# HG changeset patch # User Kevin Walker # Date 1318606515 25200 # Node ID c43f9f8fb3959f89534650a4888c1698ce552479 # Parent 084156aaee2fed4402785e96c3542344d767aa93 added some 'disk-like's in intro; other minor changes in intro; broke some lines diff -r 084156aaee2f -r c43f9f8fb395 RefereeReport.pdf Binary file RefereeReport.pdf has changed diff -r 084156aaee2f -r c43f9f8fb395 text/intro.tex --- a/text/intro.tex Fri Oct 14 07:48:41 2011 -0700 +++ b/text/intro.tex Fri Oct 14 08:35:15 2011 -0700 @@ -46,7 +46,9 @@ \subsection{Structure of the paper} The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), -summarize the formal properties of the blob complex (see \S \ref{sec:properties}), describe known specializations (see \S \ref{sec:specializations}), and outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications}). +summarize the formal properties of the blob complex (see \S \ref{sec:properties}), +describe known specializations (see \S \ref{sec:specializations}), +and outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications}). %and outline anticipated future directions (see \S \ref{sec:future}). %\nn{recheck this list after done editing intro} @@ -223,7 +225,9 @@ Here $\bc_0$ is linear combinations of fields on $W$, $\bc_1$ is linear combinations of local relations on $W$, $\bc_2$ is linear combinations of relations amongst relations on $W$, -and so on. We now have a short exact sequence of chain complexes relating resolutions of the link $L$ (c.f. Lemma \ref{lem:hochschild-exact} which shows exactness with respect to boundary conditions in the context of Hochschild homology). +and so on. We now have a short exact sequence of chain complexes relating resolutions of the link $L$ +(c.f. Lemma \ref{lem:hochschild-exact} which shows exactness +with respect to boundary conditions in the context of Hochschild homology). \subsection{Formal properties} @@ -370,36 +374,39 @@ In \S \ref{sec:ncats} we introduce the notion of disk-like $n$-categories, from which we can construct systems of fields. +Traditional $n$-categories can be converted to disk-like $n$-categories by taking string diagrams +(see \S\ref{sec:example:traditional-n-categories(fields)}). Below, when we talk about the blob complex for a disk-like $n$-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. +Further, in \S \ref{sec:ncats} we also have the notion of a disk-like $A_\infty$ $n$-category. In that section we describe how to use the blob complex to -construct $A_\infty$ $n$-categories from ordinary $n$-categories: +construct disk-like $A_\infty$ $n$-categories from ordinary disk-like $n$-categories: \newtheorem*{ex:blob-complexes-of-balls}{Example \ref{ex:blob-complexes-of-balls}} -\begin{ex:blob-complexes-of-balls}[Blob complexes of products with balls form an $A_\infty$ $n$-category] +\begin{ex:blob-complexes-of-balls}[Blob complexes of products with balls form a disk-like $A_\infty$ $n$-category] %\label{thm:blobs-ainfty} -Let $\cC$ be an ordinary $n$-category. +Let $\cC$ be an ordinary disk-like $n$-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 $0 \leq m < k$, +There is a disk-like $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ (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 +These sets have the structure of a disk-like $A_\infty$ $k$-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{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 an ordinary $n$-category. -We think of this $A_\infty$ $n$-category as a free resolution. +then we have a way of building a disk-like $A_\infty$ $n$-category from an ordinary $n$-category. % disk-like or not +We think of this disk-like $A_\infty$ $n$-category as a free resolution of the ordinary $n$-category. \end{rem} -There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category +There is a version of the blob complex for $\cC$ a disk-like $A_\infty$ $n$-category instead of an ordinary $n$-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 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 next theorem describes the blob complex for product manifolds +in terms of the $A_\infty$ blob complex of the disk-like $A_\infty$ $n$-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}} @@ -407,7 +414,7 @@ \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 +Let $\bc_*(Y;\cC)$ be the disk-like $A_\infty$ $k$-category associated to $Y$ via blob homology (see Example \ref{ex:blob-complexes-of-balls}). Then \[ @@ -418,8 +425,9 @@ (see \S \ref{ss:product-formula}). Fix a disk-like $n$-category $\cC$, which we'll omit from the notation. -Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. -(See Appendix \ref{sec:comparing-A-infty} for the translation between disk-like $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.) +Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ 1-category. +(See Appendix \ref{sec:comparing-A-infty} for the translation between disk-like $A_\infty$ $1$-categories +and the usual algebraic notion of an $A_\infty$ category.) \newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}} @@ -429,7 +437,8 @@ \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an $A_\infty$ module for $\bc_*(Y)$. -\item For any $n$-manifold $X_\text{gl} = X\bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{gl})$ is the $A_\infty$ self-tensor product of +\item For any $n$-manifold $X_\text{gl} = X\bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{gl})$ +is the $A_\infty$ self-tensor product of $\bc_*(X)$ as an $\bc_*(Y)$-bimodule: \begin{equation*} \bc_*(X_\text{gl}) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow @@ -446,11 +455,14 @@ \newtheorem*{thm:map-recon}{Theorem \ref{thm:map-recon}} \begin{thm:map-recon}[Mapping spaces] -Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps +Let $\pi^\infty_{\le n}(T)$ denote the disk-like $A_\infty$ $n$-category based on singular chains on maps $B^n \to T$. (The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) Then -$$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ +\[ + \bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}, +\] +where $C_*$ denotes singular chains. \end{thm:map-recon} This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. @@ -495,7 +507,8 @@ (see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1]; A)$, but haven't investigated the details. -Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} \nn{stabilization} \nn{stable categories, generalized cohomology theories} +Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} \nn{stabilization} +\nn{stable categories, generalized cohomology theories} } %%% end \noop %%%%%%%%%%%%%%%%%%%%% \subsection{\texorpdfstring{$n$}{n}-category terminology} @@ -529,7 +542,7 @@ the tongue as well as ``disk-like''.) Another thing we need a name for is the ability to rotate morphisms around in various ways. -For 2-categories, ``pivotal" is a standard term for what we mean. +For 2-categories, ``strict pivotal" is a standard term for what we mean. A more general term is ``duality", but duality comes in various flavors and degrees. We are mainly interested in a very strong version of duality, where the available ways of rotating $k$-morphisms correspond to all the ways of rotating $k$-balls.