--- a/blob1.tex Thu Apr 26 06:57:24 2012 -0600
+++ b/blob1.tex Fri Apr 27 22:37:14 2012 -0700
@@ -1,4 +1,4 @@
-\documentclass[11pt,leqno]{article}
+\documentclass[11pt,leqno]{gtart}
\newcommand{\pathtotrunk}{./}
\input{preamble}
@@ -15,8 +15,6 @@
\gdef\theequation{\thesection.\arabic{equation}}
\makeatother
-\maketitle
-
\begin{abstract}
Given an $n$-manifold $M$ and an $n$-category $\cC$, we define a chain complex
(the ``blob complex") $\bc_*(M; \cC)$.
@@ -28,6 +26,10 @@
is particularly well suited for work with TQFTs. This is the published version of \href{http://arxiv.org/abs/1009.5025}{arXiv:1009.5025}.
\end{abstract}
+
+\maketitle
+
+
\hypersetup{
colorlinks, linkcolor={black},
citecolor={dark-blue}, urlcolor={medium-blue}
@@ -71,8 +73,8 @@
%\input{text/comm_alg}
% ----------------------------------------------------------------
-%\newcommand{\urlprefix}{}
-\bibliographystyle{alpha}
+\newcommand{\urlprefix}{}
+\bibliographystyle{gtart}
\bibliography{bibliography/bibliography}
% ----------------------------------------------------------------
--- a/build.xml Thu Apr 26 06:57:24 2012 -0600
+++ b/build.xml Fri Apr 27 22:37:14 2012 -0700
@@ -41,7 +41,7 @@
<delete file="${arxivTarFile}"/>
<delete file="${arxivTarFile}.gz"/>
<tar destfile="${arxivTarFile}" basedir="." includes="**"
- excludes="*.synctex*,*.dvi,*.ps,blob1.pdf,*.png,${arxivTarFile},${arxivTarFile}.gz,sandbox.*,bibliography/**,papers/**,talks/**,diagrams/obsolete/**,diagrams/latex2pdf/**,pnas/**,text/obsolete/**,.hg/**,versions/**,TODO,referee/**"
+ excludes="*.synctex*,*.dvi,*.ps,blob1.pdf,*.png,${arxivTarFile},${arxivTarFile}.gz,sandbox.*,bibliography/**,papers/**,talks/**,diagrams/obsolete/**,diagrams/latex2pdf/**,pnas/**,text/obsolete/**,.hg/**,versions/**,TODO,referee/**,copyright/**"
/>
<gzip src="${arxivTarFile}" destfile="${arxivTarFile}.gz"/>
<delete file="${arxivTarFile}"/>
Binary file copyright/copyright-scott.pdf has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/gtart.bst Fri Apr 27 22:37:14 2012 -0700
@@ -0,0 +1,1340 @@
+%%
+%% This is file `gtart.bst',
+%% generated with the docstrip utility.
+%%
+%%
+%% The original source files were:
+%%
+%% merlin.mbs (with options: `ed-au,nmft,nmft-bf,dt-jnl,yr-par,xmth,yrp-x,tit-it,atit-u,jttl-rm,vnum-x,volp-blk,jdt-vs,pp-last,num-xser,numser,jnm-x,bkpg-x,pre-edn,edpar,edby-par,edbyy,blk-com,in-x,fin-bare,ppx,xedn,and-xcom,xand,eprint,url,url-blk,nfss')
+%% ----------------------------------------
+%% *** For Geometry and Topology Publications ***
+%%
+%% Copyright 1994-1999 Patrick W Daly
+%%
+%% Modified by Boris Okun 12/2001
+%%
+ % ===============================================================
+ % IMPORTANT NOTICE:
+ % This bibliographic style (bst) file has been generated from one or
+ % more master bibliographic style (mbs) files, listed above.
+ %
+ % This generated file can be redistributed and/or modified under the terms
+ % of the LaTeX Project Public License Distributed from CTAN
+ % archives in directory macros/latex/base/lppl.txt; either
+ % version 1 of the License, or any later version.
+ % ===============================================================
+ % Name and version information of the main mbs file:
+ % \ProvidesFile{merlin.mbs}[1999/03/18 3.88 (PWD)]
+ % For use with BibTeX version 0.99a or later
+ %-------------------------------------------------------------------
+ % This bibliography style file is intended for texts in ENGLISH
+ % This is a numerical citation style, and as such is standard LaTeX.
+ % It requires no extra package to interface to the main text.
+ % The form of the \bibitem entries is
+ % \bibitem{key}...
+ % Usage of \cite is as follows:
+ % \cite{key} ==>> [#]
+ % \cite[chap. 2]{key} ==>> [#, chap. 2]
+ % where # is a number determined by the ordering in the reference list.
+ % The order in the reference list is alphabetical by authors.
+ %---------------------------------------------------------------------
+
+ENTRY
+ { address
+ author
+ booktitle
+ chapter
+ edition
+ editor
+ howpublished
+ institution
+ journal
+ key
+ month
+ note
+ number
+ organization
+ pages
+ publisher
+ school
+ series
+ title
+ type
+ url
+ volume
+ year
+ }
+ {}
+ { label }
+
+INTEGERS { output.state before.all mid.sentence after.sentence after.block }
+
+FUNCTION {init.state.consts}
+{ #0 'before.all :=
+ #1 'mid.sentence :=
+ #2 'after.sentence :=
+ #3 'after.block :=
+}
+
+STRINGS { s t }
+
+FUNCTION {output.nonnull}
+{ 's :=
+ output.state mid.sentence =
+ { ", " * write$ }
+ { output.state after.block =
+ { add.period$ write$
+ newline$
+ "\newblock " write$
+ }
+ { output.state before.all =
+ 'write$
+ { add.period$ " " * write$ }
+ if$
+ }
+ if$
+ mid.sentence 'output.state :=
+ }
+ if$
+ s
+}
+
+FUNCTION {output}
+{ duplicate$ empty$
+ 'pop$
+ 'output.nonnull
+ if$
+}
+
+FUNCTION {output.check}
+{ 't :=
+ duplicate$ empty$
+ { pop$ "empty " t * " in " * cite$ * warning$ }
+ 'output.nonnull
+ if$
+}
+
+FUNCTION {fin.entry}
+{ duplicate$ empty$
+ 'pop$
+ 'write$
+ if$
+ newline$
+}
+
+FUNCTION {new.block}
+{ output.state before.all =
+ 'skip$
+ { after.block 'output.state := }
+ if$
+}
+
+FUNCTION {new.sentence}
+{ output.state after.block =
+ 'skip$
+ { output.state before.all =
+ 'skip$
+ { after.sentence 'output.state := }
+ if$
+ }
+ if$
+}
+
+FUNCTION {add.blank}
+{ " " * before.all 'output.state :=
+}
+
+FUNCTION {date.block}
+{
+ add.blank
+}
+
+FUNCTION {not}
+{ { #0 }
+ { #1 }
+ if$
+}
+
+FUNCTION {and}
+{ 'skip$
+ { pop$ #0 }
+ if$
+}
+
+FUNCTION {or}
+{ { pop$ #1 }
+ 'skip$
+ if$
+}
+
+FUNCTION {new.block.checka}
+{ empty$
+ 'skip$
+ 'new.block
+ if$
+}
+
+FUNCTION {new.block.checkb}
+{ empty$
+ swap$ empty$
+ and
+ 'skip$
+ 'new.block
+ if$
+}
+
+FUNCTION {new.sentence.checka}
+{ empty$
+ 'skip$
+ 'new.sentence
+ if$
+}
+
+FUNCTION {new.sentence.checkb}
+{ empty$
+ swap$ empty$
+ and
+ 'skip$
+ 'new.sentence
+ if$
+}
+
+FUNCTION {field.or.null}
+{ duplicate$ empty$
+ { pop$ "" }
+ 'skip$
+ if$
+}
+
+FUNCTION {emphasize}
+{ duplicate$ empty$
+ { pop$ "" }
+ { "\emph{" swap$ * "}" * }
+ if$
+}
+
+FUNCTION {bolden}
+{ duplicate$ empty$
+ { pop$ "" }
+ { "\textbf{" swap$ * "}" * }
+ if$
+}
+
+FUNCTION {quotify}
+{ duplicate$ empty$
+ { pop$ "" }
+ { "``" swap$ * "''" * }
+ if$
+}
+
+
+FUNCTION {bib.name.font}
+{ bolden }
+
+FUNCTION {bib.fname.font}
+{ bib.name.font }
+FUNCTION {capitalize}
+{ "u" change.case$ "t" change.case$ }
+
+FUNCTION {space.word}
+{ " " swap$ * " " * }
+
+ % Here are the language-specific definitions for explicit words.
+ % Each function has a name bbl.xxx where xxx is the English word.
+ % The language selected here is ENGLISH
+FUNCTION {bbl.and}
+{ "and"}
+
+FUNCTION {bbl.etal}
+{ "et~al." }
+
+FUNCTION {bbl.editors}
+{ "editors" }
+
+FUNCTION {bbl.editor}
+{ "editor" }
+
+FUNCTION {bbl.edby}
+{ "edited by" }
+
+FUNCTION {bbl.edition}
+{ "edition" }
+
+FUNCTION {bbl.volume}
+{ "volume" }
+
+FUNCTION {bbl.of}
+{ "of" }
+
+FUNCTION {bbl.number}
+{ "number" }
+
+FUNCTION {bbl.nr}
+{ "no." }
+
+FUNCTION {bbl.in}
+{ "in" }
+
+FUNCTION {bbl.pages}
+{ "" }
+
+FUNCTION {bbl.page}
+{ "" }
+
+FUNCTION {bbl.chapter}
+{ "chapter" }
+
+FUNCTION {bbl.techrep}
+{ "Technical Report" }
+
+FUNCTION {bbl.mthesis}
+{ "Master's thesis" }
+
+FUNCTION {bbl.phdthesis}
+{ "Ph.D. thesis" }
+
+MACRO {jan} {"January"}
+
+MACRO {feb} {"February"}
+
+MACRO {mar} {"March"}
+
+MACRO {apr} {"April"}
+
+MACRO {may} {"May"}
+
+MACRO {jun} {"June"}
+
+MACRO {jul} {"July"}
+
+MACRO {aug} {"August"}
+
+MACRO {sep} {"September"}
+
+MACRO {oct} {"October"}
+
+MACRO {nov} {"November"}
+
+MACRO {dec} {"December"}
+
+FUNCTION {format.url}
+{ url empty$
+ { "" }
+ { "\urlprefix\url{" url * "}" * }
+ if$
+}
+
+INTEGERS { nameptr namesleft numnames charptr}
+
+STRINGS {i j}
+
+FUNCTION {remove.periods}
+{'i :=
+ ""
+ #1 'charptr :=
+ " " 'j :=
+ {#1 j "" = - }
+ {i charptr #1 substring$
+ 'j :=
+ j "." =
+ {charptr #1 + 'charptr :=
+ i charptr #1 substring$
+ 'j :=
+ j "~" =
+ {"\," *}
+ {j *}
+ if$}
+ {j *}
+ if$
+ charptr #1 + 'charptr :=
+ }
+ while$
+}
+
+FUNCTION {format.names}
+{ 's :=
+ #1 'nameptr :=
+ s num.names$ 'numnames :=
+ numnames 'namesleft :=
+ { namesleft #0 > }
+ { s nameptr
+ "{ff }{vv~}{ll}{, jj}" format.name$
+ remove.periods
+ 't :=
+ nameptr #1 >
+ {
+ namesleft #1 >
+ { ", " * t * }
+ {
+ "," *
+ s nameptr "{ll}" format.name$ duplicate$ "others" =
+ { 't := }
+ { pop$ }
+ if$
+ t "others" =
+ {
+ " " * bbl.etal *
+% bib.name.font
+ }
+ { " " * t * }
+ if$
+ }
+ if$
+ }
+ 't
+ if$
+ nameptr #1 + 'nameptr :=
+ namesleft #1 - 'namesleft :=
+ }
+ while$
+% t "others" =
+% 'skip$
+% { bib.name.font }
+% if$
+}
+
+FUNCTION {bformat.names} % Formats names in bold, but keeps punctuation normal
+{ 's :=
+ #1 'nameptr :=
+ s num.names$ 'numnames :=
+ numnames 'namesleft :=
+ { namesleft #0 > }
+ { s nameptr
+ "{ff }{vv~}{ll}{, jj}" format.name$
+ remove.periods
+ bib.name.font
+ 't :=
+ nameptr #1 >
+ {
+ namesleft #1 >
+ { ", " * t * }
+ {
+ "," *
+ s nameptr "{ll}" format.name$ duplicate$ "others" =
+ { 't := }
+ { pop$ }
+ if$
+ t "others" =
+ {
+ " " * bbl.etal *
+% bib.name.font
+ }
+ { " " * t * }
+ if$
+ }
+ if$
+ }
+ 't
+ if$
+ nameptr #1 + 'nameptr :=
+ namesleft #1 - 'namesleft :=
+ }
+ while$
+% t "others" =
+% 'skip$
+% { bib.name.font }
+% if$
+}
+
+FUNCTION {format.names.ed}
+{ format.names }
+FUNCTION {format.authors}
+{ author empty$
+ { "" }
+ { author bformat.names }
+% bib.name.font}
+ if$
+}
+
+FUNCTION {format.editors}
+{ editor empty$
+ { "" }
+ { editor bformat.names
+% bib.name.font
+ editor num.names$ #1 >
+ { " (" * bbl.editors * ")" * }
+ { " (" * bbl.editor * ")" * }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.in.editors}
+{ editor empty$
+ { "" }
+ { editor format.names.ed
+ }
+ if$
+}
+
+FUNCTION {format.note}
+{
+ note empty$
+ { "" }
+ { note #1 #1 substring$
+ duplicate$ "{" =
+ 'skip$
+ { output.state mid.sentence =
+ { "l" }
+ { "u" }
+ if$
+ change.case$
+ }
+ if$
+ note #2 global.max$ substring$ *
+ }
+ if$
+}
+
+FUNCTION {format.title}
+{ title empty$
+ { "" }
+ { title
+ emphasize
+ }
+ if$
+}
+
+FUNCTION {output.bibitem}
+{ newline$
+ "\bibitem{" write$
+ cite$ write$
+ "}" write$
+ newline$
+ ""
+ before.all 'output.state :=
+}
+
+FUNCTION {n.dashify}
+{
+ 't :=
+ ""
+ { t empty$ not }
+ { t #1 #1 substring$ "-" =
+ { t #1 #2 substring$ "--" = not
+ { "--" *
+ t #2 global.max$ substring$ 't :=
+ }
+ { { t #1 #1 substring$ "-" = }
+ { "-" *
+ t #2 global.max$ substring$ 't :=
+ }
+ while$
+ }
+ if$
+ }
+ { t #1 #1 substring$ *
+ t #2 global.max$ substring$ 't :=
+ }
+ if$
+ }
+ while$
+}
+
+FUNCTION {word.in}
+{ "from: " }
+
+FUNCTION {format.date}
+{ year empty$
+ { "" }
+ 'year
+ if$
+ duplicate$ empty$
+ 'skip$
+ {
+ before.all 'output.state :=
+ " (" swap$ * ")" *
+ }
+ if$
+}
+
+FUNCTION{format.year}
+{ year duplicate$ empty$
+ { "empty year in " cite$ * warning$ pop$ "" }
+ { " (" swap$ * ")" * }
+ if$
+}
+
+FUNCTION {format.btitle}
+{ title emphasize
+}
+
+FUNCTION {tie.or.space.connect}
+{ duplicate$ text.length$ #3 <
+ { "~" }
+ { " " }
+ if$
+ swap$ * *
+}
+
+FUNCTION {either.or.check}
+{ empty$
+ 'pop$
+ { "can't use both " swap$ * " fields in " * cite$ * warning$ }
+ if$
+}
+
+FUNCTION {format.bvolume}
+{ volume empty$
+ { "" }
+ { bbl.volume volume tie.or.space.connect
+ series empty$
+ 'skip$
+ { bbl.of space.word * series emphasize * }
+ if$
+ "volume and number" number either.or.check
+ }
+ if$
+}
+
+FUNCTION {format.bvolume.in}
+{series empty$
+ 'format.bvolume
+ {volume empty$
+ {""}
+ {series " " volume * *
+ "volume and number" number either.or.check }
+ if$
+ }
+if$
+}
+
+FUNCTION {format.number.series}
+{ volume empty$
+ { number empty$
+ { series field.or.null }
+ { series empty$
+ { number }
+ { output.state mid.sentence =
+ { bbl.number }
+ { bbl.number capitalize }
+ if$
+ number tie.or.space.connect
+ bbl.in space.word * series *
+ }
+ if$
+ }
+ if$
+ }
+ { "" }
+ if$
+}
+
+
+FUNCTION {format.number.series.in}
+{ volume empty$
+ {series empty$
+ 'format.number.series
+ {series
+ number empty$
+ 'skip$
+ {" " number * *}
+ if$ }
+ if$
+ }
+ { "" }
+if$
+}
+
+
+
+FUNCTION {format.edition}
+{ edition empty$
+ { "" }
+ { output.state mid.sentence =
+ { edition "l" change.case$ " " * bbl.edition * }
+ { edition "t" change.case$ " " * bbl.edition * }
+ if$
+ }
+ if$
+}
+
+INTEGERS { multiresult }
+
+FUNCTION {multi.page.check}
+{ 't :=
+ #0 'multiresult :=
+ { multiresult not
+ t empty$ not
+ and
+ }
+ { t #1 #1 substring$
+ duplicate$ "-" =
+ swap$ duplicate$ "," =
+ swap$ "+" =
+ or or
+ { #1 'multiresult := }
+ { t #2 global.max$ substring$ 't := }
+ if$
+ }
+ while$
+ multiresult
+}
+
+FUNCTION {format.pages}
+{ pages empty$
+ { "" }
+ { pages multi.page.check
+ { pages n.dashify }
+ { pages }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.journal.pages}
+{ pages empty$
+ 'skip$
+ { duplicate$ empty$
+ { pop$ format.pages }
+ {
+ " " *
+ pages n.dashify *
+ }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.vol.num.pages}
+{ volume field.or.null
+ format.year *
+}
+
+FUNCTION {format.chapter.pages}
+{ chapter empty$
+ { "" }
+ { type empty$
+ { bbl.chapter }
+ { type "l" change.case$ }
+ if$
+ chapter tie.or.space.connect
+ }
+ if$
+}
+
+FUNCTION {format.in.ed.booktitle}
+{ booktitle empty$
+ { "" }
+ { editor empty$
+ { word.in booktitle quotify * }
+ { word.in booktitle quotify *
+ ", (" *
+ format.in.editors *
+ ", " *
+ editor num.names$ #1 >
+ { bbl.editors }
+ { bbl.editor }
+ if$
+ *
+ ")" *
+ }
+ if$
+ }
+ if$
+}
+
+FUNCTION {empty.misc.check}
+{ author empty$ title empty$ howpublished empty$
+ month empty$ year empty$ note empty$
+ and and and and and
+ key empty$ not and
+ { "all relevant fields are empty in " cite$ * warning$ }
+ 'skip$
+ if$
+}
+
+FUNCTION {format.thesis.type}
+{ type empty$
+ 'skip$
+ { pop$
+ type "t" change.case$
+ }
+ if$
+}
+
+FUNCTION {format.tr.number}
+{ type empty$
+ { bbl.techrep }
+ 'type
+ if$
+ number empty$
+ { "t" change.case$ }
+ { number tie.or.space.connect }
+ if$
+}
+
+FUNCTION {format.article.crossref}
+{
+ key empty$
+ { journal empty$
+ { "need key or journal for " cite$ * " to crossref " * crossref *
+ warning$
+ ""
+ }
+ { word.in journal emphasize * }
+ if$
+ }
+ { word.in key * " " *}
+ if$
+ " \cite{" * crossref * "}" *
+}
+
+FUNCTION {format.crossref.editor}
+{ editor #1 "{vv~}{ll}" format.name$
+ editor num.names$ duplicate$
+ #2 >
+ { pop$
+ " " * bbl.etal *
+ }
+ { #2 <
+ 'skip$
+ { editor #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" =
+ {
+ " " * bbl.etal *
+ }
+ { bbl.and space.word * editor #2 "{vv~}{ll}" format.name$
+ * }
+ if$
+ }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.book.crossref}
+{ volume empty$
+ { "empty volume in " cite$ * "'s crossref of " * crossref * warning$
+ word.in
+ }
+ { bbl.volume volume tie.or.space.connect
+ bbl.of space.word *
+ }
+ if$
+ editor empty$
+ editor field.or.null author field.or.null =
+ or
+ { key empty$
+ { series empty$
+ { "need editor, key, or series for " cite$ * " to crossref " *
+ crossref * warning$
+ "" *
+ }
+ { series emphasize * }
+ if$
+ }
+ { key * }
+ if$
+ }
+ { format.crossref.editor * }
+ if$
+ " \cite{" * crossref * "}" *
+}
+
+FUNCTION {format.incoll.inproc.crossref}
+{
+ editor empty$
+ editor field.or.null author field.or.null =
+ or
+ { key empty$
+ { booktitle empty$
+ { "need editor, key, or booktitle for " cite$ * " to crossref " *
+ crossref * warning$
+ ""
+ }
+ { word.in "``" booktitle "''" * * }
+ if$
+ }
+ { word.in key * " " *}
+ if$
+ }
+ { word.in format.crossref.editor * " " *}
+ if$
+ " \cite{" * crossref * "}" *
+}
+
+FUNCTION {format.org.or.pub}
+{ 't :=
+ ""
+ address empty$ t empty$ and
+ 'skip$
+ {
+ t empty$
+ { address empty$
+ 'skip$
+ { address * }
+ if$
+ }
+ { t *
+ address empty$
+ 'skip$
+ { ", " * address * }
+ if$
+ }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.publisher.address}
+{ publisher empty$
+ { "empty publisher in " cite$ * warning$
+ ""
+ }
+ { publisher }
+ if$
+ format.org.or.pub
+}
+
+FUNCTION {format.organization.address}
+{ organization empty$
+ { "" }
+ { organization }
+ if$
+ format.org.or.pub
+}
+
+FUNCTION {article}
+{ output.bibitem
+ format.authors "author" output.check
+ format.title "title" output.check
+ crossref missing$
+ { journal
+ "journal" output.check
+ add.blank
+ format.vol.num.pages output
+ }
+ { format.article.crossref output.nonnull
+% format.pages output
+ }
+ if$
+ format.journal.pages
+ format.url output
+ format.note output
+ fin.entry
+}
+
+FUNCTION {book}
+{ output.bibitem
+ author empty$
+ { format.editors "author and editor" output.check
+ }
+ { format.authors output.nonnull
+ crossref missing$
+ { "author and editor" editor either.or.check }
+ 'skip$
+ if$
+ }
+ if$
+ format.btitle "title" output.check
+ crossref missing$
+ { format.bvolume output
+ format.edition output
+ format.number.series.in output
+ format.publisher.address output
+ }
+ {
+ format.book.crossref output.nonnull
+ }
+ if$
+ format.date "year" output.check
+ format.url output
+ format.note output
+ fin.entry
+}
+
+FUNCTION {booklet}
+{ output.bibitem
+ format.authors output
+ format.title "title" output.check
+ howpublished output
+ address output
+ format.date output
+ format.url output
+ format.note output
+ fin.entry
+}
+
+FUNCTION {incollection}
+{ output.bibitem
+ format.authors "author" output.check
+ format.title "title" output.check
+ crossref missing$
+ { format.in.ed.booktitle "booktitle" output.check
+ format.bvolume.in output
+ format.edition output
+ format.number.series.in output
+ format.publisher.address output
+ }
+ { format.incoll.inproc.crossref output.nonnull
+ }
+ if$
+ format.date "year" output.check
+ date.block
+ add.blank
+ format.pages "pages" output.check
+ format.url output
+ format.note output
+ fin.entry
+}
+
+FUNCTION {inbook}{incollection}
+
+FUNCTION {inproceedings}
+{ output.bibitem
+ format.authors "author" output.check
+ format.title "title" output.check
+ crossref missing$
+ { format.in.ed.booktitle "booktitle" output.check
+ format.bvolume.in output
+ format.number.series.in output
+ publisher empty$
+ { format.organization.address output }
+ { organization output
+ format.publisher.address output
+ }
+ if$
+ }
+ { format.incoll.inproc.crossref output.nonnull
+ }
+ if$
+ format.date "year" output.check
+ date.block
+ add.blank
+ format.pages "pages" output.check
+ format.url output
+ format.note output
+ fin.entry
+}
+
+FUNCTION {conference} { inproceedings }
+
+FUNCTION {manual}
+{ output.bibitem
+ author empty$
+ { organization empty$
+ 'skip$
+ { organization output.nonnull
+ address output
+ }
+ if$
+ }
+ { format.authors output.nonnull }
+ if$
+ format.btitle "title" output.check
+ author empty$
+ { organization empty$
+ {
+ address output
+ }
+ 'skip$
+ if$
+ }
+ {
+ organization output
+ address output
+ }
+ if$
+ format.edition output
+ format.date output
+ format.url output
+ format.note output
+ fin.entry
+}
+
+FUNCTION {mastersthesis}
+{ output.bibitem
+ format.authors "author" output.check
+ format.btitle "title" output.check
+ bbl.mthesis format.thesis.type output.nonnull
+ school "school" output.check
+ address output
+ format.date "year" output.check
+ format.url output
+ format.note output
+ fin.entry
+}
+
+FUNCTION {misc}
+{ output.bibitem
+ format.authors output
+ format.title output
+ howpublished output
+ format.date output
+ format.url output
+ format.note output
+ fin.entry
+ empty.misc.check
+}
+
+FUNCTION {phdthesis}
+{ output.bibitem
+ format.authors "author" output.check
+ format.btitle "title" output.check
+ bbl.phdthesis format.thesis.type output.nonnull
+ school "school" output.check
+ address output
+ format.date "year" output.check
+ format.url output
+ format.note output
+ fin.entry
+}
+
+FUNCTION {proceedings}
+{ output.bibitem
+ editor empty$
+ { organization output }
+ { format.editors output.nonnull }
+ if$
+ format.btitle "title" output.check
+ format.bvolume output
+ editor empty$
+ { publisher empty$
+ 'skip$
+ {
+ format.number.series output
+ format.publisher.address output
+ }
+ if$
+ }
+ { publisher empty$
+ {
+ format.organization.address output }
+ {
+ organization output
+ format.publisher.address output
+ }
+ if$
+ }
+ if$
+ format.date "year" output.check
+ format.url output
+ format.note output
+ fin.entry
+}
+
+FUNCTION {techreport}
+{ output.bibitem
+ format.authors "author" output.check
+ format.title "title" output.check
+ format.tr.number output.nonnull
+ institution "institution" output.check
+ address output
+ format.date "year" output.check
+ format.url output
+ format.note output
+ fin.entry
+}
+
+FUNCTION {unpublished}
+{ output.bibitem
+ format.authors "author" output.check
+ format.title "title" output.check
+ format.date output
+ format.url output
+ format.note "note" output.check
+ fin.entry
+}
+
+FUNCTION {default.type} { misc }
+
+READ
+
+FUNCTION {sortify}
+{ purify$
+ "l" change.case$
+}
+
+INTEGERS { len }
+
+FUNCTION {chop.word}
+{ 's :=
+ 'len :=
+ s #1 len substring$ =
+ { s len #1 + global.max$ substring$ }
+ 's
+ if$
+}
+
+FUNCTION {sort.format.names}
+{ 's :=
+ #1 'nameptr :=
+ ""
+ s num.names$ 'numnames :=
+ numnames 'namesleft :=
+ { namesleft #0 > }
+ { s nameptr
+ "{vv{ } }{ll{ }}{ ff{ }}{ jj{ }}"
+ format.name$ 't :=
+ nameptr #1 >
+ {
+ " " *
+ namesleft #1 = t "others" = and
+ { "zzzzz" * }
+ { t sortify * }
+ if$
+ }
+ { t sortify * }
+ if$
+ nameptr #1 + 'nameptr :=
+ namesleft #1 - 'namesleft :=
+ }
+ while$
+}
+
+FUNCTION {sort.format.title}
+{ 't :=
+ "A " #2
+ "An " #3
+ "The " #4 t chop.word
+ chop.word
+ chop.word
+ sortify
+ #1 global.max$ substring$
+}
+
+FUNCTION {author.sort}
+{ author empty$
+ { key empty$
+ { "to sort, need author or key in " cite$ * warning$
+ ""
+ }
+ { key sortify }
+ if$
+ }
+ { author sort.format.names }
+ if$
+}
+
+FUNCTION {author.editor.sort}
+{ author empty$
+ { editor empty$
+ { key empty$
+ { "to sort, need author, editor, or key in " cite$ * warning$
+ ""
+ }
+ { key sortify }
+ if$
+ }
+ { editor sort.format.names }
+ if$
+ }
+ { author sort.format.names }
+ if$
+}
+
+FUNCTION {author.organization.sort}
+{ author empty$
+ { organization empty$
+ { key empty$
+ { "to sort, need author, organization, or key in " cite$ * warning$
+ ""
+ }
+ { key sortify }
+ if$
+ }
+ { "The " #4 organization chop.word sortify }
+ if$
+ }
+ { author sort.format.names }
+ if$
+}
+
+FUNCTION {editor.organization.sort}
+{ editor empty$
+ { organization empty$
+ { key empty$
+ { "to sort, need editor, organization, or key in " cite$ * warning$
+ ""
+ }
+ { key sortify }
+ if$
+ }
+ { "The " #4 organization chop.word sortify }
+ if$
+ }
+ { editor sort.format.names }
+ if$
+}
+
+FUNCTION {presort}
+{ type$ "book" =
+ type$ "inbook" =
+ or
+ 'author.editor.sort
+ { type$ "proceedings" =
+ 'editor.organization.sort
+ { type$ "manual" =
+ 'author.organization.sort
+ 'author.sort
+ if$
+ }
+ if$
+ }
+ if$
+ " "
+ *
+ year field.or.null sortify
+ *
+ " "
+ *
+ title field.or.null
+ sort.format.title
+ *
+ #1 entry.max$ substring$
+ 'sort.key$ :=
+}
+
+ITERATE {presort}
+
+SORT
+
+STRINGS { longest.label }
+
+INTEGERS { number.label longest.label.width }
+
+FUNCTION {initialize.longest.label}
+{ "" 'longest.label :=
+ #1 'number.label :=
+ #0 'longest.label.width :=
+}
+
+FUNCTION {longest.label.pass}
+{ number.label int.to.str$ 'label :=
+ number.label #1 + 'number.label :=
+ label width$ longest.label.width >
+ { label 'longest.label :=
+ label width$ 'longest.label.width :=
+ }
+ 'skip$
+ if$
+}
+
+EXECUTE {initialize.longest.label}
+
+ITERATE {longest.label.pass}
+
+FUNCTION {begin.bib}
+{ preamble$ empty$
+ 'skip$
+ { preamble$ write$ newline$ }
+ if$
+ "\begin{thebibliography}"
+ write$ newline$
+}
+
+EXECUTE {begin.bib}
+
+EXECUTE {init.state.consts}
+
+ITERATE {call.type$}
+
+FUNCTION {end.bib}
+{ newline$
+ "\end{thebibliography}" write$ newline$
+}
+
+EXECUTE {end.bib}
+%% End of customized bst file
+%%
+%% End of file `gtart.bst'.
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/gtart.cls Fri Apr 27 22:37:14 2012 -0700
@@ -0,0 +1,502 @@
+%%%%%%%%%%%%%%%%%% gtart.cls %%%%%%%%%%%%%%%%%%
+%
+% Format file for articles written in LaTeX for publication in
+% Geometry & Topology and Algebraic & Geometric Topology.
+%
+% For instructions see gtartins.tex and .ps and .pdf in gt/info/macros
+%
+% Version 1.3
+%
+%% Check for fairly recent version of latex2e :
+%
+\NeedsTeXFormat{LaTeX2e}[1994/12/01]
+%
+\LoadClass[11pt]{article} % Basic style
+\usepackage{amsthm} % For GT theorem style (see below)
+%
+% Basic layout :
+%
+\newskip\stdskip % standard vertical space
+\stdskip=6.6pt plus3.3pt minus3.3pt
+%
+\setlength{\textheight}{7.5in}
+\setlength{\textwidth}{5.2in}
+\flushbottom
+\setlength{\parindent}{0pt}
+\setlength{\parskip}{\stdskip}
+\setlength{\medskipamount}{\stdskip}
+\setlength{\mathsurround}{0.8pt}
+\setlength{\labelsep}{0.75em}
+\newcommand{\stdspace}{\hskip 0.75em plus 0.15em \ignorespaces}
+\let\qua\stdspace % useful abbreviation
+%
+% Some style commands (\ppar is for principal paragraph breaks, \sh is
+% for subheadings and \rk for remarks etc -- see also theorem style
+% below ) :
+%
+\newcommand{\ppar}{\par\goodbreak\vskip 8pt plus 3pt minus 3pt}
+\newcommand{\sh}[1]{\penalty-800\ppar{\bf #1}\par\medskip\nobreak}
+\newcommand{\rk}[1]{\ppar{\bf #1}\stdspace}
+%
+%
+% Theorem style. There are two recommended styles :
+%
+% plain : for theorems, corollaries etc with heading bold
+% and left justified, optional note bracketed in roman type
+% and statement in slanted type.
+%
+% definition : (alias remark) for definitions, remarks etc with
+% heading bold and left justified, optional note unbracketed in
+% slanted type and statement in roman type.
+%
+%
+% Redefine the amsthm styles plain, definition and remark to GT style:
+%
+\newtheoremstyle{plain}{14pt plus6.3pt minus6.3pt}{7.4pt plus3pt minus3pt}%
+{\sl}{}{\bf}{}{0.75em}{\thmname{#1}\thmnumber{ #2}\thmnote{\rm\stdspace(#3)}}
+%
+\newtheoremstyle{definition}{14pt plus6.3pt minus6.3pt}{7.4pt plus3pt minus3pt}%
+{\rm}{}{\bf}{}{0.75em}{\thmname{#1}\thmnumber{ #2}\thmnote{\sl\stdspace#3}}
+%
+\newtheoremstyle{remark}{14pt plus6.3pt minus6.3pt}{7.4pt plus3pt minus3pt}%
+{\rm}{}{\bf}{}{0.75em}{\thmname{#1}\thmnumber{ #2}\thmnote{\sl\stdspace#3}}
+%
+% Default theorem style :
+\theoremstyle{plain}
+%
+% Adapt the amsthm proof environment to GT style :
+%
+\renewenvironment{proof}[1][\proofname]{\par
+ \normalfont
+ \topsep\stdskip \trivlist
+ \item[\hskip\labelsep\bf
+ #1]\ignorespaces
+}{%
+ \qed\endtrivlist\par
+}
+% Knuth's \square macro :
+%
+\def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt
+ \hbox{\vrule width.#2pt height#1pt \kern#1pt \vrule width.#2pt}
+ \hrule height.#2pt}}}}
+%
+\def\sq{\sqr55} % A small square for end-of-proofs.
+\def\qedsymbol{$\sqr55$} % (Define other size squares by varing the
+% % the two numbers.)
+%
+% Some useful abbreviations :
+%
+\newcommand{\co}{\colon\thinspace} % Colon with correct spacing for maps.
+\newcommand{\np}{\newpage} % Forced page break (new page).
+\newcommand{\nl}{\hfil\break} % New line.
+\newcommand{\cl}{\centerline} % Centerline
+\def\gt{{\mathsurround=0pt\it $\cal G\mskip-2mu$eometry \&\
+$\cal T\!\!$opology}} % The journal title in recommended style
+\def\gtm{{\mathsurround=0pt\it $\cal G\mskip-2mu$eometry \&\
+$\cal T\!\!$opology $\cal M\mskip-1mu$onographs}} % for monographs
+\def\agt{{\mathsurround=0pt\it$\cal A\mskip-.7mu$lgebraic \&\
+$\cal G\mskip-2mu$eometric $\cal T\!\!$opology}} % AGT
+%
+% Define the various ingredients of the title page and cope with
+% all reasonable alternative syntax including amsart and article
+% style :
+%
+\def\title{\let\\\par\@ifnextchar[\doubletitle\singletitle}
+ \def\doubletitle[#1]#2{\def\thetitle{#2}\def\theshorttitle{#1}}
+ \def\singletitle#1{\def\thetitle{#1}}
+\def\shorttitle#1{\def\theshorttitle{#1}}
+%
+\def\author{\@ifnextchar[\doubleauthor\singleauthor}
+\def\singleauthor#1{\edef\previousauthors{\theauthors}
+ \ifx\theauthors\relax\def\theauthors{#1}\else
+ \def\theauthors{\previousauthors\par#1}\fi}
+\def\doubleauthor[#1]#2{\singleauthor{#2}}
+\let\authors\author\let\secondauthor\author % aliases
+\def\shortauthors#1{\def\theshortauthors{#1}}
+%
+\def\address#1{{\let\newline\par\xdef\previousaddresses{\theaddress}}
+ \ifx\theaddress\relax\def\theaddress{#1}\else
+ \def\theaddress{\previousaddresses\par\vskip 2pt\par#1}\fi}
+\let\addresses\address % alias
+\def\secondaddress#1{{\let\newline\par\xdef\previousaddresses{\theaddress}}
+ \ifx\theaddress\relax\def\theaddress{#1}\else
+ \def\theaddress{\previousaddresses\par{\rm and}\par#1}\fi}
+%
+\def\email#1{\edef\previousemails{\theemail}
+ \ifx\theemail\relax\def\theemail{#1}\else
+ \def\theemail{\previousemails\hskip 0.75em\relax#1}\fi}
+\let\emails\email\let\emailaddress\email\let\emailaddr\email % aliases
+\def\secondemail#1{\edef\previousemails{\theemail}
+ \ifx\theemail\relax\def\theemail{#1}\else
+ \def\theemail{\previousemails\hskip 0.75em{\rm and}\hskip 0.75em
+ \relax#1}\fi}
+%
+\def\url#1{\edef\previousurls{\theurl}
+ \ifx\theurl\relax\def\theurl{#1}\else
+ \def\theurl{\previousurls\hskip 0.75em\relax#1}\fi}
+\let\urls\url\let\urladdress\url\let\urladdr\url % aliases
+\def\secondurl#1{\edef\previousurls{\theurl}
+ \ifx\theurl\relax\def\theurl{#1}\else
+ \def\theurl{\previousurls\hskip 0.75em{\rm and}\hskip 0.75em
+ \relax#1}\fi}
+%
+\long\def\abstract#1\end#2#3\end#4%
+{\expandafter\ifx\csname#2\endcsname\abstract
+\long\gdef\theabstract{#1}\end{abstract}#3\end{#4}\else
+\long\gdef\theabstract{#1\end{#2}#3}\end{abstract}\fi}
+\def\endabstract{\relax}
+%
+\def\primaryclass#1{\def\theprimaryclass{#1}}
+\let\subjclass\primaryclass % alias
+\def\secondaryclass#1{\def\thesecondaryclass{#1}}
+\def\keywords#1{\def\thekeywords{#1}}
+%
+% Set \\ to \par and title page items to \relax to initialise macros :
+%
+\let\\\par\let\thetitle\relax\let\theauthors\relax
+\let\theaddress\relax\let\theemail\relax\let\theurl\relax
+\let\theabstract\relax\let\theprimaryclass\relax
+\let\thesecondaryclass\relax\let\thekeywords\relax
+\let\theshorttitle\relax\let\theshortauthors\relax
+%
+%%%% publication info and test defaults for authors:
+
+\def\volumenumber#1{\def\thevolumenumber{#1}}
+\def\volumename#1{\def\thevolumename{#1}}
+\def\volumeyear#1{\def\thevolumeyear{#1}}
+\def\pagenumbers#1#2{\def\startpage{#1}\def\finishpage{#2}}
+\def\published#1{\def\publishdate{#1}}
+
+\volumenumber{X}
+\volumename{Volume name goes here}
+\volumeyear{20XX}
+\pagenumbers{1}{XXX}
+\published{XX Xxxember 20XX}
+%
+%
+% Basic title page layout (edit this macro if you
+% wish to adjust the title page layout) :
+%
+\long\def\maketitlepage{ % start of definition of \maketitlepage
+%
+\vglue 0.2truein % top margin
+%
+% title :
+{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par\smallskip}{\Large
+\bf\thetitle}\par\medskip}
+%
+\vglue 0.15truein % space below title
+%
+% authors :
+{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par}{\sc\theauthors}
+\par\medskip}
+%
+\vglue 0.1truein % space below author(s)
+%
+% address(es) email's and URL's (with switches to detect whether the
+% optional items have been used) :
+{\parskip=0pt\small\let\newline\\
+{\leftskip 0pt plus 1fil\def\\{\par}{\sl\theaddress}\par}
+\ifx\theemail\relax\else % email address?
+\vglue 5pt \def\\{\ \ {\rm and}\ \ }
+\cl{Email:\ \ \tt\theemail}\fi
+\ifx\theurl\relax\else % URL given?
+\vglue 5pt \def\\{\ \ {\rm and}\ \ }
+\cl{URL:\ \ \tt\theurl}\fi\par}
+%
+\vglue 7pt % space below addresses
+%
+% Abstract:
+{\bf Abstract}\vglue 5pt\theabstract
+%
+\vglue 9pt % space below abstract
+%
+% AMS numbers and keywords:
+{\bf AMS Classification numbers}\quad Primary:\quad \theprimaryclass\par
+Secondary:\quad \thesecondaryclass\vglue 5pt
+{\bf Keywords:}\quad \thekeywords
+%
+\np % page break at the end of the title page
+} % end of definition of \maketitlepage
+%
+%
+%
+\long\def\makeshorttitle{ % start of definition of \makeshorttitle
+%
+% title :
+%
+{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par\smallskip}{\Large
+\bf\thetitle}\par\medskip}
+
+\vglue 0.05truein
+
+% authors :
+%
+{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par}{\sc\theauthors}
+\par\medskip}%
+
+\vglue 0.03truein
+
+% address(es) email's and URL's (with switches to detect whether the
+% optional items have been used) :
+%
+{\small\parskip=0pt
+{\leftskip 0pt plus 1fil\def\\{\par}{\sl\theaddress}\par}
+\ifx\theemail\relax\else % email address?
+\vglue 5pt \def\\{\stdspace{\rm and}\stdspace}
+\cl{Email:\stdspace\tt\theemail}\fi
+\ifx\theurl\relax\else % URL given?
+\vglue 5pt \def\\{\stdspace{\rm and}\stdspace}
+\cl{URL:\stdspace\tt\theurl}\fi\par}
+
+\vglue 10pt
+
+{\small\leftskip 25pt\rightskip 25pt{\bf Abstract}\stdspace\theabstract
+
+{\bf AMS Classification}\stdspace\theprimaryclass
+\ifx\thesecondaryclass\relax\else; \thesecondaryclass\fi\par
+{\bf Keywords}\stdspace \thekeywords\par}
+\vglue 7pt
+} % end of definition of \makeshorttitle
+%
+\let\maketitle\makeshorttitle %% alias
+%
+\long\def\makegtmontitle{ % start of definition of \makegtmontitle
+
+\count0=\startpage
+
+\gtm\nl % GT mongraphs (top left)
+{\small Volume \thevolumenumber: \thevolumename\nl
+Pages \startpage--\finishpage\nl}
+
+\vglue 0.1truein % top margin
+
+% title
+{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par\smallskip}{\Large
+\bf\thetitle}\par\medskip}
+\vglue 0.05truein
+
+% authors :
+%
+{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par}{\sc\theauthors}
+\par\medskip}%
+
+\vglue 0.03truein
+
+% abstract and classification numbers:
+
+{\small\leftskip 25pt\rightskip 25pt{\bf Abstract}\stdspace\theabstract
+
+{\bf AMS Classification}\stdspace\theprimaryclass
+\ifx\thesecondaryclass\relax\else; \thesecondaryclass\fi\par
+{\bf Keywords}\stdspace \thekeywords\par}\vglue 7pt
+
+} % end of definition of \makegtmontitle
+%
+\long\def\makeagttitle{ %%% start of definition of \makeagttitle
+\agt\hfill % Journal title (top left)
+% logo placeholder (top right)
+\hbox to 60pt{\vbox to 0pt{\vglue -14pt{\normalsize \bf [Logo here]}\vss}\hss}
+%
+\break
+{\small Volume \thevolumenumber\ (\thevolumeyear)
+\startpage--\finishpage\nl
+Published: \publishdate}
+
+\vglue .25truein
+
+% title
+{\parskip=0pt\leftskip 0pt plus
+1fil\def\\{\par\smallskip}{\Large\bf\thetitle}\par\medskip} \vglue
+0.05truein
+
+% authors :
+%
+{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par}{\sc\theauthors}
+\par\medskip}%
+
+\vglue 0.03truein
+
+% abstract and classification numbers:
+
+{\small\leftskip 25pt\rightskip 25pt{\bf Abstract}\stdspace\theabstract
+
+{\bf AMS Classification}\stdspace\theprimaryclass
+\ifx\thesecondaryclass\relax\else; \thesecondaryclass\fi\par
+{\bf Keywords}\stdspace \thekeywords\par}\vglue 7pt
+
+} %%%% end of definition of \makeagttitle
+%
+%%%% for addresses at the end of the paper:
+\def\Addresses{\bigskip
+{\small \parskip 0pt \leftskip 0pt \rightskip 0pt plus 1fil \def\\{\par}
+\sl\theaddress\par\medskip \rm Email:\stdspace\tt\theemail\par
+\ifx\theurl\relax\else\smallskip \rm URL:\stdspace\tt\theurl\par\fi}}
+
+\def\agtart{% Full mock-up of AGT article style (for authors to test with)
+% get print centerpage:
+\headsep 23pt
+\footskip 35pt
+\hoffset -4truemm
+\voffset 12.5truemm
+% fonts for headline and footline
+\font\lhead=cmsl9 scaled 1050
+\font\lnum=cmbx10
+\font\lfoot=cmsl9 scaled 1050
+% headline and footline
+\def\@oddhead{{\small\lhead\ifnum\count0=\startpage ISSN numbers
+are printed here\hfill {\lnum\number\count0}\else\ifodd\count0
+\def\\{ }\ifx\theshorttitle\relax \thetitle \else\theshorttitle\fi\hfill
+{\lnum\number\count0}\else\def\\{ and }{\lnum\number\count0}
+\hfill\ifx\theshortauthors\relax
+\theauthors\else\theshortauthors\fi\fi\fi}}\def\@evenhead{\@oddhead}
+\def\@oddfoot{\small\lfoot\ifnum\count0=\startpage Copyright
+declaration is printed here\hfill\else
+\agt, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}
+\def\@evenfoot{\@oddfoot}
+% force \makeagttitle
+\let\maketitlepage\makeagttitle\let\maketitle\makeagttitle
+\let\makeshorttitle\makeagttitle}
+%
+\def\gtmonart{% Full mock-up of GT monograph style (for authors to test with)
+% get print centerpage:
+\headsep 23pt
+\footskip 35pt
+\hoffset -4truemm
+\voffset 12.5truemm
+% fonts for headline and footline
+\font\lhead=cmsl9 scaled 1050
+\font\lnum=cmbx10
+\font\lfoot=cmsl9 scaled 1050
+% headline and footline
+\def\@oddhead{{\small\lhead\ifnum\count0=\startpage ISSN numbers
+are printed here\hfill {\lnum\number\count0}\else\ifodd\count0
+\def\\{ }\ifx\theshorttitle\relax \thetitle \else\theshorttitle\fi\hfill
+{\lnum\number\count0}\else\def\\{ and }{\lnum\number\count0}
+\hfill\ifx\theshortauthors\relax
+\theauthors\else\theshortauthors\fi\fi\fi}}\def\@evenhead{\@oddhead}
+\def\@oddfoot{\small\lfoot\ifnum\count0=\startpage Copyright
+declaration is printed here\hfill\else
+\gtm, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}
+\def\@evenfoot{\@oddfoot}
+% force \makegtmontitle
+\let\maketitle\makegtmontitle\let\makeshorttitle\makegtmontitle
+\let\maketitlepage\makegtmontitle}
+%
+\def\gtart{% Full mock-up of GT article style (for authors to test with)
+% get print centerpage:
+\headsep 23pt
+\footskip 35pt
+\hoffset -4truemm
+\voffset 12.5truemm
+% fonts for headline and footline
+\font\lhead=cmsl9 scaled 1050
+\font\lnum=cmbx10
+\font\lfoot=cmsl9 scaled 1050
+% headline and footline
+\def\@oddhead{{\small\lhead\ifnum\count0=\startpage ISSN numbers
+are printed here\hfill {\lnum\number\count0}\else\ifodd\count0
+\def\\{ }\ifx\theshorttitle\relax \thetitle \else\theshorttitle\fi\hfill
+{\lnum\number\count0}\else\def\\{ and }{\lnum\number\count0}
+\hfill\ifx\theshortauthors\relax
+\theauthors\else\theshortauthors\fi\fi\fi}}\def\@evenhead{\@oddhead}
+\def\@oddfoot{\small\lfoot\ifnum\count0=\startpage Copyright
+declaration is printed here\hfill\else
+\gt, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}
+\def\@evenfoot{\@oddfoot}
+% force \maketitlepage
+\let\maketitle\maketitlepage\let\makeshorttitle\maketitlepage}
+%
+% A few definitions to adapt (or disable) various items from amsart
+% style (not already covered above) :
+%
+\def\@message#1{\immediate\write16{#1}}
+\def\thanks#1{\@message{ }
+\@message{Thanks should not appear on the title page.}
+\@message{Please give thanks as acknowledgements at the end of your
+introduction.}\@message{ }\relax}
+\def\dedicatory#1{\@message{ }
+\@message{Dedications should not appear on the title page.}
+\@message{Please give these with your acknowledgements at the end of your
+introduction.}\@message{ }\relax}
+\def\bysame{\leavevmode\hbox to3em{\hrulefill}\thinspace}
+%
+% End of macros for basic title page layout
+%
+%
+% Some hacks to get various items of style correct :
+%
+% Set footnotes in 10pt type:
+%
+\let\@footnote@\footnote
+\def\footnote#1{\@footnote@{\small #1}}
+\let\fnote\footnote % useful abbreviation for \footnote
+%
+% Set captions in 10pt type (hack of excerpt from latex.ltx) :
+%
+\long\def\@caption#1[#2]#3{\par\addcontentsline{\csname
+ ext@#1\endcsname}{#1}{\protect\numberline{\csname
+ the#1\endcsname}{\ignorespaces #2}}\begingroup
+ \@parboxrestore
+ \small
+ \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par
+ \endgroup}
+%
+% Command to suppress the colon in captions (hack from article.cls) :
+%
+\def\nocolon{%
+\long\def\@makecaption##1##2{%
+ \vskip\abovecaptionskip
+ \sbox\@tempboxa{##1##2}%
+ \ifdim \wd\@tempboxa >\hsize
+ ##1##2\par
+ \else
+ \global \@minipagefalse
+ \hb@xt@\hsize{\hfil\box\@tempboxa\hfil}%
+ \fi
+ \vskip\belowcaptionskip}}
+%
+%
+% Set displayskips to correct values :
+%
+\let\@document@\document
+\def\document{\@document@%
+\setlength{\abovedisplayskip}{\stdskip}
+\setlength{\belowdisplayskip}{\stdskip}}
+%
+%
+% Get the biblio style correct (10pt with small gaps):
+%
+\let\@thebibliography@\thebibliography
+\def\thebibliography#1 {\@thebibliography@{999}\small\parskip0pt %
+plus2pt\relax}
+%
+%
+% Get item spacing reasonable :
+%
+\let\@itemize@\itemize
+\def\itemize{\@itemize@\parskip 0pt\relax}
+\def\@listi{\leftmargin28.5pt\parsep 0pt\topsep 0pt
+ \itemsep4pt plus3pt minus2pt}
+\let\@listI\@listi
+\@listi
+%
+\def\items{\bgroup\itemize} % for comptibility
+\def\enditems{\enditemize\egroup} % with gtmacros
+\let\itemb\item % (plain tex format)
+%
+% Get enumeration labels like plain or amstex :
+%
+\renewcommand{\labelenumi}{{\rm (\theenumi)}}
+%
+% and spacing to match \itemize:
+%
+\let\@enumerate@\enumerate
+\def\enumerate{\@enumerate@\parskip 0pt\relax}
+%
+\endinput
+%
+% History:
+% Version 1.1: 14 December 97
+% Version 1.2: (update for AGT) 18 October 00
+% Version 1.3: \gtart, \makegtmontitle and \gtmonart added 5.01.01
\ No newline at end of file
--- a/preamble.tex Thu Apr 26 06:57:24 2012 -0600
+++ b/preamble.tex Fri Apr 27 22:37:14 2012 -0700
@@ -169,7 +169,7 @@
\newcommand{\CD}[1]{C_*(\Diff(#1))}
\newcommand{\CH}[1]{C_*(\Homeo(#1))}
-\newcommand{\cl}[1]{\underrightarrow{#1}}
+\newcommand{\colimit}[1]{\underrightarrow{#1}}
\newcommand{\Set}{\text{\textbf{Set}}}
\newcommand{\Vect}{\text{\textbf{Vect}}}
--- a/text/a_inf_blob.tex Thu Apr 26 06:57:24 2012 -0600
+++ b/text/a_inf_blob.tex Fri Apr 27 22:37:14 2012 -0700
@@ -7,14 +7,14 @@
complex.
\begin{defn}
The blob complex $\bc_*(M;\cC)$ of an $n$-manifold $M$ with coefficients in
-an $A_\infty$ $n$-category $\cC$ is the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}.
+an $A_\infty$ $n$-category $\cC$ is the homotopy colimit $\colimit{\cC}(M)$ of \S\ref{ss:ncat_fields}.
\end{defn}
We will show below
in Corollary \ref{cor:new-old}
that when $\cC$ is obtained from a system of fields $\cE$
as the blob complex of an $n$-ball (see Example \ref{ex:blob-complexes-of-balls}),
-$\cl{\cC}(M)$ is homotopy equivalent to
+$\colimit{\cC}(M)$ is homotopy equivalent to
our original definition of the blob complex $\bc_*(M;\cE)$.
%\medskip
@@ -47,7 +47,7 @@
Then there is a homotopy equivalence between ``old-fashioned" (blob diagrams)
and ``new-fangled" (hocolimit) blob complexes
\[
- \cB_*(Y \times F) \htpy \cl{\cC_F}(Y) .
+ \cB_*(Y \times F) \htpy \colimit{\cC_F}(Y) .
\]\end{thm}
\begin{proof}
@@ -55,7 +55,7 @@
First we define a map
\[
- \psi: \cl{\cC_F}(Y) \to \bc_*(Y\times F;\cE) .
+ \psi: \colimit{\cC_F}(Y) \to \bc_*(Y\times F;\cE) .
\]
On 0-simplices of the hocolimit
we just glue together the various blob diagrams on $X_i\times F$
@@ -67,7 +67,7 @@
In the other direction, we will define (in the next few paragraphs)
a subcomplex $G_*\sub \bc_*(Y\times F;\cE)$ and a map
\[
- \phi: G_* \to \cl{\cC_F}(Y) .
+ \phi: G_* \to \colimit{\cC_F}(Y) .
\]
Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding
@@ -81,9 +81,9 @@
projections to $Y$ are contained in some disjoint union of balls.)
Note that the image of $\psi$ is equal to $G_*$.
-We will define $\phi: G_* \to \cl{\cC_F}(Y)$ using the method of acyclic models.
+We will define $\phi: G_* \to \colimit{\cC_F}(Y)$ using the method of acyclic models.
Let $a$ be a generator of $G_*$.
-Let $D(a)$ denote the subcomplex of $\cl{\cC_F}(Y)$ generated by all $(b, \ol{K})$
+Let $D(a)$ denote the subcomplex of $\colimit{\cC_F}(Y)$ generated by all $(b, \ol{K})$
where $b$ is a generator appearing
in an iterated boundary of $a$ (this includes $a$ itself)
and $b$ splits along $K_0\times F$.
@@ -198,7 +198,7 @@
\end{proof}
We are now in a position to apply the method of acyclic models to get a map
-$\phi:G_* \to \cl{\cC_F}(Y)$.
+$\phi:G_* \to \colimit{\cC_F}(Y)$.
We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is a 0-simplex
and $r$ is a sum of simplices of dimension 1 or higher.
@@ -213,7 +213,7 @@
We have $\psi(r) = 0$ since $\psi$ is zero on $(\ge 1)$-simplices.
Second, $\phi\circ\psi$ is the identity up to homotopy by another argument based on the method of acyclic models.
-To each generator $(b, \ol{K})$ of $\cl{\cC_F}(Y)$ we associate the acyclic subcomplex $D(b)$ defined above.
+To each generator $(b, \ol{K})$ of $\colimit{\cC_F}(Y)$ we associate the acyclic subcomplex $D(b)$ defined above.
Both the identity map and $\phi\circ\psi$ are compatible with this
collection of acyclic subcomplexes, so by the usual method of acyclic models argument these two maps
are homotopic.
@@ -227,7 +227,7 @@
a $j$-ball $X$ is either $\cE(X\times Y\times F)$ (if $j<m$) or $\bc_*(X\times Y\times F)$
(if $j=m$).
(See Example \ref{ex:blob-complexes-of-balls}.)
-Similarly we have an $m$-category whose value at $X$ is $\cl{\cC_F}(X\times Y)$.
+Similarly we have an $m$-category whose value at $X$ is $\colimit{\cC_F}(X\times Y)$.
These two categories are equivalent, but since we do not define functors between
disk-like $n$-categories in this paper we are unable to say precisely
what ``equivalent" means in this context.
@@ -244,7 +244,7 @@
Then for all $n$-manifolds $Y$ the old-fashioned and new-fangled blob complexes are
homotopy equivalent:
\[
- \bc^\cE_*(Y) \htpy \cl{\cC_\cE}(Y) .
+ \bc^\cE_*(Y) \htpy \colimit{\cC_\cE}(Y) .
\]
\end{cor}
@@ -272,13 +272,13 @@
(Here $p^*(E)$ denotes the pull-back bundle over $D$.)
Let $\cF_E$ denote this $k$-category over $Y$.
We can adapt the homotopy colimit construction (based on decompositions of $Y$ into balls) to
-get a chain complex $\cl{\cF_E}(Y)$.
+get a chain complex $\colimit{\cF_E}(Y)$.
\begin{thm}
Let $F \to E \to Y$ be a fiber bundle and let $\cF_E$ be the $k$-category over $Y$ defined above.
Then
\[
- \bc_*(E) \simeq \cl{\cF_E}(Y) .
+ \bc_*(E) \simeq \colimit{\cF_E}(Y) .
\]
\qed
\end{thm}
@@ -289,16 +289,16 @@
As before, we define a map
\[
- \psi: \cl{\cF_E}(Y) \to \bc_*(E) .
+ \psi: \colimit{\cF_E}(Y) \to \bc_*(E) .
\]
-The 0-simplices of the homotopy colimit $\cl{\cF_E}(Y)$ are glued up to give an element of $\bc_*(E)$.
+The 0-simplices of the homotopy colimit $\colimit{\cF_E}(Y)$ are glued up to give an element of $\bc_*(E)$.
Simplices of positive degree are sent to zero.
Let $G_* \sub \bc_*(E)$ be the image of $\psi$.
By Lemma \ref{thm:small-blobs}, $\bc_*(Y\times F; \cE)$
is homotopic to a subcomplex of $G_*$.
We will define a homotopy inverse of $\psi$ on $G_*$, using acyclic models.
-To each generator $a$ of $G_*$ we assign an acyclic subcomplex $D(a) \sub \cl{\cF_E}(Y)$ which consists of
+To each generator $a$ of $G_*$ we assign an acyclic subcomplex $D(a) \sub \colimit{\cF_E}(Y)$ which consists of
0-simplices which map via $\psi$ to $a$, plus higher simplices (as described in the proof of Theorem \ref{thm:product})
which insure that $D(a)$ is acyclic.
\end{proof}
@@ -312,14 +312,14 @@
lying above $D$.)
We can define a $k$-category $\cF_M$ based on maps of balls into $Y$ which are good with respect to $M$.
We can again adapt the homotopy colimit construction to
-get a chain complex $\cl{\cF_M}(Y)$.
+get a chain complex $\colimit{\cF_M}(Y)$.
The proof of Theorem \ref{thm:product} again goes through essentially unchanged
to show that
%\begin{thm}
%Let $M \to Y$ be a map of manifolds and let $\cF_M$ be the $k$-category over $Y$ defined above.
%Then
\[
- \bc_*(M) \simeq \cl{\cF_M}(Y) .
+ \bc_*(M) \simeq \colimit{\cF_M}(Y) .
\]
%\qed
%\end{thm}
--- a/text/intro.tex Thu Apr 26 06:57:24 2012 -0600
+++ b/text/intro.tex Fri Apr 27 22:37:14 2012 -0700
@@ -191,7 +191,7 @@
It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together
with a link $L \subset \bd W$.
The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$.
-%\todo{I'm tempted to replace $A_{Kh}$ with $\cl{Kh}$ throughout this page -S}
+%\todo{I'm tempted to replace $A_{Kh}$ with $\colimit{Kh}$ throughout this page -S}
How would we go about computing $A_{Kh}(W^4, L)$?
For the Khovanov homology of a link in $S^3$ the main tool is the exact triangle (long exact sequence)
@@ -415,7 +415,7 @@
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 definition is in fact simpler, almost tautological, and we use a different notation, $\colimit{\cC}(M)$.
The next theorem describes the blob complex for product manifolds
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.
@@ -429,7 +429,7 @@
(see Example \ref{ex:blob-complexes-of-balls}).
Then
\[
- \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W).
+ \bc_*(Y\times W; \cC) \simeq \colimit{\bc_*(Y;\cC)}(W).
\]
\end{thm:product}
The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps
--- a/text/ncat.tex Thu Apr 26 06:57:24 2012 -0600
+++ b/text/ncat.tex Fri Apr 27 22:37:14 2012 -0700
@@ -131,7 +131,7 @@
\begin{lem}
\label{lem:spheres}
-For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from
+For each $1 \le k \le n$, we have a functor $\colimit{\cC}_{k-1}$ from
the category of $k{-}1$-spheres and
homeomorphisms to the category of sets and bijections.
\end{lem}
@@ -146,13 +146,13 @@
\begin{axiom}[Boundaries]\label{nca-boundary}
-For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$.
+For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \colimit{\cC}_{k-1}(\bd X)$.
These maps, for various $X$, comprise a natural transformation of functors.
\end{axiom}
Note that the first ``$\bd$" above is part of the data for the category,
while the second is the ordinary boundary of manifolds.
-Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$.
+Given $c\in\colimit{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$.
\medskip
@@ -176,21 +176,21 @@
domain and range, but the converse meets with our approval.
That is, given compatible domain and range, we should be able to combine them into
the full boundary of a morphism.
-The following lemma will follow from the colimit construction used to define $\cl{\cC}_{k-1}$
+The following lemma will follow from the colimit construction used to define $\colimit{\cC}_{k-1}$
on spheres.
\begin{lem}[Boundary from domain and range]
\label{lem:domain-and-range}
Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$,
$B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}).
-Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the
-two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$.
+Let $\cC(B_1) \times_{\colimit{\cC}(E)} \cC(B_2)$ denote the fibered product of the
+two maps $\bd: \cC(B_i)\to \colimit{\cC}(E)$.
Then we have an injective map
\[
- \gl_E : \cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2) \into \cl{\cC}(S)
+ \gl_E : \cC(B_1) \times_{\colimit{\cC}(E)} \cC(B_2) \into \colimit{\cC}(S)
\]
which is natural with respect to the actions of homeomorphisms.
-(When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product
+(When $k=1$ we stipulate that $\colimit{\cC}(E)$ is a point, so that the above fibered product
becomes a normal product.)
\end{lem}
@@ -217,20 +217,20 @@
with respect to $E$. On the other hand, in categories based on maps to a target space (c.f. Example \ref{ex:maps-to-a-space} below) the gluing map is always surjective.
If $S$ is a 0-sphere (the case $k=1$ above), then $S$ can be identified with the {\it disjoint} union
-of two 0-balls $B_1$ and $B_2$ and the colimit construction $\cl{\cC}(S)$ can be identified
+of two 0-balls $B_1$ and $B_2$ and the colimit construction $\colimit{\cC}(S)$ can be identified
with the (ordinary, not fibered) product $\cC(B_1) \times \cC(B_2)$.
-Let $\cl{\cC}(S)\trans E$ denote the image of $\gl_E$.
-We will refer to elements of $\cl{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$".
+Let $\colimit{\cC}(S)\trans E$ denote the image of $\gl_E$.
+We will refer to elements of $\colimit{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$".
When the gluing map is surjective every such element is splittable.
If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$
-as above, then we define $\cC(X)\trans E = \bd^{-1}(\cl{\cC}(\bd X)\trans E)$.
-
-We will call the projection $\cl{\cC}(S)\trans E \to \cC(B_i)$ given by the composition
-$$\cl{\cC}(S)\trans E \xrightarrow{\gl^{-1}} \cC(B_1) \times \cC(B_2) \xrightarrow{\pr_i} \cC(B_i)$$
+as above, then we define $\cC(X)\trans E = \bd^{-1}(\colimit{\cC}(\bd X)\trans E)$.
+
+We will call the projection $\colimit{\cC}(S)\trans E \to \cC(B_i)$ given by the composition
+$$\colimit{\cC}(S)\trans E \xrightarrow{\gl^{-1}} \cC(B_1) \times \cC(B_2) \xrightarrow{\pr_i} \cC(B_i)$$
a {\it restriction} map and write $\res_{B_i}(a)$
-(or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)\trans E$.
+(or simply $\res(a)$ when there is no ambiguity), for $a\in \colimit{\cC}(S)\trans E$.
More generally, we also include under the rubric ``restriction map"
the boundary maps of Axiom \ref{nca-boundary} above,
another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition
@@ -238,7 +238,7 @@
In particular, we have restriction maps $\cC(X)\trans E \to \cC(B_i)$
defined as the composition of the boundary with the first restriction map described above:
$$
-\cC(X) \trans E \xrightarrow{\bdy} \cl{\cC}(\bdy X)\trans E \xrightarrow{\res} \cC(B_i)
+\cC(X) \trans E \xrightarrow{\bdy} \colimit{\cC}(\bdy X)\trans E \xrightarrow{\res} \cC(B_i)
.$$
These restriction maps can be thought of as
domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$.
@@ -317,7 +317,7 @@
In situations where the splitting is notationally anonymous, we will write
$\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to)
the unnamed splitting.
-If $\beta$ is a ball decomposition of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$;
+If $\beta$ is a ball decomposition of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\colimit{\cC}(\bd X)_\beta)$;
this can also be denoted $\cC(X)\spl$ if the context contains an anonymous
decomposition of $\bd X$ and no competing splitting of $X$.
@@ -995,7 +995,7 @@
Most of the examples of $n$-categories we are interested in are enriched in the following sense.
The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and
-all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some appropriate auxiliary category
+all $c\in \colimit{\cC}(\bd X)$, have the structure of an object in some appropriate auxiliary category
(e.g.\ vector spaces, or modules over some ring, or chain complexes),
and all the structure maps of the $n$-category are compatible with the auxiliary
category structure.
@@ -1021,7 +1021,7 @@
we need a preliminary definition.
Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the
category $\bbc$ of {\it $n$-balls with boundary conditions}.
-Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \cl\cC(\bd X)$ is the ``boundary condition".
+Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \colimit{\cC}(\bd X)$ is the ``boundary condition".
The morphisms from $(X, c)$ to $(X', c')$, denoted $\Homeo(X; c \to X'; c')$, are
homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$.
%Let $\pi_0(\bbc)$ denote
@@ -1038,7 +1038,7 @@
\item Composition. Let $B = B_1\cup_Y B_2$ as in Axiom \ref{axiom:composition}.
Let $Y_i = \bd B_i \setmin Y$.
Note that $\bd B = Y_1\cup Y_2$.
-Let $c_i \in \cC(Y_i)$ with $\bd c_1 = \bd c_2 = d \in \cl\cC(E)$.
+Let $c_i \in \cC(Y_i)$ with $\bd c_1 = \bd c_2 = d \in \colimit{\cC}(E)$.
Then we have a map
\[
\gl_Y : \bigoplus_c \cC(B_1; c_1 \bullet c) \otimes \cC(B_2; c_2\bullet c) \to \cC(B; c_1\bullet c_2),
@@ -1073,7 +1073,7 @@
\begin{axiom}[\textup{\textbf{[$A_\infty$ replacement for Axiom \ref{axiom:extended-isotopies}]}} Families of homeomorphisms act in dimension $n$.]
\label{axiom:families}
-For each pair of $n$-balls $X$ and $X'$ and each pair $c\in \cl{\cC}(\bd X)$ and $c'\in \cl{\cC}(\bd X')$ we have an $\cS$-morphism
+For each pair of $n$-balls $X$ and $X'$ and each pair $c\in \colimit{\cC}(\bd X)$ and $c'\in \colimit{\cC}(\bd X')$ we have an $\cS$-morphism
\[
\cJ(\Homeo(X;c \to X'; c')) \ot \cC(X; c) \to \cC(X'; c') .
\]
@@ -1184,7 +1184,7 @@
An $n$-category consists of the following data:
\begin{itemize}
\item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms});
-\item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary});
+\item boundary natural transformations $\cC_k \to \colimit{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary});
\item ``composition'' or ``gluing'' maps $\gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B_1\cup_Y B_2)\trans E$ (Axiom \ref{axiom:composition});
\item ``product'' or ``identity'' maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product});
\item if enriching in an auxiliary category, additional structure on $\cC_n(X; c)$ (Axiom \ref{axiom:enriched});
@@ -1268,7 +1268,7 @@
Let $W$ be an $n{-}j$-manifold.
Define the $j$-category $\cF(W)$ as follows.
If $X$ is a $k$-ball with $k<j$, let $\cF(W)(X) \deq \cF(W\times X)$.
-If $X$ is a $j$-ball and $c\in \cl{\cF(W)}(\bd X)$,
+If $X$ is a $j$-ball and $c\in \colimit{\cF(W)}(\bd X)$,
let $\cF(W)(X; c) \deq A_\cF(W\times X; c)$.
\end{example}
@@ -1284,7 +1284,7 @@
Given a ``traditional $n$-category with strong duality" $C$
define $\cC(X)$, for $X$ a $k$-ball with $k < n$,
to be the set of all $C$-labeled embedded cell complexes of $X$ (c.f. \S \ref{sec:fields}).
-For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X; c)$ to be finite linear
+For $X$ an $n$-ball and $c\in \colimit{\cC}(\bd X)$, define $\cC(X; c)$ to be finite linear
combinations of $C$-labeled embedded cell complexes of $X$
modulo the kernel of the evaluation map.
Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$,
@@ -1459,13 +1459,13 @@
\subsection{From balls to manifolds}
\label{ss:ncat_fields} \label{ss:ncat-coend}
In this section we show how to extend an $n$-category $\cC$ as described above
-(of either the ordinary or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$.
+(of either the ordinary or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\colimit{\cC}$.
This extension is a certain colimit, and the arrow in the notation is intended as a reminder of this.
In the case of ordinary $n$-categories, this construction factors into a construction of a
system of fields and local relations, followed by the usual TQFT definition of a
vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}.
-For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead.
+For an $A_\infty$ $n$-category, $\colimit{\cC}$ is defined using a homotopy colimit instead.
Recall that we can take an ordinary $n$-category $\cC$ and pass to the ``free resolution",
an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls
(recall Example \ref{ex:blob-complexes-of-balls} above).
@@ -1474,14 +1474,14 @@
same as the original blob complex for $M$ with coefficients in $\cC$.
Recall that we've already anticipated this construction Subsection \ref{ss:n-cat-def},
-inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls,
+inductively defining $\colimit{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls,
so that we can state the boundary axiom for $\cC$ on $k+1$-balls.
\medskip
We will first define the {\it decomposition poset} $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$.
An $n$-category $\cC$ provides a functor from this poset to the category of sets,
-and we will define $\cl{\cC}(W)$ as a suitable colimit
+and we will define $\colimit{\cC}(W)$ as a suitable colimit
(or homotopy colimit in the $A_\infty$ case) of this functor.
We'll later give a more explicit description of this colimit.
In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain
@@ -1546,8 +1546,8 @@
Since we allow decompositions in which the intersection of $X_a$ and $X_b$ might be messy
(see Example \ref{sin1x-example}), we must define $\psi_{\cC;W}(x)$ in a more roundabout way.
-Inductively, we may assume that we have already defined the colimit $\cl\cC(M)$ for $k{-}1$-manifolds $M$.
-(To start the induction, we define $\cl\cC(M)$, where $M = \du_a P_a$ is a 0-manifold and each $P_a$ is
+Inductively, we may assume that we have already defined the colimit $\colimit\cC(M)$ for $k{-}1$-manifolds $M$.
+(To start the induction, we define $\colimit\cC(M)$, where $M = \du_a P_a$ is a 0-manifold and each $P_a$ is
a 0-ball, to be $\prod_a \cC(P_a)$.)
We also assume, inductively, that we have gluing and restriction maps for colimits of $k{-}1$-manifolds.
Gluing and restriction maps for colimits of $k$-manifolds will be defined later in this subsection.
@@ -1558,17 +1558,17 @@
related to the gluings $M_{i-1} \to M_i$, $1\le i \le m$.
By Axiom \ref{nca-boundary}, we have a map
\[
- \prod_a \cC(X_a) \to \cl\cC(\bd M_0) .
+ \prod_a \cC(X_a) \to \colimit\cC(\bd M_0) .
\]
-The first condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_0)$ is splittable
-along $\bd Y_0$ and $\bd Y'_0$, and that the restrictions to $\cl\cC(Y_0)$ and $\cl\cC(Y'_0)$ agree
+The first condition is that the image of $\psi_{\cC;W}(x)$ in $\colimit\cC(\bd M_0)$ is splittable
+along $\bd Y_0$ and $\bd Y'_0$, and that the restrictions to $\colimit\cC(Y_0)$ and $\colimit\cC(Y'_0)$ agree
(with respect to the identification of $Y_0$ and $Y'_0$ provided by the gluing map).
On the subset of $\prod_a \cC(X_a)$ which satisfies the first condition above, we have a restriction
-map to $\cl\cC(N_0)$ which we can compose with the gluing map
-$\cl\cC(N_0) \to \cl\cC(\bd M_1)$.
-The second condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_1)$ is splittable
-along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\cl\cC(Y_1)$ and $\cl\cC(Y'_1)$ agree
+map to $\colimit\cC(N_0)$ which we can compose with the gluing map
+$\colimit\cC(N_0) \to \colimit\cC(\bd M_1)$.
+The second condition is that the image of $\psi_{\cC;W}(x)$ in $\colimit\cC(\bd M_1)$ is splittable
+along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\colimit\cC(Y_1)$ and $\colimit\cC(Y'_1)$ agree
(with respect to the identification of $Y_1$ and $Y'_1$ provided by the gluing map).
The $i$-th condition is defined similarly.
Note that these conditions depend only on the boundaries of elements of $\prod_a \cC(X_a)$.
@@ -1599,48 +1599,48 @@
$\cS$ and the coproduct and product in the above expression should be replaced by the appropriate
operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect).
-Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$:
+Finally, we construct $\colimit{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$:
\begin{defn}[System of fields functor]
\label{def:colim-fields}
-If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$.
+If $\cC$ is an $n$-category enriched in sets or vector spaces, $\colimit{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$.
That is, for each decomposition $x$ there is a map
-$\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps
-above, and $\cl{\cC}(W)$ is universal with respect to these properties.
+$\psi_{\cC;W}(x)\to \colimit{\cC}(W)$, these maps are compatible with the refinement maps
+above, and $\colimit{\cC}(W)$ is universal with respect to these properties.
\end{defn}
\begin{defn}[System of fields functor, $A_\infty$ case]
-When $\cC$ is an $A_\infty$ $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$
+When $\cC$ is an $A_\infty$ $n$-category, $\colimit{\cC}(W)$ for $W$ a $k$-manifold with $k < n$
is defined as above, as the colimit of $\psi_{\cC;W}$.
-When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$.
+When $W$ is an $n$-manifold, the chain complex $\colimit{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$.
\end{defn}
-%We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$
+%We can specify boundary data $c \in \colimit{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$
%with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$.
\medskip
-We must now define restriction maps $\bd : \cl{\cC}(W) \to \cl{\cC}(\bd W)$ and gluing maps.
-
-Let $y\in \cl{\cC}(W)$.
+We must now define restriction maps $\bd : \colimit{\cC}(W) \to \colimit{\cC}(\bd W)$ and gluing maps.
+
+Let $y\in \colimit{\cC}(W)$.
Choose a representative of $y$ in the colimit: a permissible decomposition $\du_a X_a \to W$ and elements
$y_a \in \cC(X_a)$.
By assumption, $\du_a (X_a \cap \bd W) \to \bd W$ can be completed to a decomposition of $\bd W$.
-Let $r(y_a) \in \cl\cC(X_a \cap \bd W)$ be the restriction.
-Choose a representative of $r(y_a)$ in the colimit $\cl\cC(X_a \cap \bd W)$: a permissible decomposition
+Let $r(y_a) \in \colimit\cC(X_a \cap \bd W)$ be the restriction.
+Choose a representative of $r(y_a)$ in the colimit $\colimit\cC(X_a \cap \bd W)$: a permissible decomposition
$\du_b Q_{ab} \to X_a \cap \bd W$ and elements $z_{ab} \in \cC(Q_{ab})$.
Then $\du_{ab} Q_{ab} \to \bd W$ is a permissible decomposition of $\bd W$ and $\{z_{ab}\}$ represents
-an element of $\cl{\cC}(\bd W)$. Define $\bd y$ to be this element.
+an element of $\colimit{\cC}(\bd W)$. Define $\bd y$ to be this element.
It is not hard to see that it is independent of the various choices involved.
Note that since we have already (inductively) defined gluing maps for colimits of $k{-}1$-manifolds,
-we can also define restriction maps from $\cl{\cC}(W)\trans{}$ to $\cl{\cC}(Y)$ where $Y$ is a codimension 0
+we can also define restriction maps from $\colimit{\cC}(W)\trans{}$ to $\colimit{\cC}(Y)$ where $Y$ is a codimension 0
submanifold of $\bd W$.
Next we define gluing maps for colimits of $k$-manifolds.
Let $W = W_1 \cup_Y W_2$.
-Let $y_i \in \cl\cC(W_i)$ and assume that the restrictions of $y_1$ and $y_2$ to $\cl\cC(Y)$ agree.
-We want to define $y_1\bullet y_2 \in \cl\cC(W)$.
+Let $y_i \in \colimit\cC(W_i)$ and assume that the restrictions of $y_1$ and $y_2$ to $\colimit\cC(Y)$ agree.
+We want to define $y_1\bullet y_2 \in \colimit\cC(W)$.
Choose a permissible decomposition $\du_a X_{ia} \to W_i$ and elements
$y_{ia} \in \cC(X_{ia})$ representing $y_i$.
It might not be the case that $\du_{ia} X_{ia} \to W$ is a permissible decomposition of $W$,
@@ -1660,14 +1660,14 @@
In the non-enriched case (e.g.\ $k<n$), where each $\cC(X_a; \beta)$ is just a set,
the colimit is
\[
- \cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) \Bigg/ \sim ,
+ \colimit{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) \Bigg/ \sim ,
\]
where $x$ runs through decompositions of $W$, and $\sim$ is the obvious equivalence relation
induced by refinement and gluing.
If $\cC$ is enriched over, for example, vector spaces and $W$ is an $n$-manifold,
we can take
\begin{equation*}
- \cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K,
+ \colimit{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K,
\end{equation*}
where $K$ is the vector space spanned by elements $a - g(a)$, with
$a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x)
@@ -1684,16 +1684,16 @@
Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$.
Such sequences (for all $m$) form a simplicial set in $\cell(W)$.
-Define $\cl{\cC}(W)$ as a vector space via
+Define $\colimit{\cC}(W)$ as a vector space via
\[
- \cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] ,
+ \colimit{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] ,
\]
where the sum is over all $m$ and all $m$-sequences $(x_i)$, and each summand is degree shifted by $m$.
Elements of a summand indexed by an $m$-sequence will be call $m$-simplices.
-We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$
+We endow $\colimit{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$
summands plus another term using the differential of the simplicial set of $m$-sequences.
More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$
-summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define
+summand of $\colimit{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define
\[
\bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) ,
\]
@@ -1728,16 +1728,16 @@
\medskip
-$\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds.
+$\colimit{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds.
Restricting to $k$-spheres, we have now proved Lemma \ref{lem:spheres}.
\begin{lem}
\label{lem:colim-injective}
Let $W$ be a manifold of dimension $j<n$. Then for each
-decomposition $x$ of $W$ the natural map $\psi_{\cC;W}(x)\to \cl{\cC}(W)$ is injective.
+decomposition $x$ of $W$ the natural map $\psi_{\cC;W}(x)\to \colimit{\cC}(W)$ is injective.
\end{lem}
\begin{proof}
-$\cl{\cC}(W)$ is a colimit of a diagram of sets, and each of the arrows in the diagram is
+$\colimit{\cC}(W)$ is a colimit of a diagram of sets, and each of the arrows in the diagram is
injective.
Concretely, the colimit is the disjoint union of the sets (one for each decomposition of $W$),
modulo the relation which identifies the domain of each of the injective maps
@@ -1745,7 +1745,7 @@
To save ink and electrons we will simplify notation and write $\psi(x)$ for $\psi_{\cC;W}(x)$.
-Suppose $a, \hat{a}\in \psi(x)$ have the same image in $\cl{\cC}(W)$ but $a\ne \hat{a}$.
+Suppose $a, \hat{a}\in \psi(x)$ have the same image in $\colimit{\cC}(W)$ but $a\ne \hat{a}$.
Then there exist
\begin{itemize}
\item decompositions $x = x_0, x_1, \ldots , x_{k-1}, x_k = x$ and $v_1,\ldots, v_k$ of $W$;
@@ -1840,21 +1840,21 @@
\begin{lem}
\label{lem:hemispheres}
-{For each $1 \le k \le n$, we have a functor $\cl\cM_{k-1}$ from
+{For each $1 \le k \le n$, we have a functor $\colimit\cM_{k-1}$ from
the category of marked $k$-hemispheres and
homeomorphisms to the category of sets and bijections.}
\end{lem}
The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details.
We use the same type of colimit construction.
-In our example, $\cl\cM(H) = \cD(H\times\bd W \cup \bd H\times W)$.
+In our example, $\colimit\cM(H) = \cD(H\times\bd W \cup \bd H\times W)$.
\begin{module-axiom}[Module boundaries]
-{For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cl\cM(\bd M)$.
+{For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \colimit\cM(\bd M)$.
These maps, for various $M$, comprise a natural transformation of functors.}
\end{module-axiom}
-Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$.
+Given $c\in\colimit\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$.
If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces),
then for each marked $n$-ball $M=(B,N)$ and $c\in \cC(\bd B \setminus N)$, the set $\cM(M; c)$ should be an object in that category.
@@ -1864,10 +1864,10 @@
{Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$),
$M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere.
Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the
-two maps $\bd: \cM(M_i)\to \cl\cM(E)$.
+two maps $\bd: \cM(M_i)\to \colimit\cM(E)$.
Then we have an injective map
\[
- \gl_E : \cM(M_1) \times_{\cl\cM(E)} \cM(M_2) \hookrightarrow \cl\cM(H)
+ \gl_E : \cM(M_1) \times_{\colimit\cM(E)} \cM(M_2) \hookrightarrow \colimit\cM(H)
\]
which is natural with respect to the actions of homeomorphisms.}
\end{lem}
@@ -1896,22 +1896,22 @@
\label{fig:module-boundary}
\end{figure}
-Let $\cl\cM(H)\trans E$ denote the image of $\gl_E$.
-We will refer to elements of $\cl\cM(H)\trans E$ as ``splittable along $E$" or ``transverse to $E$".
+Let $\colimit\cM(H)\trans E$ denote the image of $\gl_E$.
+We will refer to elements of $\colimit\cM(H)\trans E$ as ``splittable along $E$" or ``transverse to $E$".
\noop{ %%%%%%%
\begin{lem}[Module to category restrictions]
{For each marked $k$-hemisphere $H$ there is a restriction map
-$\cl\cM(H)\to \cC(H)$.
+$\colimit\cM(H)\to \cC(H)$.
($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.)
These maps comprise a natural transformation of functors.}
\end{lem}
} %%%%%%% end \noop
-It follows from the definition of the colimit $\cl\cM(H)$ that
+It follows from the definition of the colimit $\colimit\cM(H)$ that
given any (unmarked) $k{-}1$-ball $Y$ in the interior of $H$ there is a restriction map
-from a subset $\cl\cM(H)_{\trans{\bdy Y}}$ of $\cl\cM(H)$ to $\cC(Y)$.
-Combining this with the boundary map $\cM(B,N) \to \cl\cM(\bd(B,N))$, we also have a restriction
+from a subset $\colimit\cM(H)_{\trans{\bdy Y}}$ of $\colimit\cM(H)$ to $\cC(Y)$.
+Combining this with the boundary map $\cM(B,N) \to \colimit\cM(\bd(B,N))$, we also have a restriction
map from a subset $\cM(B,N)_{\trans{\bdy Y}}$ of $\cM(B,N)$ to $\cC(Y)$ whenever $Y$ is in the interior of $\bd B \setmin N$.
This fact will be used below.
@@ -2195,7 +2195,7 @@
We define the
category $\mbc$ of {\it marked $n$-balls with boundary conditions} as follows.
-Its objects are pairs $(M, c)$, where $M$ is a marked $n$-ball and $c \in \cl\cM(\bd M)$ is the ``boundary condition".
+Its objects are pairs $(M, c)$, where $M$ is a marked $n$-ball and $c \in \colimit\cM(\bd M)$ is the ``boundary condition".
The morphisms from $(M, c)$ to $(M', c')$, denoted $\Homeo(M; c \to M'; c')$, are
homeomorphisms $f:M\to M'$ such that $f|_{\bd M}(c) = c'$.
@@ -2210,7 +2210,7 @@
Retain notation from \ref{axiom:families}.
\begin{module-axiom}[Families of homeomorphisms act in dimension $n$.]
-For each pair of marked $n$-balls $M$ and $M'$ and each pair $c\in \cl{\cM}(\bd M)$ and $c'\in \cl{\cM}(\bd M')$
+For each pair of marked $n$-balls $M$ and $M'$ and each pair $c\in \colimit{\cM}(\bd M)$ and $c'\in \colimit{\cM}(\bd M')$
we have an $\cS$-morphism
\[
\cJ(\Homeo(M;c \to M'; c')) \ot \cM(M; c) \to \cM(M'; c') .
@@ -2257,7 +2257,7 @@
Define a $\cF(W)$ module $\cF(Y)$ as follows.
If $M = (B, N)$ is a marked $k$-ball with $k<j$ let
$\cF(Y)(M)\deq \cF((B\times W) \cup (N\times Y))$.
-If $M = (B, N)$ is a marked $j$-ball and $c\in \cl{\cF(Y)}(\bd M)$ let
+If $M = (B, N)$ is a marked $j$-ball and $c\in \colimit{\cF(Y)}(\bd M)$ let
$\cF(Y)(M)\deq A_\cF((B\times W) \cup (N\times Y); c)$.
\end{example}
@@ -2394,24 +2394,24 @@
So we treat this case in more detail.
First we explain the remark about derived hom above.
-Let $L$ be a marked 1-ball and let $\cl{\cX}(L)$ denote the local homotopy colimit construction
+Let $L$ be a marked 1-ball and let $\colimit{\cX}(L)$ denote the local homotopy colimit construction
associated to $L$ by $\cX$ and $\cC$.
(See \S \ref{ss:ncat_fields} and \S \ref{moddecss}.)
-Define $\cl{\cY}(L)$ similarly.
-For $K$ an unmarked 1-ball let $\cl{\cC}(K)$ denote the local homotopy colimit
+Define $\colimit{\cY}(L)$ similarly.
+For $K$ an unmarked 1-ball let $\colimit{\cC}(K)$ denote the local homotopy colimit
construction associated to $K$ by $\cC$.
Then we have an injective gluing map
\[
- \gl: \cl{\cX}(L) \ot \cl{\cC}(K) \to \cl{\cX}(L\cup K)
+ \gl: \colimit{\cX}(L) \ot \colimit{\cC}(K) \to \colimit{\cX}(L\cup K)
\]
which is also a chain map.
(For simplicity we are suppressing mention of boundary conditions on the unmarked
boundary components of the 1-balls.)
We define $\hom_\cC(\cX \to \cY)$ to be a collection of (graded linear) natural transformations
-$g: \cl{\cX}(L)\to \cl{\cY}(L)$ such that the following diagram commutes for all $L$ and $K$:
+$g: \colimit{\cX}(L)\to \colimit{\cY}(L)$ such that the following diagram commutes for all $L$ and $K$:
\[ \xymatrix{
- \cl{\cX}(L) \ot \cl{\cC}(K) \ar[r]^{\gl} \ar[d]_{g\ot \id} & \cl{\cX}(L\cup K) \ar[d]^{g}\\
- \cl{\cY}(L) \ot \cl{\cC}(K) \ar[r]^{\gl} & \cl{\cY}(L\cup K)
+ \colimit{\cX}(L) \ot \colimit{\cC}(K) \ar[r]^{\gl} \ar[d]_{g\ot \id} & \colimit{\cX}(L\cup K) \ar[d]^{g}\\
+ \colimit{\cY}(L) \ot \colimit{\cC}(K) \ar[r]^{\gl} & \colimit{\cY}(L\cup K)
} \]
The usual differential on graded linear maps between chain complexes induces a differential
@@ -2428,8 +2428,8 @@
Because we are using the {\it local} homotopy colimit, any generator
$D\ot x\ot \bar{c}\ot z$ of $\cX \ot_\cC \cZ$ can be written (perhaps non-uniquely) as a gluing
$(D'\ot x \ot \bar{c}') \bullet (D''\ot \bar{c}''\ot z)$, for some decomposition $J = L'\cup L''$
-and with $D'\ot x \ot \bar{c}'$ a generator of $\cl{\cX}(L')$ and
-$D''\ot \bar{c}''\ot z$ a generator of $\cl{\cZ}(L'')$.
+and with $D'\ot x \ot \bar{c}'$ a generator of $\colimit{\cX}(L')$ and
+$D''\ot \bar{c}''\ot z$ a generator of $\colimit{\cZ}(L'')$.
(Such a splitting exists because the blob diagram $D$ can be split into left and right halves,
since no blob can include both the leftmost and rightmost intervals in the underlying decomposition.
This step would fail if we were using the usual hocolimit instead of the local hocolimit.)
--- a/text/top_matter.tex Thu Apr 26 06:57:24 2012 -0600
+++ b/text/top_matter.tex Fri Apr 27 22:37:14 2012 -0700
@@ -11,6 +11,6 @@
%\primaryclass{57M25} \secondaryclass{57M27; 57Q45}
-%\keywords{
-
-%}
+\keywords{
+topological quantum field theory, hochschild homology, Deligne conjecture
+}