# HG changeset patch
# User Scott Morrison <scott@tqft.net>
# Date 1301430635 25200
# Node ID 59c29ecf2f66d4a37ebd44483c5314d2a4b52671
# Parent  ae93002b511e7570f82e0c7d0403ebd26f22d5ad# Parent  c48da1288047ba691f07619d02d80d75c4de0ad7
Automated merge with https://tqft.net/hg/blob

diff -r c48da1288047 -r 59c29ecf2f66 blob1.tex
--- 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}
diff -r c48da1288047 -r 59c29ecf2f66 text/ncat.tex
--- 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.