# HG changeset patch # User Kevin Walker # Date 1280535557 14400 # Node ID 4a23163843a920a01c67eb814b1e4619110cc9ba # Parent 8ed3aeb787785988d632574378ef626507e417bd misc diff -r 8ed3aeb78778 -r 4a23163843a9 text/basic_properties.tex --- a/text/basic_properties.tex Fri Jul 30 18:36:08 2010 -0400 +++ b/text/basic_properties.tex Fri Jul 30 20:19:17 2010 -0400 @@ -86,8 +86,8 @@ Note that $S$ is a disjoint union of balls. Assign to $b$ the acyclic (in positive degrees) subcomplex $T(b) \deq r\bullet\bc_*(S)$. Note that if a diagram $b'$ is part of $\bd b$ then $T(B') \sub T(b)$. -Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models), -so $f$ and the identity map are homotopic. \nn{We should actually have a section \S \ref{sec:moam} with a definition of ``compatible" and this statement as a lemma} +Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models, \S\ref{sec:moam}), +so $f$ and the identity map are homotopic. \end{proof} For the next proposition we will temporarily restore $n$-manifold boundary diff -r 8ed3aeb78778 -r 4a23163843a9 text/blobdef.tex --- a/text/blobdef.tex Fri Jul 30 18:36:08 2010 -0400 +++ b/text/blobdef.tex Fri Jul 30 20:19:17 2010 -0400 @@ -177,9 +177,20 @@ In the example above, note that $$A \sqcup B \sqcup C \sqcup D \to (A \cup B) \sqcup (C \cup D) \to A \cup B \cup C \cup D$$ is a ball decomposition, but other sequences of gluings starting from $A \sqcup B \sqcup C \sqcup D$have intermediate steps which are not manifolds. We'll now slightly restrict the possible configurations of blobs. +%%%%% oops -- I missed the similar discussion after the definition +%The basic idea is that each blob in a configuration +%is the image a ball, with embedded interior and possibly glued-up boundary; +%distinct blobs should either have disjoint interiors or be nested; +%and the entire configuration should be compatible with some gluing decomposition of $X$. \begin{defn} \label{defn:configuration} -A configuration of $k$ blobs in $X$ is an ordered collection of $k$ subsets $\{B_1, \ldots B_k\}$ of $X$ such that there exists a gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$ and for each subset $B_i$ there is some $0 \leq r \leq m$ and some connected component $M_r'$ of $M_r$ which is a ball, so $B_i$ is the image of $M_r'$ in $X$. Such a gluing decomposition is \emph{compatible} with the configuration. A blob $B_i$ is a twig blob if no other blob $B_j$ maps into the appropriate $M_r'$. \nn{that's a really clumsy way to say it, but I struggled to say it nicely and still allow boundaries to intersect -S} +A configuration of $k$ blobs in $X$ is an ordered collection of $k$ subsets $\{B_1, \ldots B_k\}$ +of $X$ such that there exists a gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$ and +for each subset $B_i$ there is some $0 \leq r \leq m$ and some connected component $M_r'$ of +$M_r$ which is a ball, so $B_i$ is the image of $M_r'$ in $X$. +We say that such a gluing decomposition +is \emph{compatible} with the configuration. +A blob $B_i$ is a twig blob if no other blob $B_j$ is a strict subset of it. \end{defn} In particular, this implies what we said about blobs above: that for any two blobs in a configuration of blobs in $X$, diff -r 8ed3aeb78778 -r 4a23163843a9 text/ncat.tex --- a/text/ncat.tex Fri Jul 30 18:36:08 2010 -0400 +++ b/text/ncat.tex Fri Jul 30 20:19:17 2010 -0400 @@ -871,7 +871,6 @@ \begin{example}[$E_n$ algebras] \rm \label{ex:e-n-alg} - 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 $\Diff(B^n)$. @@ -892,14 +891,13 @@ also comes from the $\cE\cB_n$ action on $A$. \nn{should we spell this out?} -\nn{Should remark that the associated hocolim for manifolds -agrees with Lurie's topological chiral homology construction; maybe wait -until next subsection to say that?} - Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms $\cC(X)$ are trivial (single point) for $k