--- a/blob1.tex Wed Mar 23 15:52:36 2011 -0700
+++ b/blob1.tex Tue Mar 29 13:30:35 2011 -0700
@@ -17,11 +17,6 @@
\maketitle
-%[revision $\ge$ 527; $\ge$ 30 August 2010]
-%
-%{\color[rgb]{.9,.5,.2} \large \textbf{Draft version, read with caution.}}
-%We're in the midst of revising this, and hope to have a version on the arXiv soon.
-
\begin{abstract}
Given an $n$-manifold $M$ and an $n$-category $\cC$, we define a chain complex
(the ``blob complex") $\bc_*(M; \cC)$.
@@ -46,8 +41,6 @@
}
-%\let\stdsection\section
-%\renewcommand\section{\newpage\stdsection}
\input{text/intro}
@@ -73,8 +66,6 @@
\input{text/appendixes/famodiff}
-%\input{text/appendixes/smallblobs}
-
\input{text/appendixes/comparing_defs}
%\input{text/comm_alg}
--- a/text/ncat.tex Wed Mar 23 15:52:36 2011 -0700
+++ b/text/ncat.tex Tue Mar 29 13:30:35 2011 -0700
@@ -821,15 +821,18 @@
}
-\begin{example}[The bordism $n$-category, ordinary version]
+\begin{example}[The bordism $n$-category of $d$-manifolds, ordinary version]
\label{ex:bord-cat}
\rm
\label{ex:bordism-category}
-For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional PL
+For a $k$-ball $X$, $k<n$, define $\Bord^{n,d}(X)$ to be the set of all $(d{-}n{+}k)$-dimensional PL
submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W = W \cap \bd X \times \Real^\infty$.
-For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes (rel boundary) of such $n$-dimensional submanifolds;
+For an $n$-ball $X$ define $\Bord^{n,d}(X)$ to be homeomorphism classes (rel boundary) of such $d$-dimensional submanifolds;
we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism
$W \to W'$ which restricts to the identity on the boundary.
+For $n=1$ we have the familiar bordism 1-category of $d$-manifolds.
+The case $n=d$ captures the $n$-categorical nature of bordisms.
+The case $n > 2d$ captures the full symmetric monoidal $n$-category structure.
\end{example}
\begin{remark}
Working with the smooth bordism category would require careful attention to either collars, corners or halos.
@@ -893,15 +896,14 @@
linear combinations of connected components of $T$, and the local relations are trivial.
There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$.
-\begin{example}[The bordism $n$-category, $A_\infty$ version]
+\begin{example}[The bordism $n$-category of $d$-manifolds, $A_\infty$ version]
\rm
\label{ex:bordism-category-ainf}
-As in Example \ref{ex:bord-cat}, for $X$ a $k$-ball, $k<n$, we define $\Bord^{n,\infty}(X)$
-to be the set of all $k$-dimensional
-submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W$ is
-contained in $\bd X \times \Real^\infty$.
+As in Example \ref{ex:bord-cat}, for $X$ a $k$-ball, $k<n$, we define $\Bord^{n,d}_\infty(X)$
+to be the set of all $(d{-}n{+}k)$-dimensional
+submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W = W \cap \bd X \times \Real^\infty$.
For an $n$-ball $X$ with boundary condition $c$
-define $\Bord^{n,\infty}(X; c)$ to be the space of all $k$-dimensional
+define $\Bord^{n,d}_\infty(X; c)$ to be the space of all $d$-dimensional
submanifolds $W$ of $X\times \Real^\infty$ such that
$W$ coincides with $c$ at $\bd X \times \Real^\infty$.
(The topology on this space is induced by ambient isotopy rel boundary.