# HG changeset patch # User Kevin Walker # Date 1278880728 21600 # Node ID 8aca80203f9d39d1b088aea1f102b357ce96bf94 # Parent 8f488e576afd8f45d57d01183dad40ac16f2f420 search & replace: s/((sub?)section|appendix)\s+\\ref/\S\ref/ diff -r 8f488e576afd -r 8aca80203f9d text/a_inf_blob.tex --- a/text/a_inf_blob.tex Sun Jul 11 14:31:56 2010 -0600 +++ b/text/a_inf_blob.tex Sun Jul 11 14:38:48 2010 -0600 @@ -3,7 +3,7 @@ \section{The blob complex for $A_\infty$ $n$-categories} \label{sec:ainfblob} Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the anticlimactically tautological definition of the blob -complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of Section \ref{ss:ncat_fields}. +complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}. We will show below in Corollary \ref{cor:new-old} @@ -53,7 +53,7 @@ \begin{proof} -We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. +We will use the concrete description of the colimit from \S\ref{ss:ncat_fields}. First we define a map \[ @@ -87,7 +87,7 @@ such that $a$ splits along $K_0\times F$ and $b$ is a generator appearing in an iterated boundary of $a$ (this includes $a$ itself). (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; -see Subsection \ref{ss:ncat_fields}.) +see \S\ref{ss:ncat_fields}.) By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is $b$ split according to $K_0\times F$. To simplify notation we will just write plain $b$ instead of $b^\sharp$. diff -r 8f488e576afd -r 8aca80203f9d text/appendixes/comparing_defs.tex --- a/text/appendixes/comparing_defs.tex Sun Jul 11 14:31:56 2010 -0600 +++ b/text/appendixes/comparing_defs.tex Sun Jul 11 14:38:48 2010 -0600 @@ -3,7 +3,7 @@ \section{Comparing $n$-category definitions} \label{sec:comparing-defs} -In this appendix we relate the ``topological" category definitions of Section \ref{sec:ncats} +In this appendix we relate the ``topological" category definitions of \S\ref{sec:ncats} to more traditional definitions, for $n=1$ and 2. \nn{cases to cover: (a) plain $n$-cats for $n=1,2$; (b) $n$-cat modules for $n=1$, also 2?; diff -r 8f488e576afd -r 8aca80203f9d text/basic_properties.tex --- a/text/basic_properties.tex Sun Jul 11 14:31:56 2010 -0600 +++ b/text/basic_properties.tex Sun Jul 11 14:38:48 2010 -0600 @@ -115,4 +115,4 @@ } This map is very far from being an isomorphism, even on homology. -We fix this deficit in Section \ref{sec:gluing} below. +We fix this deficit in \S\ref{sec:gluing} below. diff -r 8f488e576afd -r 8aca80203f9d text/deligne.tex --- a/text/deligne.tex Sun Jul 11 14:31:56 2010 -0600 +++ b/text/deligne.tex Sun Jul 11 14:38:48 2010 -0600 @@ -44,7 +44,7 @@ We emphasize that in $\hom(\bc^C_*(I), \bc^C_*(I))$ we are thinking of $\bc^C_*(I)$ as a module for the $A_\infty$ 1-category associated to $\bd I$, and $\hom$ means the morphisms of such modules as defined in -Subsection \ref{ss:module-morphisms}. +\S\ref{ss:module-morphisms}. We can think of a fat graph as encoding a sequence of surgeries, starting at the bottommost interval of Figure \ref{delfig1} and ending at the topmost interval. @@ -215,7 +215,7 @@ \] which satisfy the operad compatibility conditions. On $C_0(FG^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above. -When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of Section \ref{sec:evaluation}. +When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of \S\ref{sec:evaluation}. \end{thm} If, in analogy to Hochschild cochains, we define elements of $\hom(M, N)$ diff -r 8f488e576afd -r 8aca80203f9d text/evmap.tex --- a/text/evmap.tex Sun Jul 11 14:31:56 2010 -0600 +++ b/text/evmap.tex Sun Jul 11 14:38:48 2010 -0600 @@ -69,7 +69,7 @@ Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. \end{lemma} -The proof will be given in Appendix \ref{sec:localising}. +The proof will be given in \S\ref{sec:localising}. \medskip diff -r 8f488e576afd -r 8aca80203f9d text/intro.tex --- a/text/intro.tex Sun Jul 11 14:31:56 2010 -0600 +++ b/text/intro.tex Sun Jul 11 14:38:48 2010 -0600 @@ -139,7 +139,7 @@ in order to better integrate it into the current intro.} As a starting point, consider TQFTs constructed via fields and local relations. -(See Section \ref{sec:tqftsviafields} or \cite{kw:tqft}.) +(See \S\ref{sec:tqftsviafields} or \cite{kw:tqft}.) This gives a satisfactory treatment for semisimple TQFTs (i.e.\ TQFTs for which the cylinder 1-category associated to an $n{-}1$-manifold $Y$ is semisimple for all $Y$). diff -r 8f488e576afd -r 8aca80203f9d text/ncat.tex --- a/text/ncat.tex Sun Jul 11 14:31:56 2010 -0600 +++ b/text/ncat.tex Sun Jul 11 14:38:48 2010 -0600 @@ -97,7 +97,7 @@ $1\le k \le n$. At first it might seem that we need another axiom for this, but in fact once we have all the axioms in this subsection for $0$ through $k-1$ we can use a colimit -construction, as described in Subsection \ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ +construction, as described in \S\ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ to spheres (and any other manifolds): \begin{lem} @@ -746,7 +746,7 @@ to be the set of all $C$-labeled embedded cell complexes of $X\times F$. Define $\cC(X; c)$, for $X$ an $n$-ball, to be the dual Hilbert space $A(X\times F; c)$. -(See Subsection \ref{sec:constructing-a-tqft}.) +(See \S\ref{sec:constructing-a-tqft}.) \end{example} \noop{ @@ -1508,7 +1508,7 @@ \label{ss:module-morphisms} In order to state and prove our version of the higher dimensional Deligne conjecture -(Section \ref{sec:deligne}), +(\S\ref{sec:deligne}), we need to define morphisms of $A_\infty$ $1$-category modules and establish some of their elementary properties. @@ -1877,7 +1877,7 @@ of these points labeled by $n$-categories $\cA_i$ ($0\le i\le l$) and the marked points labeled by $\cA_i$-$\cA_{i+1}$ bimodules $\cM_i$. (See Figure \ref{feb21c}.) -To this data we can apply the coend construction as in Subsection \ref{moddecss} above +To this data we can apply the coend construction as in \S\ref{moddecss} above to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category. This amounts to a definition of taking tensor products of $0$-sphere module over $n$-categories. diff -r 8f488e576afd -r 8aca80203f9d text/tqftreview.tex --- a/text/tqftreview.tex Sun Jul 11 14:31:56 2010 -0600 +++ b/text/tqftreview.tex Sun Jul 11 14:38:48 2010 -0600 @@ -16,17 +16,17 @@ A system of fields is very closely related to an $n$-category. In one direction, Example \ref{ex:traditional-n-categories(fields)} shows how to construct a system of fields from a (traditional) $n$-category. -We do this in detail for $n=1,2$ (Subsection \ref{sec:example:traditional-n-categories(fields)}) +We do this in detail for $n=1,2$ (\S\ref{sec:example:traditional-n-categories(fields)}) and more informally for general $n$. In the other direction, -our preferred definition of an $n$-category in Section \ref{sec:ncats} is essentially +our preferred definition of an $n$-category in \S\ref{sec:ncats} is essentially just a system of fields restricted to balls of dimensions 0 through $n$; one could call this the ``local" part of a system of fields. Since this section is intended primarily to motivate -the blob complex construction of Section \ref{sec:blob-definition}, +the blob complex construction of \S\ref{sec:blob-definition}, we suppress some technical details. -In Section \ref{sec:ncats} the analogous details are treated more carefully. +In \S\ref{sec:ncats} the analogous details are treated more carefully. \medskip @@ -71,7 +71,7 @@ \end{example} Now for the rest of the definition of system of fields. -(Readers desiring a more precise definition should refer to Subsection \ref{ss:n-cat-def} +(Readers desiring a more precise definition should refer to \S\ref{ss:n-cat-def} and replace $k$-balls with $k$-manifolds.) \begin{enumerate} \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$,