# HG changeset patch # User Kevin Walker # Date 1283278706 25200 # Node ID 96ec10a46ee1a4cc5a529f53ee8deb4e2f1418a4 # Parent 19e58f33cdc3de4ddc7567e1b45d4319b55d6f3b minor; resolving a few \nns 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}} diff -r 19e58f33cdc3 -r 96ec10a46ee1 text/ncat.tex --- a/text/ncat.tex Mon Aug 30 13:19:05 2010 -0700 +++ b/text/ncat.tex Tue Aug 31 11:18:26 2010 -0700 @@ -17,14 +17,14 @@ The definitions presented below tie the categories more closely to the topology and avoid combinatorial questions about, for example, the minimal sufficient collections of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. -For examples of topological origin +It is easy to show that examples of topological origin (e.g.\ categories whose morphisms are maps into spaces or decorated balls), -it is easy to show that they satisfy our axioms. For examples of a more purely algebraic origin, one would typically need the combinatorial results that we have avoided here. -\nn{Say something explicit about Lurie's work here? It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen} +%\nn{Say something explicit about Lurie's work here? +%It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen} \medskip @@ -190,7 +190,8 @@ \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} Note that we insist on injectivity above. -The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. \nn{we might want a more official looking proof...} +The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. +%\nn{we might want a more official looking proof...} Let $\cl{\cC}(S)_E$ denote the image of $\gl_E$. We will refer to elements of $\cl{\cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$". @@ -890,12 +891,12 @@ The remaining data for the $A_\infty$ $n$-category --- composition and $\Diff(X\to X')$ action --- also comes from the $\cE\cB_n$ action on $A$. -\nn{should we spell this out?} +%\nn{should we spell this out?} Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms $\cC(X)$ are trivial (single point) for $k