186,14 @@
a diagram like the one in Proposition \ref{CHprop} commutes.
\nn{repeat diagram here?}
\nn{restate this with $\Homeo(M\to M')$? what about boundary fixing property?}}
+\end{module-axiom}
\medskip
Note that the above axioms imply that an $n$-category module has the structure
of an $n{-}1$-category.
More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$,
-where $X$ is a $k$-ball or $k{-}1$-sphere and in the product $X\times J$ we pinch
+where $X$ is a $k$-ball and in the product $X\times J$ we pinch
above the non-marked boundary component of $J$.
(More specifically, we collapse $X\times P$ to a single point, where
$P$ is the non-marked boundary component of $J$.)
@@ -1024,13 +1217,22 @@
\end{example}
\begin{example}
-Suppose $S$ is a topological space, with a subspace $T$. We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs $(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all such maps modulo homotopies fixed on $\bdy B \setminus N$. This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and \ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains.
+Suppose $S$ is a topological space, with a subspace $T$.
+We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$
+for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs
+$(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all
+such maps modulo homotopies fixed on $\bdy B \setminus N$.
+This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}.
+Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and
+\ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to
+Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains.
\end{example}
\subsection{Modules as boundary labels (colimits for decorated manifolds)}
\label{moddecss}
-Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. Let $W$ be a $k$-manifold ($k\le n$),
+Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$.
+Let $W$ be a $k$-manifold ($k\le n$),
let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$,
and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$.
@@ -1039,7 +1241,8 @@
%component $\bd_i W$ of $W$.
%(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.)
-We will define a set $\cC(W, \cN)$ using a colimit construction similar to the one appearing in \S \ref{ss:ncat_fields} above.
+We will define a set $\cC(W, \cN)$ using a colimit construction similar to
+the one appearing in \S \ref{ss:ncat_fields} above.
(If $k = n$ and our $n$-categories are enriched, then
$\cC(W, \cN)$ will have additional structure; see below.)
@@ -1054,7 +1257,8 @@
\begin{figure}[!ht]\begin{equation*}
\mathfig{.4}{ncat/mblabel}
\end{equation*}\caption{A permissible decomposition of a manifold
-whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure}
+whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$.
+Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure}
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$.
This defines a partial ordering $\cell(W)$, which we will think of as a category.
@@ -1080,7 +1284,8 @@
If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define
$\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold
-$D\times Y_i \sub \bd(D\times W)$. It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$
+$D\times Y_i \sub \bd(D\times W)$.
+It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$
has the structure of an $n{-}k$-category, which we call $\cT(W, \cN)(D)$.
\medskip
@@ -1094,7 +1299,8 @@
a left module and the other a right module.)
Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$.
Define the tensor product $\cM_1 \tensor \cM_2$ to be the
-$n{-}1$-category $\cT(J, \{\cM_1, \cM_2\})$. This of course depends (functorially)
+$n{-}1$-category $\cT(J, \{\cM_1, \cM_2\})$.
+This of course depends (functorially)
on the choice of 1-ball $J$.
We will define a more general self tensor product (categorified coend) below.
@@ -1116,7 +1322,8 @@
we need to define morphisms of $A_\infty$ $1$-category modules and establish
some of their elementary properties.
-To motivate the definitions which follow, consider algebras $A$ and $B$, right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction
+To motivate the definitions which follow, consider algebras $A$ and $B$,
+right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction
\begin{eqnarray*}
\hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\
f &\mapsto& [x \mapsto f(x\ot -)] \\
@@ -1244,7 +1451,9 @@
\[
\olD\ot x \ot \cbar \mapsto g(\olD\ot x \ot \cbar) \in \cY(I_1\cup\cdots\cup I_{p-1}) .
\]
-For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$, where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and $\cbar''$ corresponds to the subintervals
+For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$,
+where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and
+$\cbar''$ corresponds to the subintervals
which are dropped off the right side.
(Either $\cbar'$ or $\cbar''$ might be empty.)
\nn{surely $\cbar'$ can't be empy: we don't allow $D_1$ to be empty.}
@@ -1252,10 +1461,11 @@
we have
\begin{eqnarray*}
(\bd g)(\olD\ot x \ot \cbar) &=& \bd(g(\olD\ot x \ot \cbar)) + g(\olD\ot\bd(x\ot\cbar)) + \\
- & & \;\; g((\bd_+\olD)\ot x\ot\cbar) + \gl(g((\bd_0\olD)\ot x\ot\cbar')\ot\cbar'') .
+ & & \;\; g((\bd_+\olD)\ot x\ot\cbar) + \gl''(g((\bd_0\olD)\ot \gl'(x\ot\cbar'))\ot\cbar'') .
\end{eqnarray*}
\nn{put in signs, rearrange terms to match order in previous formulas}
-Here $\gl$ denotes the module action in $\cY_\cC$.
+Here $\gl''$ denotes the module action in $\cY_\cC$
+and $\gl'$ denotes the module action in $\cX_\cC$.
This completes the definition of $\hom_\cC(\cX_\cC \to \cY_\cC)$.
Note that if $\bd g = 0$, then each
@@ -1292,14 +1502,24 @@
If $\deg(\olD) = 0$, $(\bd g) = 0$ is equivalent to the fact
that each $h_K$ is a chain map.
+We can think of a general closed element $g\in \hom_\cC(\cX_\cC \to \cY_\cC)$
+as a collection of chain maps which commute with the module action (gluing) up to coherent homotopy.
+\nn{ideally should give explicit examples of this in low degrees,
+but skip that for now.}
+\nn{should also say something about composition of morphisms; well-defined up to homotopy, or maybe
+should make some arbitrary choice}
\medskip
Given $_\cC\cZ$ and $g: \cX_\cC \to \cY_\cC$ with $\bd g = 0$ as above, we next define a chain map
\[
g\ot\id : \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ .
\]
-\nn{this is fairly straightforward, but the details are messy enough that I'm inclined
-to postpone writing it up, in the hopes that I'll think of a better way to organize things.}
+
+\nn{not sure whether to do low degree examples or try to state the general case; ideally both,
+but maybe just low degrees for now.}
+
+
+\nn{...}
@@ -1307,13 +1527,10 @@
\medskip
-\nn{do we need to say anything about composing morphisms of modules?}
-
-\nn{should we define functors between $n$-cats in a similar way?}
-
-
-\nn{...}
-
+\nn{should we define functors between $n$-cats in a similar way? i.e.\ natural transformations
+of the $\cC$ functors which commute with gluing only up to higher morphisms?
+perhaps worth having both definitions available.
+certainly the simple kind (strictly commute with gluing) arise in nature.}
@@ -1365,8 +1582,10 @@
\label{feb21a}
\end{figure}
-The $0$-marked balls can be cut into smaller balls in various ways. 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).
+The $0$-marked balls can be cut into smaller balls in various ways.
+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.
Fix $n$-categories $\cA$ and $\cB$.
--- a/text/tqftreview.tex Sun Jun 06 09:49:06 2010 -0700
+++ b/text/tqftreview.tex Sun Jun 06 09:49:57 2010 -0700
@@ -5,7 +5,15 @@
\label{sec:tqftsviafields}
In this section we review the notion of a ``system of 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}. 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.
+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}.
+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.
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
@@ -21,7 +29,9 @@
oriented, topological, smooth, spin, etc. --- but for definiteness 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 $\cS = \Vect$. The presentation here requires that the objects of $\cS$ have an underlying set, but this could probably be avoided if desired.
+Fix a symmetric monoidal category $\cS$.
+While reading the definition, you should just think about the cases $\cS = \Set$ 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 $\cS$
is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$
@@ -38,30 +48,35 @@
\begin{example}
\label{ex:traditional-n-categories(fields)}
Fix an $n$-category $C$, and let $\cC(X)$ be
-the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by
+the set of embedded cell complexes in $X$ with codimension-$j$ cells labeled by
$j$-morphisms of $C$.
-One can think of such sub-cell-complexes as dual to pasting diagrams for $C$.
+One can think of such embedded cell complexes as dual to pasting diagrams for $C$.
This is described in more detail in \S \ref{sec:example:traditional-n-categories(fields)}.
\end{example}
Now for the rest of the definition of system of fields.
+(Readers desiring a more precise definition should refer to Subsection \ref{ss:n-cat-def}
+and replace $k$-balls with $k$-manifolds.)
\begin{enumerate}
\item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$,
-and these maps are a natural
+and these maps comprise a natural
transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$.
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 $\Kom$), 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 $\cS$.
+\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 $\Kom$),
+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 $\cS$.
\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, restriction to boundary, and orientation reversal.
+compatibly with homeomorphisms and restriction to boundary.
We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$
restriction maps.
\item Gluing without corners.
-Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds.
-Let $X\sgl$ denote $X$ glued to itself along $\pm Y$.
-Using the boundary restriction, disjoint union, and (in one case) orientation reversal
+Let $\bd X = Y \du Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds.
+Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$.
+Using the boundary restriction and disjoint union
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.
@@ -70,15 +85,15 @@
\Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) ,
\]
and this gluing map is compatible with all of the above structure (actions
-of homeomorphisms, boundary restrictions, orientation reversal, disjoint union).
+of homeomorphisms, boundary restrictions, disjoint union).
Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity,
the gluing map is surjective.
-From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the
-gluing surface, we say that fields in the image of the gluing map
+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 $\pm Y$ and $W$ might intersect along their boundaries.
-Let $X\sgl$ denote $X$ glued to itself along $\pm Y$.
+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 $Y$.
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
@@ -97,8 +112,7 @@
of homeomorphisms, boundary restrictions, orientation reversal, disjoint union).
Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity,
the gluing map is surjective.
-From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the
-gluing surface, we say that fields in the image of the gluing map
+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
$c \mapsto c\times I$.
@@ -137,8 +151,8 @@
on $M$ generated by isotopy plus all instance of the above construction
(for all appropriate $Y$ and $x$).
-\nn{should also say something about pseudo-isotopy}
-
+\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(?)}
@@ -184,11 +198,12 @@
\subsection{Systems of fields from $n$-categories}
\label{sec:example:traditional-n-categories(fields)}
-We now describe in more detail Example \ref{ex:traditional-n-categories(fields)}, systems of fields coming from sub-cell-complexes labeled
+We now describe in more detail Example \ref{ex:traditional-n-categories(fields)},
+systems of fields coming from embedded cell complexes labeled
by $n$-category morphisms.
Given an $n$-category $C$ with the right sort of duality
-(e.g. a pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category),
+(e.g. a pivotal 2-category, *-1-category),
we can construct a system of fields as follows.
Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$
with codimension $i$ cells labeled by $i$-morphisms of $C$.
@@ -196,7 +211,8 @@
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. Put another way, the cell decompositions we consider are dual to standard cell
+transversely, so cells meeting the boundary are mere half-cells.
+Put another way, the cell decompositions we consider are dual to standard cell
decompositions of $X$.
We will always assume that our $n$-categories have linear $n$-morphisms.
@@ -269,7 +285,8 @@
\subsection{Local relations}
\label{sec:local-relations}
-Local relations are certain subspaces of the fields on balls, which form an ideal under gluing. Again, we give the examples first.
+Local relations are certain subspaces of the fields on balls, which form an ideal under gluing.
+Again, we give the examples first.
\addtocounter{prop}{-2}
\begin{example}[contd.]
@@ -291,12 +308,12 @@
homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$,
satisfying the following properties.
\begin{enumerate}
-\item functoriality:
+\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:
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:
+\item Ideal with respect to gluing:
if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$
\end{enumerate}
\end{defn}
@@ -352,7 +369,8 @@
Let $Y$ be an $n{-}1$-manifold.
Define a (linear) 1-category $A(Y)$ as follows.
The objects of $A(Y)$ are $\cC(Y)$.
-The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$, where $a$ and $b$ label the two boundary components of the cylinder $Y\times I$.
+The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$,
+where $a$ and $b$ label the two boundary components of the cylinder $Y\times I$.
Composition is given by gluing of cylinders.
Let $X$ be an $n$-manifold with boundary and consider the collection of vector spaces