# HG changeset patch # User Kevin Walker # Date 1284963321 25200 # Node ID 0bc6fa29b62ab7fdaaa143ce58dd57e307f76a4a # Parent 4f142fcd386e1c81aa52e029f2d36b255df785aa# Parent 3baa4e4d395e4434da0acd328e480ecbfbf7479f Automated merge with https://tqft.net/hg/blob/ diff -r 3baa4e4d395e -r 0bc6fa29b62a preamble.tex --- a/preamble.tex Sun Sep 19 23:14:41 2010 -0700 +++ b/preamble.tex Sun Sep 19 23:15:21 2010 -0700 @@ -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 3baa4e4d395e -r 0bc6fa29b62a text/a_inf_blob.tex --- a/text/a_inf_blob.tex Sun Sep 19 23:14:41 2010 -0700 +++ b/text/a_inf_blob.tex Sun Sep 19 23:15:21 2010 -0700 @@ -10,7 +10,7 @@ that when $\cC$ is obtained from a system of fields $\cD$ as the blob complex of an $n$-ball (see Example \ref{ex:blob-complexes-of-balls}), $\cl{\cC}(M)$ is homotopy equivalent to -our original definition of the blob complex $\bc_*^\cD(M)$. +our original definition of the blob complex $\bc_*(M;\cD)$. %\medskip @@ -33,7 +33,7 @@ Given a system of fields $\cE$ and a $n{-}k$-manifold $F$, recall from Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $\cC_F$ defined by $\cC_F(X) = \cE(X\times F)$ if $\dim(X) < k$ and -$\cC_F(X) = \bc_*^\cE(X\times F)$ if $\dim(X) = k$. +$\cC_F(X) = \bc_*(X\times F;\cE)$ if $\dim(X) = k$. \begin{thm} \label{thm:product} diff -r 3baa4e4d395e -r 0bc6fa29b62a text/deligne.tex --- a/text/deligne.tex Sun Sep 19 23:14:41 2010 -0700 +++ b/text/deligne.tex Sun Sep 19 23:15:21 2010 -0700 @@ -107,8 +107,8 @@ (See Figure \ref{xdfig3}.) \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} +\caption{Conjugating by a homeomorphism.} +\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 @@ -122,7 +122,7 @@ (See Figure \ref{xdfig1}.) \begin{figure}[t] $$\mathfig{.3}{deligne/dfig1a} \leftarrow \mathfig{.3}{deligne/dfig1b} \rightarrow \mathfig{.3}{deligne/dfig1c}$$ -\caption{Changing the order of a surgery}\label{xdfig1} +\caption{Changing the order of a surgery.}\label{xdfig1} \end{figure} \end{itemize} diff -r 3baa4e4d395e -r 0bc6fa29b62a text/evmap.tex --- a/text/evmap.tex Sun Sep 19 23:14:41 2010 -0700 +++ b/text/evmap.tex Sun Sep 19 23:15:21 2010 -0700 @@ -21,12 +21,12 @@ introduce a homotopy equivalent alternate version of the blob complex, $\btc_*(X)$, which is more amenable to this sort of action. Recall from Remark \ref{blobsset-remark} that blob diagrams -have the structure of a sort-of-simplicial set. +have the structure of a sort-of-simplicial set. \nn{need a more conventional sounding name: `polyhedral set'?} Blob diagrams can also be equipped with a natural topology, which converts this 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. @@ -70,17 +70,17 @@ \medskip Fix $\cU$, an open cover of $X$. -Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(X)$ +Define the ``small blob complex" $\bc^{\cU}_*(X)$ to be the subcomplex of $\bc_*(X)$ of all blob diagrams in which every blob is contained in some open set of $\cU$, and moreover each field labeling a region cut out by the blobs is splittable into fields on smaller regions, each of which is contained in some open set of $\cU$. \begin{lemma}[Small blobs] \label{small-blobs-b} \label{thm:small-blobs} -The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. +The inclusion $i: \bc^{\cU}_*(X) \into \bc_*(X)$ is a homotopy equivalence. \end{lemma} \begin{proof} -It suffices to show that for any finitely generated pair of subcomplexes +It suffices \nn{why? we should spell this out somewhere} to show that for any finitely generated pair $(C_*, D_*)$, with $D_*$ a subcomplex of $C_*$ such that \[ (C_*, D_*) \sub (\bc_*(X), \sbc_*(X)) \] @@ -92,20 +92,20 @@ for all $x\in C_*$. For simplicity we will assume that all fields are splittable into small pieces, so that -$\sbc_0(X) = \bc_0$. +$\sbc_0(X) = \bc_0(X)$. (This is true for all of the examples presented in this paper.) Accordingly, we define $h_0 = 0$. Next we define $h_1$. Let $b\in C_1$ be a 1-blob diagram. Let $B$ be the blob of $b$. -We will construct a 1-chain $s(b)\in \sbc_1$ such that $\bd(s(b)) = \bd b$ +We will construct a 1-chain $s(b)\in \sbc_1(X)$ such that $\bd(s(b)) = \bd b$ and the support of $s(b)$ is contained in $B$. -(If $B$ is not embedded in $X$, then we implicitly work in some term of a decomposition +(If $B$ is not embedded in $X$, then we implicitly work in some stage of a decomposition of $X$ where $B$ is embedded. -See \ref{defn:configuration} and preceding discussion.) -It then follows from \ref{disj-union-contract} that we can choose -$h_1(b) \in \bc_1(X)$ such that $\bd(h_1(b)) = s(b) - b$. +See Definition \ref{defn:configuration} and preceding discussion.) +It then follows from Corollary \ref{disj-union-contract} that we can choose +$h_1(b) \in \bc_2(X)$ such that $\bd(h_1(b)) = s(b) - b$. Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series of small collar maps, plus a shrunken version of $b$. @@ -113,9 +113,9 @@ Let $\cV_1$ be an auxiliary open cover of $X$, subordinate to $\cU$ and also satisfying conditions specified below. -Let $b = (B, u, r)$, $u = \sum a_i$ be the label of $B$, $a_i\in \bc_0(B)$. +Let $b = (B, u, r)$, with $u = \sum a_i$ the label of $B$, and $a_i\in \bc_0(B)$. Choose a sequence of collar maps $\bar{f}_j:B\cup\text{collar}\to B$ satisfying conditions which we cannot express -until introducing more notation. +until introducing more notation. \nn{needs some rewriting, I guess} Let $f_j:B\to B$ be the restriction of $\bar{f}_j$ to $B$; $f_j$ maps $B$ homeomorphically to a slightly smaller submanifold of $B$. Let $g_j = f_1\circ f_2\circ\cdots\circ f_j$. @@ -125,13 +125,13 @@ $g_{j-1}(|f_j|)$ is also contained is an open set of $\cV_1$. There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ -(more specifically, $|c_{ij}| = g_{j-1}(|f_j|)$) +(more specifically, $|c_{ij}| = g_{j-1}(|f_j|)$ \nn{doesn't strictly make any sense}) and $\bd c_{ij} = g_{j-1}(a_i) - g_{j}(a_i)$. Define \[ s(b) = \sum_{i,j} c_{ij} + g(b) \] -and choose $h_1(b) \in \bc_1(X)$ such that +and choose $h_1(b) \in \bc_2(X)$ such that \[ \bd(h_1(b)) = s(b) - b . \] @@ -141,12 +141,12 @@ Let $B = |b|$, either a ball or a union of two balls. By possibly working in a decomposition of $X$, we may assume that the ball(s) of $B$ are disjointly embedded. -We will construct a 2-chain $s(b)\in \sbc_2$ such that +We will construct a 2-chain $s(b)\in \sbc_2(X)$ such that \[ \bd(s(b)) = \bd(h_1(\bd b) + b) = s(\bd b) \] and the support of $s(b)$ is contained in $B$. -It then follows from \ref{disj-union-contract} that we can choose +It then follows from Corollary \ref{disj-union-contract} that we can choose $h_2(b) \in \bc_2(X)$ such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$. Similarly to the construction of $h_1$ above, @@ -156,7 +156,7 @@ disjoint union of balls. Let $\cV_2$ be an auxiliary open cover of $X$, subordinate to $\cU$ and -also satisfying conditions specified below. +also satisfying conditions specified below. \nn{This happens sufficiently far below (i.e. not in this paragraph) that we probably should give better warning.} As before, choose a sequence of collar maps $f_j$ such that each has support contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms @@ -168,7 +168,7 @@ Let $s(\bd b) = \sum e_k$, and let $\{p_m\}$ be the 0-blob diagrams appearing in the boundaries of the $e_k$. As in the construction of $h_1$, we can choose 1-blob diagrams $q_m$ such that -$\bd q_m = g_{j-1}(p_m) - g_j(p_m)$ and $\supp(q_m)$ is contained in an open set of $\cV_1$. +$\bd q_m = g_{j-1}(p_m) - g_j(p_m)$ and $|q_m|$ is contained in an open set of $\cV_1$. If $x$ is a sum of $p_m$'s, we denote the corresponding sum of $q_m$'s by $q(x)$. Now consider, for each $k$, $g_{j-1}(e_k) - q(\bd e_k)$. @@ -183,7 +183,7 @@ (In this case there are either one or two balls in the disjoint union.) For any fixed open cover $\cV_2$ this condition can be satisfied by choosing $\cV_1$ to be a sufficiently fine cover. -It follows from \ref{disj-union-contract} that we can choose +It follows from Corollary \ref{disj-union-contract} that we can choose $x_k \in \bc_2(X)$ with $\bd x_k = g_{j-1}(e_k) - g_j(e_k) - q(\bd e_k)$ and with $\supp(x_k) = U$. We can now take $d_j \deq \sum x_k$. @@ -219,24 +219,25 @@ We give $\BD_k$ the finest topology such that \begin{itemize} \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. +\item \nn{don't we need something for collaring maps?} \item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous. \item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous, where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on -$\bc_0(B)$ comes from the generating set $\BD_0(B)$. +$\bc_0(B)$ comes from the generating set $\BD_0(B)$. \nn{don't we need to say more to specify a topology on an $\infty$-dimensional vector space} \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 -$P\to \Homeo(M)$, 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 $r(p)$, +$P\to \BD_k(X)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ and $r(p)$ are induced by a map +$P\to \Homeo(X)$, with the twig blob labels $u_i(p)$ varying independently. +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 $\btc_*(M)$. +In particular, such a definition of $\btc_*(X)$ would result in a homotopy equivalent complex. Next we define $\btc_*(X)$ to be the total complex of the double complex (denoted $\btc_{**}$) whose $(i,j)$ entry is $C_j(\BD_i)$, the singular $j$-chains on the space of $i$-blob diagrams. The vertical boundary of the double complex, denoted $\bd_t$, is the singular boundary, and the horizontal boundary, denoted $\bd_b$, is -the blob boundary. +the blob boundary. Following the usual sign convention, we have $\bd = \bd_b + (-1)^i \bd_t$. We will regard $\bc_*(X)$ as the subcomplex $\btc_{*0}(X) \sub \btc_{**}(X)$. The main result of this subsection is @@ -251,7 +252,7 @@ $\btc_*(B^n)$ is contractible (acyclic in positive degrees). \end{lemma} \begin{proof} -We will construct a contracting homotopy $h: \btc_*(B^n)\to \btc_*(B^n)$. +We will construct a contracting homotopy $h: \btc_*(B^n)\to \btc_{*+1}(B^n)$. We will assume a splitting $s:H_0(\btc_*(B^n))\to \btc_0(B^n)$ of the quotient map $q:\btc_0(B^n)\to H_0(\btc_*(B^n))$. @@ -266,9 +267,10 @@ e: \btc_{ij}\to\btc_{i+1,j} \] adds an outermost blob, equal to all of $B^n$, to the $j$-parameter family of blob diagrams. +Note that for fixed $i$, $e$ is a chain map, i.e. $\bd_t e = e \bd_t$. A generator $y\in \btc_{0j}$ is a map $y:P\to \BD_0$, where $P$ is some $j$-dimensional polyhedron. -We define $r(y)\in \btc_{0j}$ to be the constant function $r\circ y : P\to \BD_0$. +We define $r(y)\in \btc_{0j}$ to be the constant function $r\circ y : P\to \BD_0$. \nn{I found it pretty confusing to reuse the letter $r$ here.} Let $c(r(y))\in \btc_{0,j+1}$ be the constant map from the cone of $P$ to $\BD_0$ taking the same value (namely $r(y(p))$, for any $p\in P$). Let $e(y - r(y)) \in \btc_{1j}$ denote the $j$-parameter family of 1-blob diagrams @@ -277,39 +279,36 @@ \[ h(y) = e(y - r(y)) + c(r(y)) . \] -\nn{up to sign, at least} We must now verify that $h$ does the job it was intended to do. For $x\in \btc_{ij}$ with $i\ge 2$ we have -\nn{ignoring signs} \begin{align*} - \bd h(x) + h(\bd x) &= \bd(e(x)) + e(\bd x) \\ - &= \bd_b(e(x)) + \bd_t(e(x)) + e(\bd_b x) + e(\bd_t x) \\ - &= \bd_b(e(x)) + e(\bd_b x) \quad\quad\text{(since $\bd_t(e(x)) = e(\bd_t x)$)} \\ - &= x . + \bd h(x) + h(\bd x) &= \bd(e(x)) + e(\bd x) && \\ + &= \bd_b(e(x)) + (-1)^{i+1} \bd_t(e(x)) + e(\bd_b x) + (-1)^i e(\bd_t x) && \\ + &= \bd_b(e(x)) + e(\bd_b x) && \text{(since $\bd_t(e(x)) = e(\bd_t x)$)} \\ + &= x . && \end{align*} For $x\in \btc_{1j}$ we have -\nn{ignoring signs} \begin{align*} - \bd h(x) + h(\bd x) &= \bd_b(e(x)) + \bd_t(e(x)) + e(\bd_b x - r(\bd_b x)) + c(r(\bd_b x)) + e(\bd_t x) \\ - &= \bd_b(e(x)) + e(\bd_b x) \quad\quad\text{(since $r(\bd_b x) = 0$)} \\ - &= x . + \bd h(x) + h(\bd x) &= \bd_b(e(x)) + \bd_t(e(x)) + e(\bd_b x - r(\bd_b x)) + c(r(\bd_b x)) - e(\bd_t x) && \\ + &= \bd_b(e(x)) + e(\bd_b x) && \text{(since $r(\bd_b x) = 0$)} \\ + &= x . && \end{align*} For $x\in \btc_{0j}$ with $j\ge 1$ we have -\nn{ignoring signs} \begin{align*} - \bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) + \bd_t(e(x - r(x))) + \bd_t(c(r(x))) + + \bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) - \bd_t(e(x - r(x))) - \bd_t(c(r(x))) + e(\bd_t x - r(\bd_t x)) + c(r(\bd_t x)) \\ - &= x - r(x) + \bd_t(c(r(x))) + c(r(\bd_t x)) \\ + &= x - r(x) - \bd_t(c(r(x))) + c(r(\bd_t x)) \\ &= x - r(x) + r(x) \\ &= x. \end{align*} +Here we have used the fact that $\bd_b(c(r(x))) = 0$ since $c(r(x))$ is a $0$-blob diagram, as well as that $\bd_t(e(r(x))) = e(r(\bd_t x))$ \nn{explain why this is true?} and $c(r(\bd_t x)) - \bd_t(c(r(x))) = r(x)$ \nn{explain?}. + For $x\in \btc_{00}$ we have -\nn{ignoring signs} \begin{align*} \bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) + \bd_t(c(r(x))) \\ &= x - r(x) + r(x) - r(x)\\ - &= x - r(x). + &= x - r(x). \qedhere \end{align*} \end{proof} @@ -317,10 +316,10 @@ For manifolds $X$ and $Y$, we have $\btc_*(X\du Y) \simeq \btc_*(X)\otimes\btc_*(Y)$. \end{lemma} \begin{proof} -This follows from the Eilenber-Zilber theorem and the fact that -\[ - \BD_k(X\du Y) \cong \coprod_{i+j=k} \BD_i(X)\times\BD_j(Y) . -\] +This follows from the Eilenberg-Zilber theorem and the fact that +\begin{align*} + \BD_k(X\du Y) & \cong \coprod_{i+j=k} \BD_i(X)\times\BD_j(Y) . \qedhere +\end{align*} \end{proof} For $S\sub X$, we say that $a\in \btc_k(X)$ is {\it supported on $S$} @@ -358,13 +357,13 @@ \end{proof} -\begin{proof}[Proof of \ref{lem:bc-btc}] -Armed with the above lemmas, we can now proceed similarly to the proof of \ref{small-blobs-b}. +\begin{proof}[Proof of Lemma \ref{lem:bc-btc}] +Armed with the above lemmas, we can now proceed similarly to the proof of Lemma \ref{small-blobs-b}. It suffices to show that for any finitely generated pair of subcomplexes $(C_*, D_*) \sub (\btc_*(X), \bc_*(X))$ -we can find a homotopy $h:C_*\to \btc_*(X)$ such that $h(D_*) \sub \bc_*(X)$ -and $x + h\bd(x) + \bd h(X) \in \bc_*(X)$ for all $x\in C_*$. +we can find a homotopy $h:C_*\to \btc_{*+1}(X)$ such that $h(D_*) \sub \bc_{*+1}(X)$ +and $x + h\bd(x) + \bd h(x) \in \bc_*(X)$ for all $x\in C_*$. By Lemma \ref{small-top-blobs}, we may assume that $C_* \sub \btc_*^\cU(X)$ for some cover $\cU$ of our choosing. @@ -376,22 +375,22 @@ Let $b \in C_1$ be a generator. Since $b$ is supported in a disjoint union of balls, we can find $s(b)\in \bc_1$ with $\bd (s(b)) = \bd b$ -(by \ref{disj-union-contract}), and also $h_1(b) \in \btc_2$ +(by Corollary \ref{disj-union-contract}), and also $h_1(b) \in \btc_2(X)$ such that $\bd (h_1(b)) = s(b) - b$ -(by \ref{bt-contract} and \ref{btc-prod}). +(by Lemmas \ref{bt-contract} and \ref{btc-prod}). Now let $b$ be a generator of $C_2$. If $\cU$ is fine enough, there is a disjoint union of balls $V$ on which $b + h_1(\bd b)$ is supported. -Since $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_2$, we can find -$s(b)\in \bc_2$ with $\bd(s(b)) = \bd(b + h_1(\bd b))$ (by \ref{disj-union-contract}). -By \ref{bt-contract} and \ref{btc-prod}, we can now find -$h_2(b) \in \btc_3$, also supported on $V$, such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$ +Since $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_2(X)$, we can find +$s(b)\in \bc_2(X)$ with $\bd(s(b)) = \bd(b + h_1(\bd b))$ (by Corollary \ref{disj-union-contract}). +By Lemmas \ref{bt-contract} and \ref{btc-prod}, we can now find +$h_2(b) \in \btc_3(X)$, also supported on $V$, such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$ The general case, $h_k$, is similar. \end{proof} -The proof of \ref{lem:bc-btc} constructs a homotopy inverse to the inclusion +The proof of Lemma \ref{lem:bc-btc} constructs a homotopy inverse to the inclusion $\bc_*(X)\sub \btc_*(X)$. One might ask for more: a contractible set of possible homotopy inverses, or at least an $m$-connected set for arbitrarily large $m$. @@ -440,7 +439,7 @@ \begin{proof} In light of Lemma \ref{lem:bc-btc}, it suffices to prove the theorem with $\bc_*$ replaced by $\btc_*$. -And in fact for $\btc_*$ we get a sharper result: we can omit +In fact, for $\btc_*$ we get a sharper result: we can omit the ``up to homotopy" qualifiers. Let $f\in CH_k(X, Y)$, $f:P^k\to \Homeo(X \to Y)$ and $a\in \btc_{ij}(X)$, diff -r 3baa4e4d395e -r 0bc6fa29b62a text/intro.tex --- a/text/intro.tex Sun Sep 19 23:14:41 2010 -0700 +++ b/text/intro.tex Sun Sep 19 23:15:21 2010 -0700 @@ -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 3baa4e4d395e -r 0bc6fa29b62a text/ncat.tex --- a/text/ncat.tex Sun Sep 19 23:14:41 2010 -0700 +++ b/text/ncat.tex Sun Sep 19 23:15:21 2010 -0700 @@ -1118,8 +1118,7 @@ \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.