diff -r 19e58f33cdc3 -r 96ec10a46ee1 text/intro.tex --- a/text/intro.tex Mon Aug 30 13:19:05 2010 -0700 +++ b/text/intro.tex Tue Aug 31 11:18:26 2010 -0700 @@ -76,9 +76,9 @@ In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category (using a colimit along certain decompositions of a manifold into balls). -With this in hand, we freely write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$ +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 $\cC$. -\nn{KW: I don't think we use this notational convention any more, right?} +%\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 topological $n$-category as an @@ -127,7 +127,7 @@ \draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Theorem \ref{thm:skein-modules}} (A); \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); -\draw (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); +\draw[<->] (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); \end{tikzpicture} } @@ -139,8 +139,8 @@ Section \S \ref{sec:deligne} gives a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cochains) in terms of the blob complex. -The appendixes prove technical results about $\CH{M}$ and the ``small blob complex", -and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, +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. Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, thought of as a topological $n$-category, in terms of the topology of $M$. @@ -343,9 +343,9 @@ } \end{equation*} \end{enumerate} -Moreover any such chain map is unique, up to an iterated homotopy. -(That is, any pair of homotopies have a homotopy between them, and so on.) -\nn{revisit this after proof below has stabilized} +%Moreover any such chain map is unique, up to an iterated homotopy. +%(That is, any pair of homotopies have a homotopy between them, and so on.) +%\nn{revisit this after proof below has stabilized} \end{thm:CH} \newtheorem*{thm:CH-associativity}{Theorem \ref{thm:CH-associativity}}