# HG changeset patch # User Kevin Walker # Date 1316983478 21600 # Node ID 70e947e15f5791c6ea5f122ee4064a567220faea # Parent ab0b4827c89c43dec926f8e4d3fc3a6bf12f4fa5# Parent a0fd6e6209260dd5f43ebef3f2925699a01a8060 undoing the 'disk-like's diff -r a0fd6e620926 -r 70e947e15f57 RefereeReport.pdf Binary file RefereeReport.pdf has changed diff -r a0fd6e620926 -r 70e947e15f57 blob to-do --- a/blob to-do Sun Sep 25 14:33:30 2011 -0600 +++ b/blob to-do Sun Sep 25 14:44:38 2011 -0600 @@ -3,8 +3,10 @@ * better discussion of systems of fields from disk-like n-cats (Is this done by now?) + +* ?? say clearly that certain lemmas don't work for TOP; we're only claiming DIFF and PL (requires small changes in many places) -* need to fix fam-o-homeo argument per discussion with Rob +* need to fix fam-o-homeo argument per discussion with Rob (or just remove it) * need to change module axioms to follow changes in n-cat axioms; search for and destroy all the "Homeo_\bd"'s, add a v-cone axiom diff -r a0fd6e620926 -r 70e947e15f57 diagrams/ncat/boundary-collar.pdf Binary file diagrams/ncat/boundary-collar.pdf has changed diff -r a0fd6e620926 -r 70e947e15f57 preamble.tex --- a/preamble.tex Sun Sep 25 14:33:30 2011 -0600 +++ b/preamble.tex Sun Sep 25 14:44:38 2011 -0600 @@ -50,6 +50,9 @@ \theoremstyle{plain} %\newtheorem*{fact}{Fact} \newtheorem{prop}{Proposition}[subsection] +\makeatletter +\@addtoreset{prop}{section} +\makeatother \newtheorem{conj}[prop]{Conjecture} \newtheorem{thm}[prop]{Theorem} \newtheorem{lem}[prop]{Lemma} diff -r a0fd6e620926 -r 70e947e15f57 sandbox.tex --- a/sandbox.tex Sun Sep 25 14:33:30 2011 -0600 +++ b/sandbox.tex Sun Sep 25 14:44:38 2011 -0600 @@ -1,15 +1,12 @@ -\documentclass[11pt,leqno]{amsart} - -%\usepackage{amsthm} +\documentclass[11pt,leqno]{article} \newcommand{\pathtotrunk}{./} +\input{preamble} \input{text/article_preamble} -\input{text/top_matter} \input{text/kw_macros} %\title{Blob Homology} \title{Sandbox} - \begin{document} diff -r a0fd6e620926 -r 70e947e15f57 text/a_inf_blob.tex --- a/text/a_inf_blob.tex Sun Sep 25 14:33:30 2011 -0600 +++ b/text/a_inf_blob.tex Sun Sep 25 14:44:38 2011 -0600 @@ -418,14 +418,31 @@ \begin{rem} Lurie has shown in \cite[Theorem 3.8.6]{0911.0018} that the topological chiral homology of an $n$-manifold $M$ with coefficients in a certain $E_n$ algebra constructed from $T$ recovers -the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected. +the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n{-}1$-connected. This extra hypothesis is not surprising, in view of the idea described in Example \ref{ex:e-n-alg} that an $E_n$ algebra is roughly equivalent data to an $A_\infty$ $n$-category which is trivial at levels 0 through $n-1$. Ricardo Andrade also told us about a similar result. + +Specializing still further, Theorem \ref{thm:map-recon} is related to the classical result that for connected spaces $T$ +we have $HH_*(C_*(\Omega T)) \cong H_*(LT)$, that is, the Hochschild homology of based loops in $T$ is isomorphic +to the homology of the free loop space of $T$ (see \cite{MR793184} and \cite{MR842427}). +Theorem \ref{thm:map-recon} says that for any space $T$ (connected or not) we have +$\bc_*(S^1; C_*(\pi^\infty_{\le 1}(T))) \simeq C_*(LT)$. +Here $C_*(\pi^\infty_{\le 1}(T))$ denotes the singular chain version of the fundamental infinity-groupoid of $T$, +whose objects are points in $T$ and morphism chain complexes are $C_*(\paths(t_1 \to t_2))$ for $t_1, t_2 \in T$. +If $T$ is connected then the $A_\infty$ 1-category $C_*(\pi^\infty_{\le 1}(T))$ is Morita equivalent to the +$A_\infty$ algebra $C_*(\Omega T)$; +the bimodule for the equivalence is the singular chains of the space of paths which start at the base point of $T$. +Theorem \ref{thm:hochschild} holds for $A_\infty$ 1-categories (though we do not prove that in this paper), +which then implies that +\[ + Hoch_*(C_*(\Omega T)) \simeq Hoch_*(C_*(\pi^\infty_{\le 1}(T))) + \simeq \bc_*(S^1; C_*(\pi^\infty_{\le 1}(T))) \simeq C_*(LT) . +\] \end{rem} -\begin{proof} +\begin{proof}[Proof of Theorem \ref{thm:map-recon}] The proof is again similar to that of Theorem \ref{thm:product}. We begin by constructing a chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$. diff -r a0fd6e620926 -r 70e947e15f57 text/appendixes/comparing_defs.tex --- a/text/appendixes/comparing_defs.tex Sun Sep 25 14:33:30 2011 -0600 +++ b/text/appendixes/comparing_defs.tex Sun Sep 25 14:44:38 2011 -0600 @@ -4,8 +4,8 @@ \label{sec:comparing-defs} In \S\ref{sec:example:traditional-n-categories(fields)} we showed how to construct -a topological $n$-category from a traditional $n$-category; the morphisms of the -topological $n$-category are string diagrams labeled by the traditional $n$-category. +a disk-like $n$-category from a traditional $n$-category; the morphisms of the +disk-like $n$-category are string diagrams labeled by the traditional $n$-category. In this appendix we sketch how to go the other direction, for $n=1$ and 2. The basic recipe, given a disk-like $n$-category $\cC$, is to define the $k$-morphisms of the corresponding traditional $n$-category to be $\cC(B^k)$, where @@ -585,6 +585,7 @@ The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$ ($\cC$ applied to the standard interval with boundary labeled by $x$ and $y$). For simplicity we will now assume there is only one object and suppress it from the notation. +Henceforth $A$ will also denote its unique morphism space. A choice of homeomorphism $I\cup I \to I$ induces a chain map $m_2: A\otimes A\to A$. We now have two different homeomorphisms $I\cup I\cup I \to I$, but they are isotopic. @@ -610,7 +611,7 @@ We define a $\Homeo(J)$ action on $\cC(J)$ via $g_*(f, a) = (g\circ f, a)$. The $C_*(\Homeo(J))$ action is defined similarly. -Let $J_1$ and $J_2$ be intervals. +Let $J_1$ and $J_2$ be intervals, and let $J_1\cup J_2$ denote their union along a single boundary point. We must define a map $\cC(J_1)\ot\cC(J_2)\to\cC(J_1\cup J_2)$. Choose a homeomorphism $g:I\to J_1\cup J_2$. Let $(f_i, a_i)\in \cC(J_i)$. diff -r a0fd6e620926 -r 70e947e15f57 text/appendixes/famodiff.tex --- a/text/appendixes/famodiff.tex Sun Sep 25 14:33:30 2011 -0600 +++ b/text/appendixes/famodiff.tex Sun Sep 25 14:44:38 2011 -0600 @@ -236,14 +236,13 @@ Since $X$ is compact, we may assume without loss of generality that the cover $\cU$ is finite. Let $\cU = \{U_\alpha\}$, $1\le \alpha\le N$. -We will need some wiggle room, so for each $\alpha$ choose a large finite number of open sets +We will need some wiggle room, so for each $\alpha$ choose $2N$ additional open sets \[ - U_\alpha = U_\alpha^0 \supset U_\alpha^1 \supset U_\alpha^2 \supset \cdots + U_\alpha = U_\alpha^0 \supset U_\alpha^\frac12 \supset U_\alpha^1 \supset U_\alpha^\frac32 \supset \cdots \supset U_\alpha^N \] -so that for each fixed $i$ the sets $\{U_\alpha^i\}$ are an open cover of $X$, and also so that -the closure $\ol{U_\alpha^i}$ is compact and $U_\alpha^{i-1} \supset \ol{U_\alpha^i}$. -\nn{say specifically how many we need?} - +so that for each fixed $i$ the set $\cU^i = \{U_\alpha^i\}$ is an open cover of $X$, and also so that +the closure $\ol{U_\alpha^i}$ is compact and $U_\alpha^{i-\frac12} \supset \ol{U_\alpha^i}$. +%\nn{say specifically how many we need?} Let $P$ be some $k$-dimensional polyhedron and $f:P\to \Homeo(X)$. After subdividing $P$, we may assume that there exists $g\in \Homeo(X)$ @@ -252,34 +251,97 @@ The sense of ``small" we mean will be explained below. It depends only on $\cU$ and the choice of $U_\alpha^i$'s. -We may assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU$. +Our goal is to homotope $P$, rel boundary, so that it is adapted to $\cU$. +By the local contractibility of $\Homeo(X)$ (Corollary 1.1 of \cite{MR0283802}), +it suffices to find $f':P\to \Homeo(X)$ such that $f' = f$ on $\bd P$ and $f'$ is adapted to $\cU$. + +We may assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU^N$. +So $\bd P = \sum Q_\beta$, and the support of $f$ restricted to $Q_\beta$ is $V_\beta^N$, the union of $k-1$ of +the $U_\alpha^N$'s. Define $V_\beta^i \sup V_\beta^N$ to be the corresponding union of $k-1$ +of the $U_\alpha^i$'s. + +Define +\[ + W_j^i = U_1^i \cup U_2^i \cup \cdots \cup U_j^i . +\] + +By the local contractibility of $\Homeo(X)$ (Corollary 1.1 of \cite{MR0283802}), + +We will construct a sequence of maps $f_i : \bd P\to \Homeo(X)$, for $i = 0, 1, \ldots, N$, with the following properties: +\begin{itemize} +\item[(A)] $f_0 = f|_{\bd P}$; +\item[(B)] $f_i = g$ on $W_i^i$; +\item[(C)] $f_i$ restricted to $Q_\beta$ has support contained in $V_\beta^{N-i}$; and +\item[(D)] there is a homotopy $F_i : \bd P\times I \to \Homeo(X)$ from $f_{i-1}$ to $f_i$ such that the +support of $F_i$ restricted to $Q_\beta\times I$ is contained in $U_i^i\cup V_\beta^{N-i}$. +\nn{check this when done writing} +\end{itemize} + +Once we have the $F_i$'s as in (D), we can finish the argument as follows. +Assemble the $F_i$'s into a map $F: \bd P\times [0,N] \to \Homeo(X)$. +$F$ is adapted to $\cU$ by (D). +$F$ restricted to $\bd P\times\{N\}$ is constant on $W_N^N = X$ by (B). +We can therefore view $F$ as a map $f'$ from $\Cone(\bd P) \cong P$ to $\Homeo(X)$ +which is adapted to $\cU$. + +The homotopies $F_i$ will be composed of three types of pieces, $A_\beta$, $B_\beta$ and $C$, % NOT C_\beta +as illustrated in Figure \nn{xxxx}. +($A_\beta$, $B_\beta$ and $C$ also depend on $i$, but we are suppressing that from the notation.) +The homotopy $A_\beta : Q_\beta \times I \to \Homeo(X)$ will arrange that $f_i$ agrees with $g$ +on $U_i^i \setmin V_\beta^{N-i+1}$. +The homotopy $B_\beta : Q_\beta \times I \to \Homeo(X)$ will extend the agreement with $g$ to all of $U_i^i$. +The homotopies $C$ match things up between $\bd Q_\beta \times I$ and $\bd Q_{\beta'} \times I$ when +$Q_\beta$ and $Q_{\beta'}$ are adjacent. + +Assume inductively that we have defined $f_{i-1}$. + +Now we define $A_\beta$. +Choose $q_0\in Q_\beta$. +Theorem 5.1 of \cite{MR0283802} implies that we can choose a homotopy $h:I \to \Homeo(X)$, with $h(0)$ the identity, such that +\begin{itemize} +\item[(E)] the support of $h$ is contained in $U_i^{i-1} \setmin W_{i-1}^{i-\frac12}$; and +\item[(F)] $h(1) \circ f_{i-1}(q_0) = g$ on $U_i^i$. +\end{itemize} +Define $A_\beta$ by +\[ + A_\beta(q, t) = h(t) \circ f_{i-1}(q) . +\] +It follows that +\begin{itemize} +\item[(G)] $A_\beta(q,1) = A(q',1)$, for all $q,q' \in Q_\beta$, on $X \setmin V_\beta^{N-i+1}$; +\item[(H)] $A_\beta(q, 1) = g$ on $(U_i^i \cup W_{i-1}^{i-\frac12})\setmin V_\beta^{N-i+1}$; and +\item[(I)] the support of $A_\beta$ is contained in $U_i^{i-1} \cup V_\beta^{N-i+1}$. +\end{itemize} + +Next we define $B_\beta$. +Theorem 5.1 of \cite{MR0283802} implies that we can choose a homotopy $B_\beta:Q_\beta\times I\to \Homeo(X)$ +such that +\begin{itemize} +\item[(J)] $B_\beta(\cdot, 0) = A_\beta(\cdot, 1)$; +\item[(K)] $B_\beta(q,1) = g$ on $W_i^i$; +\item[(L)] the support of $B_\beta(\cdot,1)$ is contained in $V_\beta^{N-i}$; and +\item[(M)] the support of $B_\beta$ is contained in $U_i^i \cup V_\beta^{N-i}$. +\end{itemize} + +All that remains is to define the ``glue" $C$ which interpolates between adjacent $Q_\beta$ and $Q_{\beta'}$. +First consider the $k=2$ case. +(In this case Figure \nn{xxxx} is literal rather than merely schematic.) +Let $q = Q_\beta \cap Q_{\beta'}$ be a point on the boundaries of both $Q_\beta$ and $Q_{\beta'}$. +We have an arc of Homeomorphisms, composed of $B_\beta(q, \cdot)$, $A_\beta(q, \cdot)$, +$A_{\beta'}(q, \cdot)$ and $B_{\beta'}(q, \cdot)$, which connects $B_\beta(q, 1)$ to $B_{\beta'}(q, 1)$. + +\nn{Hmmmm..... I think there's a problem here} -Theorem 5.1 of \cite{MR0283802}, applied $N$ times, allows us -to choose a homotopy $h:P\times [0,N] \to \Homeo(X)$ with the following properties: -\begin{itemize} -\item $h(p, 0) = f(p)$ for all $p\in P$. -\item The restriction of $h(p, i)$ to $U_0^i \cup \cdots \cup U_i^i$ is equal to the homeomorphism $g$, -for all $p\in P$. -\item For each fixed $p\in P$, the family of homeomorphisms $h(p, [i-1, i])$ is supported on -$U_i^{i-1} \setmin (U_0^i \cup \cdots \cup U_{i-1}^i)$ -(and hence supported on $U_i$). -\end{itemize} + +\nn{resume revising here} + + +\nn{scraps:} + To apply Theorem 5.1 of \cite{MR0283802}, the family $f(P)$ must be sufficiently small, and the subdivision mentioned above is chosen fine enough to insure this. -By reparametrizing, we can view $h$ as a homotopy (rel boundary) from $h(\cdot,0) = f: P\to\Homeo(X)$ -to the family -\[ - h(\bd(P\times [0,N]) \setmin P\times \{0\}) = h(\bd P \times [0,N]) \cup h(P \times \{N\}) . -\] -We claim that the latter family of homeomorphisms is adapted to $\cU$. -By the second bullet above, $h(P\times \{N\})$ is the constant family $g$, and is therefore supported on the empty set. -Via an inductive assumption and a preliminary homotopy, we have arranged above that $f(\bd P)$ is -adapted to $\cU$, which means that $\bd P = \cup_j Q_j$ with $f(Q_j)$ supported on the union of $k-1$ -of the $U_\alpha$'s for each $j$. -It follows (using the third bullet above) that $h(Q_j \times [i-1,i])$ is supported on the union of $k$ -of the $U_\alpha$'s, specifically, the support of $f(Q_j)$ union $U_i$. \end{proof} diff -r a0fd6e620926 -r 70e947e15f57 text/article_preamble.tex --- a/text/article_preamble.tex Sun Sep 25 14:33:30 2011 -0600 +++ b/text/article_preamble.tex Sun Sep 25 14:44:38 2011 -0600 @@ -18,7 +18,7 @@ \usetikzlibrary{decorations,decorations.pathreplacing} \usetikzlibrary{fit,calc,through} -\pgfrealjobname{blob1} +%\pgfrealjobname{blob1} \makeatletter \@ifclassloaded{beamer}{}{% diff -r a0fd6e620926 -r 70e947e15f57 text/basic_properties.tex --- a/text/basic_properties.tex Sun Sep 25 14:33:30 2011 -0600 +++ b/text/basic_properties.tex Sun Sep 25 14:44:38 2011 -0600 @@ -90,7 +90,7 @@ $r$ be the restriction of $b$ to $X\setminus S$. 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 $T(B') \sub T(b)$. +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, \S\ref{sec:moam}), so $f$ and the identity map are homotopic. \end{proof} diff -r a0fd6e620926 -r 70e947e15f57 text/blobdef.tex --- a/text/blobdef.tex Sun Sep 25 14:33:30 2011 -0600 +++ b/text/blobdef.tex Sun Sep 25 14:44:38 2011 -0600 @@ -43,7 +43,7 @@ ``the space of all local relations that can be imposed on $\bc_0(X)$". Thus we say a $1$-blob diagram consists of: \begin{itemize} -\item An closed ball in $X$ (``blob") $B \sub X$. +\item A closed ball in $X$ (``blob") $B \sub X$. \item A boundary condition $c \in \cF(\bdy B) = \cF(\bd(X \setmin B))$. \item A field $r \in \cF(X \setmin B; c)$. \item A local relation field $u \in U(B; c)$. @@ -156,7 +156,7 @@ \end{itemize} Combining these two operations can give rise to configurations of blobs whose complement in $X$ is not a manifold. -Thus will need to be more careful when speaking of a field $r$ on the complement of the blobs. +Thus we will need to be more careful when speaking of a field $r$ on the complement of the blobs. \begin{example} \label{sin1x-example} Consider the four subsets of $\Real^3$, @@ -208,7 +208,7 @@ %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 $\{B_1, \ldots B_k\}$ +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 $0 \leq l \leq m$ and some connected component $M_l'$ of $M_l$ which is a ball, so $B_i$ is the image of $M_l'$ in $X$. @@ -238,7 +238,7 @@ \label{defn:blob-diagram} A $k$-blob diagram on $X$ consists of \begin{itemize} -\item a configuration $\{B_1, \ldots B_k\}$ of $k$ blobs in $X$, +\item a configuration $\{B_1, \ldots, B_k\}$ of $k$ blobs in $X$, \item and a field $r \in \cF(X)$ which is splittable along some gluing decomposition compatible with that configuration, \end{itemize} such that diff -r a0fd6e620926 -r 70e947e15f57 text/deligne.tex --- a/text/deligne.tex Sun Sep 25 14:33:30 2011 -0600 +++ b/text/deligne.tex Sun Sep 25 14:44:38 2011 -0600 @@ -160,7 +160,7 @@ We have now defined a map from the little $n{+}1$-balls operad to the $n$-SC operad, with contractible fibers. (The fibers correspond to moving the $D_i$'s in the $x_{n+1}$ -direction without changing their ordering.) +direction while keeping them disjoint.) %\nn{issue: we've described this by varying the $R_i$'s, but above we emphasize varying the $f_i$'s. %does this need more explanation?} @@ -178,7 +178,8 @@ p(\ol{f}): \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) \to \hom(\bc_*(M_0), \bc_*(N_0)) . \] -Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define $p(\ol{f}$) to be the composition +Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define +$p(\ol{f})(\alpha_1\ot\cdots\ot\alpha_k)$ to be the composition \[ \bc_*(M_0) \stackrel{f_0}{\to} \bc_*(R_1\cup M_1) \stackrel{\id\ot\alpha_1}{\to} \bc_*(R_1\cup N_1) @@ -201,7 +202,7 @@ \label{thm:deligne} There is a collection of chain maps \[ - C_*(SC^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes + C_*(SC^n_{\ol{M}\ol{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) \] which satisfy the operad compatibility conditions. @@ -216,7 +217,7 @@ a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disks operad. \begin{proof} -As described above, $SC^n_{\overline{M}, \overline{N}}$ is equal to the disjoint +As described above, $SC^n_{\ol{M}\ol{N}}$ is equal to the disjoint union of products of homeomorphism spaces, modulo some relations. By Theorem \ref{thm:CH} and the Eilenberg-Zilber theorem, we have for each such product $P$ a chain map @@ -225,7 +226,7 @@ \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) . \] It suffices to show that the above maps are compatible with the relations whereby -$SC^n_{\overline{M}, \overline{N}}$ is constructed from the various $P$'s. +$SC^n_{\ol{M}\ol{N}}$ is constructed from the various $P$'s. This in turn follows easily from the fact that the actions of $C_*(\Homeo(\cdot\to\cdot))$ are local (compatible with gluing) and associative. %\nn{should add some detail to above} diff -r a0fd6e620926 -r 70e947e15f57 text/evmap.tex --- a/text/evmap.tex Sun Sep 25 14:33:30 2011 -0600 +++ b/text/evmap.tex Sun Sep 25 14:44:38 2011 -0600 @@ -123,7 +123,7 @@ Let $g_j = f_1\circ f_2\circ\cdots\circ f_j$. Let $g$ be the last of the $g_j$'s. Choose the sequence $\bar{f}_j$ so that -$g(B)$ is contained is an open set of $\cV_1$ and +$g(B)$ is contained in an open set of $\cV_1$ and $g_{j-1}(|f_j|)$ is also contained in an open set of $\cV_1$. There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ @@ -325,7 +325,7 @@ \end{proof} For $S\sub X$, we say that $a\in \btc_k(X)$ is {\it supported on $S$} -if there exists $a'\in \btc_k(S)$ +if there exist $a'\in \btc_k(S)$ and $r\in \btc_0(X\setmin S)$ such that $a = a'\bullet r$. \newcommand\sbtc{\btc^{\cU}} @@ -385,7 +385,7 @@ 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 $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_2(X)$, we can find +Since $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_1(X)$, 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 $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$ diff -r a0fd6e620926 -r 70e947e15f57 text/hochschild.tex --- a/text/hochschild.tex Sun Sep 25 14:33:30 2011 -0600 +++ b/text/hochschild.tex Sun Sep 25 14:44:38 2011 -0600 @@ -12,7 +12,7 @@ Hochschild complex of the 1-category $\cC$. (Recall from \S \ref{sec:example:traditional-n-categories(fields)} that a $1$-category gives rise to a $1$-dimensional system of fields; as usual, -talking about the blob complex with coefficients in a $n$-category means +talking about the blob complex with coefficients in an $n$-category means first passing to the corresponding $n$ dimensional system of fields.) Thus the blob complex is a natural generalization of something already known to be interesting in higher homological degrees. @@ -293,7 +293,7 @@ $\widetilde{q} \in \ker(C \tensor E \tensor C \to E)$. Further, \begin{align*} -\hat{g}(\widetilde{q}) & = \sum_i \left(a_i \tensor g(\widetilde{q_i}) \tensor b_i\right) - 1 \tensor \left(\sum_i a_i g(\widetilde{q_i}\right) b_i) \tensor 1 \\ +\hat{g}(\widetilde{q}) & = \sum_i \left(a_i \tensor g(\widetilde{q_i}) \tensor b_i\right) - 1 \tensor \left(\sum_i a_i g(\widetilde{q_i}) b_i\right) \tensor 1 \\ & = q - 0 \end{align*} (here we used that $g$ is a map of $C$-$C$ bimodules, and that $\sum_i a_i q_i b_i = 0$). @@ -341,11 +341,11 @@ $j$-th term have labelled points $d_{j,i}$ to the left of $*$, $m_j$ at $*$, and $d_{j,i}'$ to the right of $*$, and there are labels $c_i$ at the labeled points outside the blob. We know that -$$\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \tensor m_j \tensor d_{j,1}' \tensor \cdots \tensor d_{j,k'_j}' \in \ker(\DirectSum_{k,k'} C^{\tensor k} \tensor M \tensor C^{\tensor k'} \tensor \to M),$$ +$$\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \tensor m_j \tensor d_{j,1}' \tensor \cdots \tensor d_{j,k'_j}' \in \ker(\DirectSum_{k,k'} C^{\tensor k} \tensor M \tensor C^{\tensor k'} \to M),$$ and so \begin{align*} -\ev(\bdy y) & = \sum_j m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k d_{j,1} \cdots d_{j,k_j} \\ - & = \sum_j d_{j,1} \cdots d_{j,k_j} m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k \\ +\pi\left(\ev(\bdy y)\right) & = \pi\left(\sum_j m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k d_{j,1} \cdots d_{j,k_j}\right) \\ + & = \pi\left(\sum_j d_{j,1} \cdots d_{j,k_j} m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k\right) \\ & = 0 \end{align*} where this time we use the fact that we're mapping to $\coinv{M}$, not just $M$. diff -r a0fd6e620926 -r 70e947e15f57 text/intro.tex --- a/text/intro.tex Sun Sep 25 14:33:30 2011 -0600 +++ b/text/intro.tex Sun Sep 25 14:44:38 2011 -0600 @@ -260,8 +260,7 @@ Note that this includes the case of gluing two disjoint manifolds together. \begin{property}[Gluing map] \label{property:gluing-map}% -Given a gluing $X \to X_\mathrm{gl}$, there is -a natural map +Given a gluing $X \to X_\mathrm{gl}$, there is an injective natural map \[ \bc_*(X) \to \bc_*(X_\mathrm{gl}) \] diff -r a0fd6e620926 -r 70e947e15f57 text/kw_macros.tex --- a/text/kw_macros.tex Sun Sep 25 14:33:30 2011 -0600 +++ b/text/kw_macros.tex Sun Sep 25 14:44:38 2011 -0600 @@ -64,7 +64,7 @@ % \DeclareMathOperator{\pr}{pr} etc. \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}} -\applytolist{declaremathop}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}{Compat}{Coll}{Cone}; +\applytolist{declaremathop}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}{Compat}{Coll}{Cone}{pr}{paths}; \DeclareMathOperator*{\colim}{colim} \DeclareMathOperator*{\hocolim}{hocolim} diff -r a0fd6e620926 -r 70e947e15f57 text/ncat.tex --- a/text/ncat.tex Sun Sep 25 14:33:30 2011 -0600 +++ b/text/ncat.tex Sun Sep 25 14:44:38 2011 -0600 @@ -214,12 +214,14 @@ with the (ordinary, not fibered) product $\cC(B_1) \times \cC(B_2)$. Let $\cl{\cC}(S)\trans E$ denote the image of $\gl_E$. -We will refer to elements of $\cl{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$". When the gluing map is surjective every such element is splittable. +We will refer to elements of $\cl{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$". +When the gluing map is surjective every such element is splittable. If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ as above, then we define $\cC(X)\trans E = \bd^{-1}(\cl{\cC}(\bd X)\trans E)$. -We will call the projection $\cl{\cC}(S)\trans E \to \cC(B_i)$ +We will call the projection $\cl{\cC}(S)\trans E \to \cC(B_i)$ given by the composition +$$\cl{\cC}(S)\trans E \xrightarrow{\gl^{-1}} \cC(B_1) \times \cC(B_2) \xrightarrow{\pr_i} \cC(B_i)$$ a {\it restriction} map and write $\res_{B_i}(a)$ (or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)\trans E$. More generally, we also include under the rubric ``restriction map" @@ -227,9 +229,14 @@ another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition of restriction maps. In particular, we have restriction maps $\cC(X)\trans E \to \cC(B_i)$ -($i = 1, 2$, notation from previous paragraph). +defined as the composition of the boundary with the first restriction map described above: +$$ +\cC(X) \trans E \xrightarrow{\bdy} \cl{\cC}(\bdy X)\trans E \xrightarrow{\res} \cC(B_i) +.$$ These restriction maps can be thought of as domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. +These restriction maps in fact have their image in the subset $\cC(B_i)\trans E$, +and so to emphasize this we will sometimes write the restriction map as $\cC(X)\trans E \to \cC(B_i)\trans E$. Next we consider composition of morphisms. @@ -977,7 +984,7 @@ There are two differences. First, for the $n$-category definition we restrict our attention to balls (and their boundaries), while for fields we consider all manifolds. -Second, in category definition we directly impose isotopy +Second, in the category definition we directly impose isotopy invariance in dimension $n$, while in the fields definition we instead remember a subspace of local relations which contain differences of isotopic fields. (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) @@ -994,6 +1001,8 @@ In the $n$-category axioms above we have intermingled data and properties for expository reasons. Here's a summary of the definition which segregates the data from the properties. +We also remind the reader of the inductive nature of the definition: All the data for $k{-}1$-morphisms must be in place +before we can describe the data for $k$-morphisms. An $n$-category consists of the following data: \begin{itemize} @@ -1164,8 +1173,8 @@ \label{ex:blob-complexes-of-balls} Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. We will define an $A_\infty$ $k$-category $\cC$. -When $X$ is a $m$-ball, with $m