diff -r 296fcf7e5914 -r 7abe7642265e text/intro.tex --- a/text/intro.tex Tue Aug 09 23:22:07 2011 -0700 +++ b/text/intro.tex Tue Aug 09 23:55:13 2011 -0700 @@ -64,34 +64,34 @@ definition of an $n$-category, or rather a definition of an $n$-category with strong duality. (Removing the duality conditions from our definition would make it more complicated rather than less.) We call these ``disk-like $n$-categories'', to differentiate them from previous versions. -Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms. +Moreover, we find that we need analogous $A_\infty$ disk-like $n$-categories, and we define these as well following very similar axioms. (See \S \ref{n-cat-names} below for a discussion of $n$-category terminology.) The basic idea is that each potential definition of an $n$-category makes a choice about the ``shape" of morphisms. We try to be as lax as possible: a disk-like $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball. These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid. -For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to +For an $A_\infty$ disk-like $n$-category, we associate a chain complex instead of a vector space to each such $B$ and ask that the action of homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. -The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: +The axioms for an $A_\infty$ disk-like $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a disk-like $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. -In \S \ref{ssec:spherecat} we explain how $n$-categories can be viewed as objects in an $n{+}1$-category +In \S \ref{ssec:spherecat} we explain how disk-like $n$-categories can be viewed as objects in a disk-like $n{+}1$-category of sphere modules. When $n=1$ this just the familiar 2-category of 1-categories, bimodules and intertwiners. In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a disk-like $n$-category (using a colimit along certain decompositions of a manifold into balls). With this in hand, we write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$ -with the system of fields constructed from the $n$-category $\cC$. +with the system of fields constructed from the disk-like $n$-category $\cC$. %\nn{KW: I don't think we use this notational convention any more, right?} In \S \ref{sec:ainfblob} we give an alternative definition -of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). -Using these definitions, we show how to use the blob complex to ``resolve" any ordinary $n$-category as an -$A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. -We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), +of the blob complex for an $A_\infty$ disk-like $n$-category on an $n$-manifold (analogously, using a homotopy colimit). +Using these definitions, we show how to use the blob complex to ``resolve" any ordinary disk-like $n$-category as an +$A_\infty$ disk-like $n$-category, and relate the first and second definitions of the blob complex. +We use the blob complex for $A_\infty$ disk-like $n$-categories to establish important properties of the blob complex (in both variants), in particular the ``gluing formula" of Theorem \ref{thm:gluing} below. The relationship between all these ideas is sketched in Figure \ref{fig:outline}. @@ -155,8 +155,8 @@ a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cochains) in terms of the blob complex. The appendices prove technical results about $\CH{M}$ and -make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, -as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. +make connections between our definitions of disk-like $n$-categories and familiar definitions for $n=1$ and $n=2$, +as well as relating the $n=1$ case of our $A_\infty$ disk-like $n$-categories with usual $A_\infty$ algebras. %Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, %thought of as a disk-like $n$-category, in terms of the topology of $M$. @@ -373,42 +373,42 @@ from which we can construct systems of 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 an $A_\infty$ disk-like $n$-category. In that section we describe how to use the blob complex to -construct $A_\infty$ $n$-categories from ordinary $n$-categories: +construct $A_\infty$ disk-like $n$-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 an $A_\infty$ disk-like $n$-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 an $A_\infty$ disk-like $k$-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 an $A_\infty$ disk-like $k$-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 an $A_\infty$ disk-like $n$-category from an ordinary disk-like $n$-category. +We think of this $A_\infty$ disk-like $n$-category as a free resolution. \end{rem} -There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category -instead of an ordinary $n$-category; this is described in \S \ref{sec:ainfblob}. +There is a version of the blob complex for $\cC$ an $A_\infty$ disk-like $n$-category +instead of an ordinary disk-like $n$-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. +in terms of the $A_\infty$ blob complex of the $A_\infty$ disk-like $n$-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 $\cC$ be a disk-like $n$-category. +Let $\bc_*(Y;\cC)$ be the $A_\infty$ disk-like $k$-category associated to $Y$ via blob homology (see Example \ref{ex:blob-complexes-of-balls}). Then \[ @@ -420,7 +420,7 @@ 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.) +(See Appendix \ref{sec:comparing-A-infty} for the translation between $A_\infty$ disk-like $1$-categories and the usual algebraic notion of an $A_\infty$ category.) \newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}} @@ -447,7 +447,7 @@ \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 $A_\infty$ disk-like $n$-category based on maps $B^n \to T$. (The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) Then @@ -512,11 +512,11 @@ since we think of the higher homotopies not as morphisms of the $n$-category but rather as belonging to some auxiliary category (like chain complexes) that we are enriching in. -We have decided to call them ``$A_\infty$ $n$-categories", since they are a natural generalization +We have decided to call them ``$A_\infty$ disk-like $n$-categories", since they are a natural generalization of the familiar $A_\infty$ 1-categories. We also considered the names ``homotopy $n$-categories" and ``infinity $n$-categories". When we need to emphasize that we are talking about an $n$-category which is not $A_\infty$ in this sense -we will say ``ordinary $n$-category". +we will say ``ordinary disk-like $n$-category". % small problem: our n-cats are of course strictly associative, since we have more morphisms. % when we say ``associative only up to homotopy" above we are thinking about % what would happen we we tried to convert to a more traditional n-cat with fewer morphisms