# HG changeset patch # User Scott Morrison # Date 1284575627 18000 # Node ID 0b5c9bc2519162b9820476a3f4ad417cbbc8a38f # Parent 9ca95f2627f861efcdddea95c7150f02939499aa# Parent df1f7400d6efe56c7190299670ef730c76b48df6 Automated merge with https://tqft.net/hg/blob/ diff -r 9ca95f2627f8 -r 0b5c9bc25191 preamble.tex --- a/preamble.tex Wed Sep 15 11:27:12 2010 -0700 +++ b/preamble.tex Wed Sep 15 13:33:47 2010 -0500 @@ -62,8 +62,8 @@ \newtheorem*{defn*}{Definition} % unnumbered definition \newtheorem{question}{Question} \newtheorem{property}{Property} -\newtheorem{axiom}{Axiom} -\newtheorem{module-axiom}{Module Axiom} +\newtheorem{axiom}{Axiom}[section] +\newtheorem{module-axiom}{Module Axiom}[section] \newenvironment{rem}{\noindent\textsl{Remark.}}{} % perhaps looks better than rem above? \newtheorem{rem*}[prop]{Remark} \newtheorem{remark}[prop]{Remark} diff -r 9ca95f2627f8 -r 0b5c9bc25191 text/deligne.tex --- a/text/deligne.tex Wed Sep 15 11:27:12 2010 -0700 +++ b/text/deligne.tex Wed Sep 15 13:33:47 2010 -0500 @@ -108,7 +108,7 @@ \begin{figure}[t] $$\mathfig{.4}{deligne/dfig3a} \to \mathfig{.4}{deligne/dfig3b} $$ \caption{Conjugating by a homeomorphism -\nn{change right $R_i$ to $R'_i$}}\label{xdfig3} +\label{xdfig3} \end{figure} \item If $M_i = M'_i \du M''_i$ and $N_i = N'_i \du N''_i$ (and there is a compatible disjoint union of $\bd M = \bd N$), we can replace diff -r 9ca95f2627f8 -r 0b5c9bc25191 text/evmap.tex --- a/text/evmap.tex Wed Sep 15 11:27:12 2010 -0700 +++ b/text/evmap.tex Wed Sep 15 13:33:47 2010 -0500 @@ -26,7 +26,7 @@ sort-of-simplicial set into a sort-of-simplicial space. Taking singular chains of this space we get $\btc_*(X)$. The details are in \S \ref{ss:alt-def}. -We also prove a useful lemma (\ref{small-blobs-b}) which says that we can assume that +We also prove a useful result (Lemma \ref{small-blobs-b}) which says that we can assume that blobs are small with respect to any fixed open cover. @@ -226,9 +226,9 @@ \end{itemize} We can summarize the above by saying that in the typical continuous family -$P\to \BD_k(M)$, $p\mapsto (B_i(p), u_i(p), r(p)$, $B_i(p)$ and $r(p)$ are induced by a map +$P\to \BD_k(M)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ and $r(p)$ are induced by a map $P\to \Homeo(M)$, with the twig blob labels $u_i(p)$ varying independently. -We note that while have no need to allow the blobs $B_i(p)$ to vary independently of the field $r(p)$, +We note that while we've decided not to allow the blobs $B_i(p)$ to vary independently of the field $r(p)$, if we did allow this it would not affect the truth of the claims we make below. In particular, we would get a homotopy equivalent complex $\btc_*(M)$. diff -r 9ca95f2627f8 -r 0b5c9bc25191 text/intro.tex --- a/text/intro.tex Wed Sep 15 11:27:12 2010 -0700 +++ b/text/intro.tex Wed Sep 15 13:33:47 2010 -0500 @@ -3,7 +3,7 @@ \section{Introduction} We construct a chain complex $\bc_*(M; \cC)$ --- the ``blob complex'' --- -associated to an $n$-manifold $M$ and a linear $n$-category with strong duality $\cC$. +associated to an $n$-manifold $M$ and a linear $n$-category $\cC$ with strong duality. This blob complex provides a simultaneous generalization of several well known constructions: \begin{itemize} \item The 0-th homology $H_0(\bc_*(M; \cC))$ is isomorphic to the usual @@ -124,7 +124,7 @@ } (FU.100); \draw[->] (C) -- node[above left=3pt] {restrict to \\ standard balls} (tC); \draw[->] (FU.80) -- node[right] {restrict \\ to balls} (C.-80); -\draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Theorem \ref{thm:skein-modules}} (A); +\draw[->] (BC) -- node[right] {$H_0$ \\ c.f. Theorem \ref{thm:skein-modules}} (A); \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); \draw[<->] (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); @@ -367,17 +367,14 @@ for any homeomorphic pair $X$ and $Y$, satisfying corresponding conditions. -\nn{KW: the next paragraph seems awkward to me} - -\nn{KW: also, I'm not convinced that all of these (above and below) should be called theorems} +In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. +Below, when we talk about the blob complex for a topological $n$-category, we are implicitly passing first to this associated system of fields. +Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. In that section we describe how to use the blob complex to construct $A_\infty$ $n$-categories from topological $n$-categories: -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. +\newtheorem*{ex:blob-complexes-of-balls}{Example \ref{ex:blob-complexes-of-balls}} -\todo{Give this a number inside the text} -\begin{thm}[Blob complexes of products with balls form an $A_\infty$ $n$-category] -\label{thm:blobs-ainfty} +\begin{ex:blob-complexes-of-balls}[Blob complexes of products with balls form an $A_\infty$ $n$-category] +%\label{thm:blobs-ainfty} 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 $0 \leq m < k$, @@ -386,17 +383,15 @@ (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 Theorem \ref{thm:evaluation}. -\end{thm} +\end{ex:blob-complexes-of-balls} \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. \end{rem} -Theorem \ref{thm:blobs-ainfty} appears as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats}. - 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, $\cl{\cC}(M)$. +The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. The next theorem describes the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as in the previous example. %The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. \newtheorem*{thm:product}{Theorem \ref{thm:product}} @@ -404,7 +399,7 @@ \begin{thm:product}[Product formula] 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 Theorem \ref{thm:blobs-ainfty}). +Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Example \ref{ex:blob-complexes-of-balls}). Then \[ \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). diff -r 9ca95f2627f8 -r 0b5c9bc25191 text/ncat.tex --- a/text/ncat.tex Wed Sep 15 11:27:12 2010 -0700 +++ b/text/ncat.tex Wed Sep 15 13:33:47 2010 -0500 @@ -1093,8 +1093,7 @@ \end{itemize} In other words, we have a zig-zag of equivalences starting at $a$ and ending at $\hat{a}$. The idea of the proof is to produce a similar zig-zag where everything antirefines to the same -disjoint union of balls, and then invoke the associativity axiom \ref{nca-assoc}. -\nn{hmmm... it would be nicer if this were ``7.xx" instead of ``4"} +disjoint union of balls, and then invoke Axiom \ref{nca-assoc} which ensures associativity. Let $z$ be a decomposition of $W$ which is in general position with respect to all of the $x_i$'s and $v_i$'s.