blob: comparison text/intro.tex

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 We could also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories).
 \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}
 The first part of the paper (sections \S \ref{sec:fields}--\S \ref{sec:evaluation}) gives the definition of the blob complex,
 and establishes some of its properties.
 	\cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) .
 \]
 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}
 \label{sec:properties}
 The blob complex enjoys the following list of formal properties.
 for any homeomorphic pair $X$ and $Y$,
 satisfying corresponding conditions.
 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.
+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 $A_\infty$ $n$-category as a free resolution.
+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,
+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.
+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}}
 \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
 \[
 	\bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W).
 \]
 \end{thm:product}
 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps
 (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.
+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.)
+(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}}
 \begin{thm:gluing}[Gluing formula]
 \mbox{}% <-- gets the indenting right
 \begin{itemize}
 \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
 \end{equation*}
 \end{itemize}
 Finally, we give two applications of the above machinery.
 \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.
 Note that there is no restriction on the connectivity of $T$.
 The proof appears in \S \ref{sec:map-recon}.
 Many results in Hochschild homology can be understood ``topologically" via the blob complex.
 For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$
 (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}
 \label{n-cat-names}
 play a prominent role in the definition.
 (In general we prefer ``$k$-ball" to ``$k$-disk", but ``ball-like" doesn't roll off
 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.
 We sometimes refer to this as ``strong duality", and sometimes we consider it to be implied
 by ``disk-like".

changeset 912	c43f9f8fb395
parent 905	7afa2ffbbac8
child 913	75c1e11d0f25