# HG changeset patch # User Scott Morrison # Date 1275016452 25200 # Node ID a798a1e00cb306dac7785816c76a2b33486c0303 # Parent 0b3e76167461790c5eb3c89b56ce081332ce9df8# Parent ff867bfc8e9c4c91237b6e37c1e8b9dca368c6bd Automated merge with https://tqft.net/hg/blob/ diff -r 0b3e76167461 -r a798a1e00cb3 blob1.tex --- a/blob1.tex Thu May 27 17:52:46 2010 -0700 +++ b/blob1.tex Thu May 27 20:14:12 2010 -0700 @@ -46,6 +46,32 @@ \nn{maybe to do: add appendix on various versions of acyclic models} +\paragraph{To do list} +\begin{itemize} +\item[1] (K) tweak intro +\item[2] (S) needs explanation that this will be superseded by the n-cat +definitions in \S 7. +\item[2] (S) incorporate improvements from later +\item[2.3] (S) foreshadow generalising; quotient to resolution +\item[3] (K) look over blob homology section again +\item[4] (S) basic properties, not much to do +\item[5] (K) finish the lemmas in the Hochschild section +\item[6] (K) proofs need finishing, then (S) needs to confirm details and try +to make more understandable +\item[7] (S) do some work here -- identity morphisms are still imperfect. Say something about the cobordism and stabilization hypotheses \cite{MR1355899} in this setting? Say something about $E_n$ algebras? +\item[7.6] is new! (S) read +\item[8] improve the beginning, (K) small blobs, finish proof for products, +check the argument about maps +\item[9] (K) proofs trail off +\item[10] (S) read what's already here +\item[A] may need to weaken statement to get boundaries working (K) finish +\item[B] (S) look at this, decide what to keep + +\item Make clear exactly what counts as a "blob diagram", and search for +"blob diagram" + +\end{itemize} + \tableofcontents diff -r 0b3e76167461 -r a798a1e00cb3 text/a_inf_blob.tex --- a/text/a_inf_blob.tex Thu May 27 17:52:46 2010 -0700 +++ b/text/a_inf_blob.tex Thu May 27 20:14:12 2010 -0700 @@ -15,6 +15,12 @@ \medskip +\subsection{The small blob complex} + +\input{text/smallblobs} + +\subsection{A product formula} + Let $M^n = Y^k\times F^{n-k}$. Let $C$ be a plain $n$-category. Let $\cF$ be the $A_\infty$ $k$-category which assigns to a $k$-ball @@ -25,7 +31,7 @@ new-fangled blob complex $\bc_*^\cF(Y)$. \end{thm} -\input{text/smallblobs} + \begin{proof}[Proof of Theorem \ref{product_thm}] We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. @@ -213,6 +219,9 @@ \medskip +\subsection{A gluing theorem} +\label{sec:gluing} + Next we prove a gluing theorem. Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$. We will need an explicit collar on $Y$, so rewrite this as @@ -230,6 +239,7 @@ \end{itemize} \begin{thm} +\label{thm:gluing} $\bc(X) \cong \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. \end{thm} @@ -254,6 +264,8 @@ \medskip +\subsection{Reconstructing mapping spaces} + The next theorem shows how to reconstruct a mapping space from local data. Let $T$ be a topological space, let $M$ be an $n$-manifold, and recall the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$ diff -r 0b3e76167461 -r a798a1e00cb3 text/hochschild.tex --- a/text/hochschild.tex Thu May 27 17:52:46 2010 -0700 +++ b/text/hochschild.tex Thu May 27 20:14:12 2010 -0700 @@ -176,6 +176,7 @@ \ref{lem:hochschild-free}. \end{proof} +\subsection{Technical details} \begin{proof}[Proof of Lemma \ref{lem:module-blob}] We show that $K_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$. $K_*(C)$ differs from $\bc_*(S^1)$ only in that the base point * @@ -468,7 +469,7 @@ It follows that $H_0(K''_*) \cong C$. \end{proof} -\medskip +\subsection{An explicit chain map in low degrees} For purposes of illustration, we describe an explicit chain map $\HC_*(M) \to K_*(M)$ diff -r 0b3e76167461 -r a798a1e00cb3 text/ncat.tex --- a/text/ncat.tex Thu May 27 17:52:46 2010 -0700 +++ b/text/ncat.tex Thu May 27 20:14:12 2010 -0700 @@ -3,7 +3,7 @@ \def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip} \def\mmpar#1#2#3{\smallskip\noindent{\bf #1} (#2). {\it #3} \smallskip} -\section{$n$-categories} +\section{Definitions of $n$-categories} \label{sec:ncats} \subsection{Definition of $n$-categories} @@ -1025,9 +1025,8 @@ %component $\bd_i W$ of $W$. %(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.) -We will define a set $\cC(W, \cN)$ using a colimit construction similar to above. -\nn{give ref} -(If $k = n$ and our $k$-categories are enriched, then +We will define a set $\cC(W, \cN)$ using a colimit construction similar to the one appearing in \S \ref{ss:ncat_fields} above. +(If $k = n$ and our $n$-categories are enriched, then $\cC(W, \cN)$ will have additional structure; see below.) Define a permissible decomposition of $W$ to be a decomposition @@ -1039,7 +1038,7 @@ with $M_{ib}\cap Y_i$ being the marking. (See Figure \ref{mblabel}.) \begin{figure}[!ht]\begin{equation*} -\mathfig{.6}{ncat/mblabel} +\mathfig{.4}{ncat/mblabel} \end{equation*}\caption{A permissible decomposition of a manifold whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure} Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement @@ -1048,7 +1047,7 @@ (The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) -$\cN$ determines +The collection of modules $\cN$ determines a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets (possibly with additional structure if $k=n$). For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset @@ -1057,20 +1056,18 @@ \] such that the restrictions to the various pieces of shared boundaries amongst the $X_a$ and $M_{ib}$ all agree. -(Think fibered product.) +(That is, the fibered product over the boundary maps.) If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. -Finally, define $\cC(W, \cN)$ to be the colimit of $\psi_\cN$. -(Recall that if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means +We now define the set $\cC(W, \cN)$ to be the colimit of the functor $\psi_\cN$. +(As usual, if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means homotopy colimit.) If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define $\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold -$D\times Y_i \sub \bd(D\times W)$. - -It is not hard to see that the assignment $D \mapsto \cT(W, \cN)(D) \deq \cC(D\times W, \cN)$ -has the structure of an $n{-}k$-category. +$D\times Y_i \sub \bd(D\times W)$. It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$ +has the structure of an $n{-}k$-category, which we call $\cT(W, \cN)(D)$. \medskip @@ -1079,15 +1076,11 @@ construction to define tensor products of modules. Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. -(If $k=1$ and manifolds are oriented, then one should be +(If $k=1$ and our manifolds are oriented, then one should be a left module and the other a right module.) Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. -Define the tensor product of $\cM_1$ and $\cM_2$ to be the -$n{-}1$-category $\cT(J, \cM_1, \cM_2)$, -\[ - \cM_1\otimes \cM_2 \deq \cT(J, \cM_1, \cM_2) . -\] -This of course depends (functorially) +Define the tensor product $\cM_1 \tensor \cM_2$ to be the +$n{-}1$-category $\cT(J, \{\cM_1, \cM_2\})$. This of course depends (functorially) on the choice of 1-ball $J$. We will define a more general self tensor product (categorified coend) below. @@ -1105,11 +1098,10 @@ In order to state and prove our version of the higher dimensional Deligne conjecture (Section \ref{sec:deligne}), -we need to define morphisms of $A_\infty$ 1-cat modules and establish +we need to define morphisms of $A_\infty$ 1-category modules and establish some of their elementary properties. -To motivate the definitions which follow, consider algebras $A$ and $B$, right/bi/left modules -$X_B$, $_BY_A$ and $Z_A$, and the familiar adjunction +To motivate the definitions which follow, consider algebras $A$ and $B$, right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction \begin{eqnarray*} \hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\ f &\mapsto& [x \mapsto f(x\ot -)] \\ @@ -1125,43 +1117,43 @@ (\cM_\cC\ot {_\cC\cN})^* \cong \hom_\cC(\cM_\cC \to (_\cC\cN)^*) . \] -In the next few paragraphs we define the things appearing in the above equation: +In the next few paragraphs we define the objects appearing in the above equation: $\cM_\cC\ot {_\cC\cN}$, $(\cM_\cC\ot {_\cC\cN})^*$, $(_\cC\cN)^*$ and finally $\hom_\cC$. -In the previous subsection we defined a tensor product of $A_\infty$ $n$-cat modules + +\def\olD{{\overline D}} +\def\cbar{{\bar c}} +In the previous subsection we defined a tensor product of $A_\infty$ $n$-category modules for general $n$. For $n=1$ this definition is a homotopy colimit indexed by subdivisions of a fixed interval $J$ and their gluings (antirefinements). -(The tensor product will depend (functorially) on the choice of $J$.) -To a subdivision +(This tensor product depends functorially on the choice of $J$.) +To a subdivision $D$ \[ J = I_1\cup \cdots\cup I_p \] we associate the chain complex \[ - \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{m-1})\ot\cN(I_m) . + \psi(D) = \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{m-1})\ot\cN(I_m) . \] -(If $D$ denotes the subdivision of $J$, then we denote this complex by $\psi(D)$.) To each antirefinement we associate a chain map using the composition law of $\cC$ and the module actions of $\cC$ on $\cM$ and $\cN$. -\def\olD{{\overline D}} -\def\cbar{{\bar c}} The underlying graded vector space of the homotopy colimit is \[ \bigoplus_l \bigoplus_{\olD} \psi(D_0)[l] , \] where $l$ runs through the natural numbers, $\olD = (D_0\to D_1\to\cdots\to D_l)$ -runs through chains of antirefinements, and $[l]$ denotes a grading shift. +runs through chains of antirefinements of length $l+1$, and $[l]$ denotes a grading shift. We will denote an element of the summand indexed by $\olD$ by $\olD\ot m\ot\cbar\ot n$, where $m\ot\cbar\ot n \in \psi(D_0)$. -The boundary map is given (ignoring signs) by -\begin{eqnarray*} - \bd(\olD\ot m\ot\cbar\ot n) &=& \olD\ot\bd(m\ot\cbar)\ot n + \olD\ot m\ot\cbar\ot \bd n + \\ - & & \;\; (\bd_+ \olD)\ot m\ot\cbar\ot n + (\bd_0 \olD)\ot \rho(m\ot\cbar\ot n) , -\end{eqnarray*} -where $\bd_+ \olD = \sum_{i>0} (D_0, \cdots \widehat{D_i} \cdots , D_l)$ (the part of the simplicial -boundary which retains $D_0$), $\bd_0 \olD = (D_1, \cdots , D_l)$, +The boundary map is given by +\begin{align*} + \bd(\olD\ot m\ot\cbar\ot n) &= (\bd_0 \olD)\ot \rho(m\ot\cbar\ot n) + (\bd_+ \olD)\ot m\ot\cbar\ot n \; + \\ + & \qquad + (-1)^l \olD\ot\bd(m\ot\cbar\ot n) +\end{align*} +where $\bd_+ \olD = \sum_{i>0} (-1)^i (D_0\to \cdots \to \widehat{D_i} \to \cdots \to D_l)$ (those parts of the simplicial +boundary which retain $D_0$), $\bd_0 \olD = (D_1 \to \cdots \to D_l)$, and $\rho$ is the gluing map associated to the antirefinement $D_0\to D_1$. $(\cM_\cC\ot {_\cC\cN})^*$ is just the dual chain complex to $\cM_\cC\ot {_\cC\cN}$: @@ -1175,7 +1167,7 @@ f(\olD\ot m\ot\cbar\ot \bd n) + \\ & & \;\; f((\bd_+ \olD)\ot m\ot\cbar\ot n) + f((\bd_0 \olD)\ot \rho(m\ot\cbar\ot n)) . \end{eqnarray*} -(Again, we are ignoring signs.) +(Again, we are ignoring signs.) \nn{put signs in} Next we define the dual module $(_\cC\cN)^*$. This will depend on a choice of interval $J$, just as the tensor product did. @@ -1205,7 +1197,7 @@ We are almost ready to give the definition of morphisms between arbitrary modules $\cX_\cC$ and $\cY_\cC$. Note that the rightmost interval $I_m$ does not appear above, except implicitly in $\olD$. -To fix this, we define subdivisions are antirefinements of left-marked intervals. +To fix this, we define subdivisions as antirefinements of left-marked intervals. Subdivisions are just the obvious thing, but antirefinements are defined to mimic the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always omitted.