--- a/text/a_inf_blob.tex Thu May 27 15:06:48 2010 -0700
+++ b/text/a_inf_blob.tex Thu May 27 20:09:47 2010 -0700
@@ -15,6 +15,12 @@
\medskip
+\subsection{The small blob complex}
+
+\input{text/smallblobs}
+
+\subsection{A product formula}
+
Let $M^n = Y^k\times F^{n-k}$.
Let $C$ be a plain $n$-category.
Let $\cF$ be the $A_\infty$ $k$-category which assigns to a $k$-ball
@@ -25,7 +31,7 @@
new-fangled blob complex $\bc_*^\cF(Y)$.
\end{thm}
-\input{text/smallblobs}
+
\begin{proof}[Proof of Theorem \ref{product_thm}]
We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}.
@@ -213,6 +219,9 @@
\medskip
+\subsection{A gluing theorem}
+\label{sec:gluing}
+
Next we prove a gluing theorem.
Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$.
We will need an explicit collar on $Y$, so rewrite this as
@@ -230,6 +239,7 @@
\end{itemize}
\begin{thm}
+\label{thm:gluing}
$\bc(X) \cong \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
\end{thm}
@@ -254,6 +264,8 @@
\medskip
+\subsection{Reconstructing mapping spaces}
+
The next theorem shows how to reconstruct a mapping space from local data.
Let $T$ be a topological space, let $M$ be an $n$-manifold,
and recall the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$
--- a/text/ncat.tex Thu May 27 15:06:48 2010 -0700
+++ b/text/ncat.tex Thu May 27 20:09:47 2010 -0700
@@ -3,7 +3,7 @@
\def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip}
\def\mmpar#1#2#3{\smallskip\noindent{\bf #1} (#2). {\it #3} \smallskip}
-\section{$n$-categories}
+\section{Definitions of $n$-categories}
\label{sec:ncats}
\subsection{Definition of $n$-categories}
@@ -1025,9 +1025,8 @@
%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 above.
-\nn{give ref}
-(If $k = n$ and our $k$-categories are enriched, then
+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.)
Define a permissible decomposition of $W$ to be a decomposition
@@ -1039,7 +1038,7 @@
with $M_{ib}\cap Y_i$ being the marking.
(See Figure \ref{mblabel}.)
\begin{figure}[!ht]\begin{equation*}
-\mathfig{.6}{ncat/mblabel}
+\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}
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement
@@ -1048,7 +1047,7 @@
(The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique
morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.)
-$\cN$ determines
+The collection of modules $\cN$ determines
a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets
(possibly with additional structure if $k=n$).
For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset
@@ -1057,20 +1056,18 @@
\]
such that the restrictions to the various pieces of shared boundaries amongst the
$X_a$ and $M_{ib}$ all agree.
-(Think fibered product.)
+(That is, the fibered product over the boundary maps.)
If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$
via the gluing (composition or action) maps from $\cC$ and the $\cN_i$.
-Finally, define $\cC(W, \cN)$ to be the colimit of $\psi_\cN$.
-(Recall that if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means
+We now define the set $\cC(W, \cN)$ to be the colimit of the functor $\psi_\cN$.
+(As usual, if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means
homotopy colimit.)
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 $D \mapsto \cT(W, \cN)(D) \deq \cC(D\times W, \cN)$
-has the structure of an $n{-}k$-category.
+$D\times Y_i \sub \bd(D\times W)$. 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
@@ -1079,15 +1076,11 @@
construction to define tensor products
of modules.
Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$.
-(If $k=1$ and manifolds are oriented, then one should be
+(If $k=1$ and our manifolds are oriented, then one should be
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 of $\cM_1$ and $\cM_2$ to be the
-$n{-}1$-category $\cT(J, \cM_1, \cM_2)$,
-\[
- \cM_1\otimes \cM_2 \deq \cT(J, \cM_1, \cM_2) .
-\]
-This of course depends (functorially)
+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)
on the choice of 1-ball $J$.
We will define a more general self tensor product (categorified coend) below.
@@ -1105,11 +1098,10 @@
In order to state and prove our version of the higher dimensional Deligne conjecture
(Section \ref{sec:deligne}),
-we need to define morphisms of $A_\infty$ 1-cat modules and establish
+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/bi/left modules
-$X_B$, $_BY_A$ and $Z_A$, 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 -)] \\
@@ -1125,43 +1117,43 @@
(\cM_\cC\ot {_\cC\cN})^* \cong \hom_\cC(\cM_\cC \to (_\cC\cN)^*) .
\]
-In the next few paragraphs we define the things appearing in the above equation:
+In the next few paragraphs we define the objects appearing in the above equation:
$\cM_\cC\ot {_\cC\cN}$, $(\cM_\cC\ot {_\cC\cN})^*$, $(_\cC\cN)^*$ and finally
$\hom_\cC$.
-In the previous subsection we defined a tensor product of $A_\infty$ $n$-cat modules
+
+\def\olD{{\overline D}}
+\def\cbar{{\bar c}}
+In the previous subsection we defined a tensor product of $A_\infty$ $n$-category modules
for general $n$.
For $n=1$ this definition is a homotopy colimit indexed by subdivisions of a fixed interval $J$
and their gluings (antirefinements).
-(The tensor product will depend (functorially) on the choice of $J$.)
-To a subdivision
+(This tensor product depends functorially on the choice of $J$.)
+To a subdivision $D$
\[
J = I_1\cup \cdots\cup I_p
\]
we associate the chain complex
\[
- \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{m-1})\ot\cN(I_m) .
+ \psi(D) = \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{m-1})\ot\cN(I_m) .
\]
-(If $D$ denotes the subdivision of $J$, then we denote this complex by $\psi(D)$.)
To each antirefinement we associate a chain map using the composition law of $\cC$ and the
module actions of $\cC$ on $\cM$ and $\cN$.
-\def\olD{{\overline D}}
-\def\cbar{{\bar c}}
The underlying graded vector space of the homotopy colimit is
\[
\bigoplus_l \bigoplus_{\olD} \psi(D_0)[l] ,
\]
where $l$ runs through the natural numbers, $\olD = (D_0\to D_1\to\cdots\to D_l)$
-runs through chains of antirefinements, and $[l]$ denotes a grading shift.
+runs through chains of antirefinements of length $l+1$, and $[l]$ denotes a grading shift.
We will denote an element of the summand indexed by $\olD$ by
$\olD\ot m\ot\cbar\ot n$, where $m\ot\cbar\ot n \in \psi(D_0)$.
-The boundary map is given (ignoring signs) by
-\begin{eqnarray*}
- \bd(\olD\ot m\ot\cbar\ot n) &=& \olD\ot\bd(m\ot\cbar)\ot n + \olD\ot m\ot\cbar\ot \bd n + \\
- & & \;\; (\bd_+ \olD)\ot m\ot\cbar\ot n + (\bd_0 \olD)\ot \rho(m\ot\cbar\ot n) ,
-\end{eqnarray*}
-where $\bd_+ \olD = \sum_{i>0} (D_0, \cdots \widehat{D_i} \cdots , D_l)$ (the part of the simplicial
-boundary which retains $D_0$), $\bd_0 \olD = (D_1, \cdots , D_l)$,
+The boundary map is given by
+\begin{align*}
+ \bd(\olD\ot m\ot\cbar\ot n) &= (\bd_0 \olD)\ot \rho(m\ot\cbar\ot n) + (\bd_+ \olD)\ot m\ot\cbar\ot n \; + \\
+ & \qquad + (-1)^l \olD\ot\bd(m\ot\cbar\ot n)
+\end{align*}
+where $\bd_+ \olD = \sum_{i>0} (-1)^i (D_0\to \cdots \to \widehat{D_i} \to \cdots \to D_l)$ (those parts of the simplicial
+boundary which retain $D_0$), $\bd_0 \olD = (D_1 \to \cdots \to D_l)$,
and $\rho$ is the gluing map associated to the antirefinement $D_0\to D_1$.
$(\cM_\cC\ot {_\cC\cN})^*$ is just the dual chain complex to $\cM_\cC\ot {_\cC\cN}$:
@@ -1175,7 +1167,7 @@
f(\olD\ot m\ot\cbar\ot \bd n) + \\
& & \;\; f((\bd_+ \olD)\ot m\ot\cbar\ot n) + f((\bd_0 \olD)\ot \rho(m\ot\cbar\ot n)) .
\end{eqnarray*}
-(Again, we are ignoring signs.)
+(Again, we are ignoring signs.) \nn{put signs in}
Next we define the dual module $(_\cC\cN)^*$.
This will depend on a choice of interval $J$, just as the tensor product did.
@@ -1205,7 +1197,7 @@
We are almost ready to give the definition of morphisms between arbitrary modules
$\cX_\cC$ and $\cY_\cC$.
Note that the rightmost interval $I_m$ does not appear above, except implicitly in $\olD$.
-To fix this, we define subdivisions are antirefinements of left-marked intervals.
+To fix this, we define subdivisions as antirefinements of left-marked intervals.
Subdivisions are just the obvious thing, but antirefinements are defined to mimic
the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always
omitted.