--- a/ant-arxiv.bat Thu Mar 18 19:40:46 2010 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1 +0,0 @@
-ant arxiv
--- a/ant-arxiv.sh Thu Mar 18 19:40:46 2010 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1 +0,0 @@
-ant arxiv
--- a/ant-copy-pdf.bat Thu Mar 18 19:40:46 2010 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1 +0,0 @@
-ant scott-copy-pdf
--- a/ant-eps-diagrams.bat Thu Mar 18 19:40:46 2010 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1 +0,0 @@
-ant eps-diagrams
--- a/ant-pdf.bat Thu Mar 18 19:40:46 2010 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1 +0,0 @@
-ant pdf
--- a/ant-pdf.sh Thu Mar 18 19:40:46 2010 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1 +0,0 @@
-ant pdf
--- a/blob1.tex Thu Mar 18 19:40:46 2010 +0000
+++ b/blob1.tex Sat Mar 27 03:07:45 2010 +0000
@@ -21,13 +21,10 @@
\maketitle
-[version $>$ 214; $>$ 23 Feb 2010]
+[revision $>$ 222; $>$ 25 March 2010]
\textbf{Draft version, read with caution.}
-%\versioninfo
-%[19 October 2009, rev. $>$ 129]
-
\medskip
\noindent
--- a/diagrams/latex2pdf/defontify.bat Thu Mar 18 19:40:46 2010 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,4 +0,0 @@
-latex defontify
-dvips -R0 defontify
-ps2pdf defontify.ps
-gs-native -r9600 -sDEVICE=pswrite -dNOCACHE -sOutputFile=nofonts.ps -q -dbatch -dNOPAUSE defontify.pdf -c quit
--- a/diagrams/latex2pdf/defontify.tex Thu Mar 18 19:40:46 2010 +0000
+++ b/diagrams/latex2pdf/defontify.tex Sat Mar 27 03:07:45 2010 +0000
@@ -14,7 +14,8 @@
$n$-category composition
-$B B_1 B_2 u_1 u_2 u r r u_i, u_j u_k u_l$
+\newcommand{\cN}{\mathcal{N}}
+$\cN_1 \cN_2 \cN_3$
\begin{align*}
abmab & \\
Binary file diagrams/pdf/ncat/mblabel.pdf has changed
Binary file diagrams/pdf/ncat/zz0.pdf has changed
Binary file diagrams/pdf/ncat/zz1.pdf has changed
Binary file diagrams/pdf/ncat/zz2.pdf has changed
Binary file diagrams/pdf/ncat/zz3.pdf has changed
Binary file diagrams/pdf/tempkw/blah3.pdf has changed
Binary file diagrams/pdf/tempkw/mblabel.pdf has changed
Binary file diagrams/pdf/tempkw/zz0.pdf has changed
Binary file diagrams/pdf/tempkw/zz1.pdf has changed
Binary file diagrams/pdf/tempkw/zz1b.pdf has changed
Binary file diagrams/pdf/tempkw/zz2.pdf has changed
Binary file diagrams/pdf/tempkw/zz3.pdf has changed
--- a/gtart.bst Thu Mar 18 19:40:46 2010 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1340 +0,0 @@
-%%
-%% 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'.
--- a/gtart.cls Thu Mar 18 19:40:46 2010 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,502 +0,0 @@
-%%%%%%%%%%%%%%%%%% 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/talks/20091108-Riverside/riverside1.tex Thu Mar 18 19:40:46 2010 +0000
+++ b/talks/20091108-Riverside/riverside1.tex Sat Mar 27 03:07:45 2010 +0000
@@ -101,6 +101,14 @@
\begin{block}{Pasting diagrams}
Fix an $n$-category with strong duality $\cC$. A \emph{field} on $\cM$ is a pasting diagram drawn on $\cM$, with cells labelled by morphisms from $\cC$.
\end{block}
+\begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category]
+$$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF^{\text{TL}_d}\left(T^2\right)$$
+\end{example}
+\begin{block}{}
+Given a pasting diagram on a ball, we can evaluate it to a morphism. We call the kernel the \emph{null fields}.
+\vspace{-3mm}
+$$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$
+\end{block}
\end{frame}
\begin{frame}{Background: TQFT invariants}
@@ -139,7 +147,7 @@
\begin{block}{}
\vspace{-1mm}
-$$\bc_1(\cM; \cC) = \setcr{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$
+$$\bc_1(\cM; \cC) = \Complex\setcr{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$
\end{block}
\vspace{-3.5mm}
$$\mathfig{.5}{definition/single-blob}$$
@@ -160,7 +168,7 @@
\begin{block}{}
\vspace{-5mm}
\begin{align*}
-\bc_2^{\text{disjoint}} & = \setcl{\roundframe{\mathfig{0.5}{definition/disjoint-blobs}}}{\text{ev}_{B_i}(u_i) = 0}
+\bc_2^{\text{disjoint}} & = \Complex\setcl{\roundframe{\mathfig{0.5}{definition/disjoint-blobs}}}{\text{ev}_{B_i}(u_i) = 0}
\end{align*}
\vspace{-4mm}
$$d_2 : (B_1, B_2, u_1, u_2, r) \mapsto (B_2, u_2, r \circ u_1) - (B_1, u_1, r \circ u_2)$$
@@ -168,7 +176,7 @@
\begin{block}{}
\vspace{-5mm}
\begin{align*}
-\bc_2^{\text{nested}} & = \setcl{\roundframe{\mathfig{0.5}{definition/nested-blobs}}}{\text{ev}_{B_1}(u)=0}
+\bc_2^{\text{nested}} & = \Complex\setcl{\roundframe{\mathfig{0.5}{definition/nested-blobs}}}{\text{ev}_{B_1}(u)=0}
\end{align*}
\vspace{-4mm}
$$d_2 : (B_1, B_2, u, r', r) \mapsto (B_2, u \circ r', r) - (B_1, u, r \circ r')$$
@@ -177,7 +185,7 @@
\begin{frame}{Definition, general case}
\begin{block}{}
-$$\bc_k = \set{\roundframe{\mathfig{0.7}{definition/k-blobs}}}$$
+$$\bc_k = \Complex\set{\roundframe{\mathfig{0.7}{definition/k-blobs}}}$$
$k$ blobs, properly nested or disjoint, with ``innermost'' blobs labelled by pasting diagrams that evaluate to zero.
\end{block}
\begin{block}{}
@@ -219,6 +227,22 @@
\end{block}
\end{frame}
+\begin{frame}{Higher Deligne conjecture}
+\begin{block}{Deligne conjecture}
+Chains on the little discs operad acts on Hochschild cohomology.
+\end{block}
+
+\begin{block}{}
+Call $\Hom{A_\infty}{\bc_*(\cM)}{\bc_*(\cM)}$ `blob cochains on $\cM$'.
+\end{block}
+
+\begin{block}{Theorem* (Higher Deligne conjecture)}
+\scalebox{0.96}{Chains on the $n$-dimensional fat graph operad acts on blob cochains.}
+\vspace{-3mm}
+$$\mathfig{.85}{tempkw/delfig2}$$
+\end{block}
+\end{frame}
+
\begin{frame}{Gluing}
\begin{block}{$\bc_*(Y \times [0,1])$ is naturally an $A_\infty$ category}
\begin{itemize}
--- a/text/a_inf_blob.tex Thu Mar 18 19:40:46 2010 +0000
+++ b/text/a_inf_blob.tex Sat Mar 27 03:07:45 2010 +0000
@@ -249,6 +249,7 @@
Theorem \ref{product_thm}.
\end{proof}
+This establishes Property \ref{property:gluing}.
\medskip
--- a/text/basic_properties.tex Thu Mar 18 19:40:46 2010 +0000
+++ b/text/basic_properties.tex Sat Mar 27 03:07:45 2010 +0000
@@ -3,10 +3,11 @@
\section{Basic properties of the blob complex}
\label{sec:basic-properties}
-\begin{prop} \label{disjunion}
-There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$.
-\end{prop}
-\begin{proof}
+In this section we complete the proofs of Properties 1-4. Throughout the paper, where possible, we prove results using Properties 1-4, rather than the actual definition of blob homology. This allows the possibility of future improvements to or alternatives on our definition. In fact, we hope that there may be a characterisation of blob homology in terms of Properties 1-4, but at this point we are unaware of one.
+
+Recall Property \ref{property:disjoint-union}, that there is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$.
+
+\begin{proof}[Proof of Property \ref{property:disjoint-union}]
Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them
(putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a
blob diagram $(b_1, b_2)$ on $X \du Y$.
@@ -14,10 +15,8 @@
In the other direction, any blob diagram on $X\du Y$ is equal (up to sign)
to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines
a pair of blob diagrams on $X$ and $Y$.
-These two maps are compatible with our sign conventions.
+These two maps are compatible with our sign conventions. (We follow the usual convention for tensors products of complexes, as in e.g. \cite{MR1438306}: $d(a \tensor b) = da \tensor b + (-1)^{\deg(a)} a \tensor db$.)
The two maps are inverses of each other.
-\nn{should probably say something about sign conventions for the differential
-in a tensor product of chain complexes; ask Scott}
\end{proof}
For the next proposition we will temporarily restore $n$-manifold boundary
@@ -44,8 +43,8 @@
Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to
the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$.
\end{proof}
-
-Note that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy
+This proves Property \ref{property:contractibility} (the second half of the statement of this Property was immediate from the definitions).
+Note that even when there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy
equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$.
For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$,
@@ -56,7 +55,7 @@
\end{cor}
\begin{proof}
-This follows from \ref{disjunion} and \ref{bcontract}.
+This follows from Properties \ref{property:disjoint-union} and \ref{property:contractibility}.
\end{proof}
Define the {\it support} of a blob diagram to be the union of all the
@@ -84,37 +83,6 @@
so $f$ and the identity map are homotopic.
\end{proof}
-
-\medskip
-
-\nn{Maybe there is no longer a need to repeat the next couple of props here, since we also state them in the introduction.
-But I think it's worth saying that the Diff actions will be enhanced later.
-Maybe put that in the intro too.}
-
-As we noted above,
-\begin{prop}
-There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$.
-\qed
-\end{prop}
-
-
-\begin{prop}
-For fixed fields ($n$-cat), $\bc_*$ is a functor from the category
-of $n$-manifolds and homeomorphisms to the category of chain complexes and
-(chain map) isomorphisms.
-\qed
-\end{prop}
-
-In particular,
-\begin{prop} \label{diff0prop}
-There is an action of $\Homeo(X)$ on $\bc_*(X)$.
-\qed
-\end{prop}
-
-The above will be greatly strengthened in Section \ref{sec:evaluation}.
-
-\medskip
-
For the next proposition we will temporarily restore $n$-manifold boundary
conditions to the notation.
@@ -124,9 +92,9 @@
Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$,
we have the blob complex $\bc_*(X; a, b, c)$.
If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on
-$X$ to get blob diagrams on $X\sgl$:
+$X$ to get blob diagrams on $X\sgl$. This proves Property \ref{property:gluing-map}, which we restate here in more detail.
-\begin{prop}
+\textbf{Property \ref{property:gluing-map}.}\emph{
There is a natural chain map
\eq{
\gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl).
@@ -134,22 +102,7 @@
The sum is over all fields $a$ on $Y$ compatible at their
($n{-}2$-dimensional) boundaries with $c$.
`Natural' means natural with respect to the actions of diffeomorphisms.
-\qed
-\end{prop}
-
-The above map is very far from being an isomorphism, even on homology.
-This will be fixed in Section \ref{sec:gluing} below.
-
-%\nn{Next para not needed, since we already use bullet = gluing notation above(?)}
+}
-%An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$
-%and $X\sgl = X_1 \cup_Y X_2$.
-%(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.)
-%For $x_i \in \bc_*(X_i)$, we introduce the notation
-%\eq{
-% x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) .
-%}
-%Note that we have resumed our habit of omitting boundary labels from the notation.
-
-
-
+This map is very far from being an isomorphism, even on homology.
+We fix this deficit in Section \ref{sec:gluing} below.
--- a/text/blobdef.tex Thu Mar 18 19:40:46 2010 +0000
+++ b/text/blobdef.tex Sat Mar 27 03:07:45 2010 +0000
@@ -59,7 +59,7 @@
(but keeping the blob label $u$).
Note that the skein space $A(X)$
-is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.
+is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. This is Property \ref{property:skein-modules}, and also used in the second half of Property \ref{property:contractibility}.
$\bc_2(X)$ is, roughly, the space of all relations (redundancies, syzygies) among the
local relations encoded in $\bc_1(X)$.
--- a/text/intro.tex Thu Mar 18 19:40:46 2010 +0000
+++ b/text/intro.tex Sat Mar 27 03:07:45 2010 +0000
@@ -4,15 +4,15 @@
We construct the ``blob complex'' $\bc_*(M; \cC)$ associated to an $n$-manifold $M$ and a ``linear $n$-category with strong duality'' $\cC$. This blob complex provides a simultaneous generalisation of several well-understood constructions:
\begin{itemize}
-\item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. (See \S \ref{sec:fields} \nn{more specific}.)
-\item When $n=1$, $\cC$ is just a 1-category (e.g.\ an associative algebra), and $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. (See \S \ref{sec:hochschild}.)
+\item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. (See Property \ref{property:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.)
+\item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. (See Property \ref{property:hochschild} and \S \ref{sec:hochschild}.)
\item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have
that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains
on the configurations space of unlabeled points in $M$.
%$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$
\end{itemize}
The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space (replacing quotient of fields by local relations with some sort of resolution),
-and for a generalization of Hochschild homology to higher $n$-categories. We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. The blob complex allows us to do all of these! More detailed motivations are described in \S \ref{sec:motivations}.
+and for a generalization of Hochschild homology to higher $n$-categories. We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. The blob complex gives us all of these! More detailed motivations are described in \S \ref{sec:motivations}.
The blob complex has good formal properties, summarized in \S \ref{sec:properties}. These include an action of $\CH{M}$,
extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Property \ref{property:evaluation}) and a gluing formula allowing calculations by cutting manifolds into smaller parts (see Property \ref{property:gluing}).
@@ -128,8 +128,9 @@
\end{equation*}
is a functor from $n$-manifolds and homeomorphisms between them to chain complexes and isomorphisms between them.
\end{property}
+As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*^\cC(X)$; this action is extended to all of $C_*(\Homeo(X))$ in Property \ref{property:evaluation} below.
-The blob complex is also functorial with respect to $\cC$, although we will not address this in detail here.
+The blob complex is also functorial with respect to $\cC$, although we will not address this in detail here. \todo{exact w.r.t $\cC$?}
\begin{property}[Disjoint union]
\label{property:disjoint-union}
@@ -139,7 +140,7 @@
\end{equation*}
\end{property}
-If an $n$-manifold $X_\text{cut}$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$-submanifold of its boundary, write $X_\text{glued} = X_\text{cut} \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. Note that this includes the case of gluing two disjoint manifolds together.
+If an $n$-manifold $X_\text{cut}$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, write $X_\text{glued} = X_\text{cut} \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. Note that this includes the case of gluing two disjoint manifolds together.
\begin{property}[Gluing map]
\label{property:gluing-map}%
%If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map
@@ -156,9 +157,7 @@
\begin{property}[Contractibility]
\label{property:contractibility}%
-\nn{this holds with field coefficients, or more generally when
-the map to 0-th homology has a splitting; need to fix statement}
-The blob complex on an $n$-ball is contractible in the sense that it is quasi-isomorphic to its $0$-th homology. Moreover, the $0$-th homology of balls can be canonically identified with the original $n$-category $\cC$.
+With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology. Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cC$ to balls.
\begin{equation}
\xymatrix{\bc_*^{\cC}(B^n) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*^{\cC}(B^n)) \ar[r]^(0.6)\iso & \cC(B^n)}
\end{equation}
@@ -213,21 +212,22 @@
$$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$
satisfying corresponding conditions.
-In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields, as well as the notion of an $A_\infty$ $n$-category.
+In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing to the system of fields. Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category.
\begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category]
\label{property:blobs-ainfty}
Let $\cC$ be a topological $n$-category. Let $Y$ be an $n{-}k$-manifold.
-Define $A_*(Y, \cC)$ on each $m$-ball $D$, for $0 \leq m < k$, to be the set $$A_*(Y,\cC)(D) = A^\cC(Y \times D)$$ and on $k$-balls $D$ to be the set $$A_*(Y, \cC)(D) = \bc_*(Y \times D, \cC).$$ (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) These sets have the structure of an $A_\infty$ $k$-category.
+There is an $A_\infty$ $k$-category $A_*(Y, \cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, to be the set $$A_*(Y,\cC)(D) = A^\cC(Y \times D)$$ and on $k$-balls $D$ to be the set $$A_*(Y, \cC)(D) = \bc_*(Y \times D, \cC).$$ (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Property \ref{property:evaluation}.
\end{property}
\begin{rem}
-Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from a topological $n$-category.
+Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from a topological $n$-category. We think of this $A_\infty$ $n$-category as a free resolution.
\end{rem}
There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category
instead of a garden variety $n$-category; this is described in \S \ref{sec:ainfblob}.
\begin{property}[Product formula]
+\label{property:product}
Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. Let $\cC$ be an $n$-category.
Let $A_*(Y,\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}).
Then
@@ -242,14 +242,14 @@
\label{property:gluing}%
\mbox{}% <-- gets the indenting right
\begin{itemize}
-\item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is
+\item For any $(n-1)$-manifold $Y$, the blob complex of $Y \times I$ is
naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below.
-\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an
+\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an
$A_\infty$ module for $\bc_*(Y \times I)$.
\item For any $n$-manifold $X_\text{glued} = X_\text{cut} \bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{glued})$ is the $A_\infty$ self-tensor product of
-$\bc_*(X_\text{cut})$ as an $\bc_*(Y \times I)$-bimodule.
+$\bc_*(X_\text{cut})$ as an $\bc_*(Y \times I)$-bimodule:
\begin{equation*}
\bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \selfarrow
\end{equation*}
@@ -261,7 +261,7 @@
\begin{property}[Mapping spaces]
Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps
$B^n \to T$.
-(The case $n=1$ is the usual $A_\infty$ category of paths in $T$.)
+(The case $n=1$ is the usual $A_\infty$-category of paths in $T$.)
Then
$$\bc_*(X, \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$
\end{property}
@@ -277,20 +277,18 @@
Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in
\S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.}
Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.
-Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation},
-and Property \ref{property:gluing} in \S \ref{sec:gluing}.
-\nn{need to say where the remaining properties are proved.}
+Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, Property \ref{property:blobs-ainfty} as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats},
+and Properties \ref{property:product} and \ref{property:gluing} in \S \ref{sec:ainfblob} as consequences of Theorem \ref{product_thm}.
\subsection{Future directions}
\label{sec:future}
Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard).
-In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do.
-\nn{maybe make similar remark about chain complexes and $(\infty, 0)$-categories}
+In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do. We could presumably also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories), and likely it will prove useful to think about the connections between what we do here and $(\infty,k)$-categories.
More could be said about finite characteristic (there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$, for example). Much more could be said about other types of manifolds, in particular oriented, $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; there may be some differences for topological manifolds and smooth manifolds.
Many results in Hochschild homology can be understood `topologically' via the blob complex. For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$, but haven't investigated the details.
-Most importantly, however, \nn{applications!} \nn{$n=2$ cases, contact, Kh}
+Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh}
\subsection{Thanks and acknowledgements}
--- a/text/kw_macros.tex Thu Mar 18 19:40:46 2010 +0000
+++ b/text/kw_macros.tex Sat Mar 27 03:07:45 2010 +0000
@@ -57,7 +57,7 @@
% \DeclareMathOperator{\pr}{pr} etc.
\def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}}
-\applytolist{declaremathop}{pr}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res};
+\applytolist{declaremathop}{pr}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad};
\DeclareMathOperator{\kone}{cone}
--- a/text/ncat.tex Thu Mar 18 19:40:46 2010 +0000
+++ b/text/ncat.tex Sat Mar 27 03:07:45 2010 +0000
@@ -156,18 +156,17 @@
\begin{figure}[!ht]
$$
-\begin{tikzpicture}[every label/.style={green}]
-\node[fill=black, circle, label=below:$E$](S) at (0,0) {};
-\node[fill=black, circle, label=above:$E$](N) at (0,2) {};
+\begin{tikzpicture}[%every label/.style={green}
+ ]
+\node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {};
+\node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {};
\draw (S) arc (-90:90:1);
\draw (N) arc (90:270:1);
\node[left] at (-1,1) {$B_1$};
\node[right] at (1,1) {$B_2$};
\end{tikzpicture}
$$
-$$\mathfig{.4}{tempkw/blah3}$$
-\caption{Combining two balls to get a full boundary
-\nn{maybe smaller dots for $E$ in pdf fig}}\label{blah3}\end{figure}
+\caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure}
Note that we insist on injectivity above.
@@ -215,8 +214,21 @@
\end{axiom}
\begin{figure}[!ht]
+$$
+\begin{tikzpicture}[%every label/.style={green},
+ x=1.5cm,y=1.5cm]
+\node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {};
+\node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {};
+\draw (S) arc (-90:90:1);
+\draw (N) arc (90:270:1);
+\draw (N) -- (S);
+\node[left] at (-1/4,1) {$B_1$};
+\node[right] at (1/4,1) {$B_2$};
+\node at (1/6,3/2) {$Y$};
+\end{tikzpicture}
+$$
$$\mathfig{.4}{tempkw/blah5}$$
-\caption{From two balls to one ball}\label{blah5}\end{figure}
+\caption{From two balls to one ball.}\label{blah5}\end{figure}
\begin{axiom}[Strict associativity] \label{nca-assoc}
The composition (gluing) maps above are strictly associative.
@@ -224,7 +236,7 @@
\begin{figure}[!ht]
$$\mathfig{.65}{tempkw/blah6}$$
-\caption{An example of strict associativity}\label{blah6}\end{figure}
+\caption{An example of strict associativity.}\label{blah6}\end{figure}
\nn{figure \ref{blah6} (blah6) needs a dotted line in the south split ball}
@@ -263,7 +275,7 @@
\begin{figure}[!ht]
$$\mathfig{.8}{tempkw/blah7}$$
-\caption{Operadish composition and associativity}\label{blah7}\end{figure}
+\caption{Operad composition and associativity}\label{blah7}\end{figure}
The next axiom is related to identity morphisms, though that might not be immediately obvious.
@@ -520,8 +532,7 @@
\label{ex:traditional-n-categories}
Given a `traditional $n$-category with strong duality' $C$
define $\cC(X)$, for $X$ a $k$-ball or $k$-sphere with $k < n$,
-to be the set of all $C$-labeled sub cell complexes of $X$.
-(See Subsection \ref{sec:fields}.)
+to be the set of all $C$-labeled sub cell complexes of $X$ (c.f. \S \ref{sec:fields}).
For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear
combinations of $C$-labeled sub cell complexes of $X$
modulo the kernel of the evaluation map.
@@ -628,7 +639,7 @@
\begin{figure}[!ht]
\begin{equation*}
-\mathfig{.63}{tempkw/zz2}
+\mathfig{.63}{ncat/zz2}
\end{equation*}
\caption{A small part of $\cJ(W)$}
\label{partofJfig}
@@ -733,8 +744,7 @@
\subsection{Modules}
-Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations,
-a.k.a.\ actions).
+Next we define topological and $A_\infty$ $n$-category modules.
The definition will be very similar to that of $n$-categories,
but with $k$-balls replaced by {\it marked $k$-balls,} defined below.
\nn{** need to make sure all revisions of $n$-cat def are also made to module def.}
@@ -745,10 +755,10 @@
Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$.
This will be explained in more detail as we present the axioms.
-Fix an $n$-category $\cC$.
+Throughout, we fix an $n$-category $\cC$. For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. We state the final axiom, on actions of homeomorphisms, differently in the two cases.
Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair
-(standard $k$-ball, northern hemisphere in boundary of standard $k$-ball).
+$$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$
We call $B$ the ball and $N$ the marking.
A homeomorphism between marked $k$-balls is a homeomorphism of balls which
restricts to a homeomorphism of markings.
@@ -831,16 +841,16 @@
\begin{figure}[!ht]
\begin{equation*}
-\mathfig{.63}{tempkw/zz3}
+\mathfig{.4}{ncat/zz3}
\end{equation*}
-\caption{Module composition (top); $n$-category action (bottom)}
+\caption{Module composition (top); $n$-category action (bottom).}
\label{zzz3}
\end{figure}
First, we can compose two module morphisms to get another module morphism.
\mmpar{Module axiom 6}{Module composition}
-{Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls ($0\le k\le n$)
+{Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls (with $0\le k\le n$)
and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball.
Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere.
Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$.
@@ -886,7 +896,7 @@
\begin{figure}[!ht]
\begin{equation*}
-\mathfig{1}{tempkw/zz1b}
+\mathfig{0.49}{ncat/zz0} \mathfig{0.49}{ncat/zz1}
\end{equation*}
\caption{Two examples of mixed associativity}
\label{zzz1b}
@@ -1015,9 +1025,9 @@
with $M_{ib}\cap Y_i$ being the marking.
(See Figure \ref{mblabel}.)
\begin{figure}[!ht]\begin{equation*}
-\mathfig{.9}{tempkw/mblabel}
+\mathfig{.6}{ncat/mblabel}
\end{equation*}\caption{A permissible decomposition of a manifold
-whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$.}\label{mblabel}\end{figure}
+whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure}
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$.
This defines a partial ordering $\cJ(W)$, which we will think of as a category.
@@ -1087,9 +1097,8 @@
In this subsection we define an $n{+}1$-category $\cS$ of ``sphere modules"
whose objects correspond to $n$-categories.
This is a version of the familiar algebras-bimodules-intertwiners 2-category.
-(Terminology: It is clearly appropriate to call an $S^0$ modules a bimodule,
-since a 0-sphere has an obvious bi-ness.
-This is much less true for higher dimensional spheres,
+(Terminology: It is clearly appropriate to call an $S^0$ module a bimodule,
+but this is much less true for higher dimensional spheres,
so we prefer the term ``sphere module" for the general case.)
The $0$- through $n$-dimensional parts of $\cC$ are various sorts of modules, and we describe
@@ -1146,7 +1155,7 @@
\medskip
-Part of the structure of an $n$-cat 0-sphere module is captured my saying it is
+Part of the structure of an $n$-category 0-sphere module is captured by saying it is
a collection $\cD^{ab}$ of $n{-}1$-categories, indexed by pairs $(a, b)$ of objects (0-morphisms)
of $\cA$ and $\cB$.
Let $J$ be some standard 0-marked 1-ball (i.e.\ an interval with a marked point in its interior).
--- a/text/smallblobs.tex Thu Mar 18 19:40:46 2010 +0000
+++ b/text/smallblobs.tex Sat Mar 27 03:07:45 2010 +0000
@@ -3,12 +3,37 @@
Fix $\cU$, an open cover of $M$. Define the `small blob complex' $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$.
-\begin{lem}[Small blobs]
+\begin{thm}[Small blobs]
The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence.
-\end{lem}
+\end{thm}
\begin{proof}
-Given a blob diagram $b \in \bc_k(M)$, denote by $b_\cS$ for $\cS \subset \{1, \ldots, k\}$ the blob diagram obtained by erasing the corresponding blobs. In particular, $b_\eset = b$, $b_{\{1,\ldots,k\}} \in \bc_0(M)$, and $d b_\cS = \sum_{\cS' = \cS'\sqcup\{i\}} \text{some sign} b_{\cS'}$.
-Similarly, for a configuration of $k$ blobs $\beta$ (that is, an choice of embeddings of balls in $M$, satisfying the disjointness rules for blobs, rather than a blob diagram, which is additionally labelled by appropriate fields), $\beta_\cS$ denotes the result of erasing a subset of blobs. We'll write $\beta' \prec \beta$ if $\beta' = \beta_\cS$ for some $\cS$. Finally, for finite sequences, we'll write $i \prec i'$ if $i$ is subsequence of $i'$, and $i \prec_1 i$ if the lengths differ by exactly 1.
+We begin by describing the homotopy inverse in small degrees, to illustrate the general technique.
+We will construct a chain map $s: \bc_*(M) \to \bc^{\cU}_*(M)$ and a homotopy $h:\bc_*(M) \to \bc_{*+1}(M)$ so that $\bdy h+h \bdy=\id - i\circ s$. The composition $s \circ i$ will just be the identity.
+
+On $0$-blobs, $s$ is just the identity; a blob diagram without any blobs is compatible with any open cover. Nevertheless, we'll begin introducing nomenclature at this point: for configuration $\beta$ of disjoint embedded balls in $M$ we'll associate a one parameter family of homeomorphisms $\phi_\beta : \Delta^1 \to \Homeo(M)$ (here $\Delta^m$ is the standard simplex $\setc{\mathbf{x} \in \Real^{m+1}}{\sum_i x_i = 1}$). For $0$-blobs, where $\beta = \eset$, all these homeomorphisms are just the identity.
+
+On a $1$-blob $b$, with ball $\beta$, $s$ is defined as the sum of two terms. Essentially, the first term `makes $\beta$ small', while the other term `gets the boundary right'. First, pick a one-parameter family $\phi_\beta : \Delta^1 \to \Homeo(M)$ of homeomorphisms, so $\phi_\beta(0,1)$ is the identity and $\phi_\beta(1,0)$ makes the ball $\beta$ small. Next, pick a two-parameter family $\phi_{\eset \prec \beta} : \Delta^2 \to \Homeo(M)$ so that $\phi_{\eset \prec \beta}(s,t,0)$ makes the ball $\beta$ small for all $s+t=1$, while $\phi_{\eset \prec \beta}(0,t,u) = \phi_\eset(t,u)$ and $\phi_{\eset \prec \beta}(s,0,u) = \phi_\beta(s,u)$. (It's perhaps not obvious that this is even possible --- see Lemma \ref{lem:extend-small-homeomorphisms} below.) We now define $s$ by
+$$s(b) = \phi_\beta(1,0)(b) + \restrict{\phi_{\eset \prec \beta}}{u=0}(\bdy b).$$
+Here, $\phi_\beta(1,0)$ is just a homeomorphism, which we apply to $b$, while $\restrict{\phi_{\eset \prec \beta}}{u=0}$ is a one parameter family of homeomorphisms which acts on the $0$-blob $\bdy b$ to give a $1$-blob. We now check that $s$, as defined so far, is a chain map, calculating
+\begin{align*}
+\bdy (s(b)) & = \phi_\beta(1,0)(\bdy b) + (\bdy \restrict{\phi_{\eset \prec \beta}}{u=0})(\bdy b) \\
+ & = \phi_\beta(1,0)(\bdy b) + \phi_\eset(1,0)(\bdy b) - \phi_\beta(1,0)(\bdy b) \\
+ & = \phi_\eset(1,0)(\bdy b) \\
+ & = s(\bdy b)
+\end{align*}
+Next, we compute the compositions $s \circ i$ and $i \circ s$. If we start with a small $1$-blob diagram $b$, first include it up to the full blob complex then apply $s$, we get exactly back to $b$, at least assuming we adopt the convention that for any ball $\beta$ which is already small, we choose the families of homeomorphisms $\phi_\beta$ and $\phi_{\eset \prec \beta}$ to always be the identity. In the other direction, $i \circ s$, we will need to construct the homotopy $h:\bc_*(M) \to \bc_{*+1}(M)$ for $*=0$ or $1$. This is defined by $h(b) = \phi_\eset(b)$ when $b$ is a $0$-blob (here $\phi_\eset$ is a one parameter family of homeomorphisms, so this is a $1$-blob), and $h(b) = \phi_\beta(b) - \phi_{\eset \prec \beta}(\bdy b)$ when $b$ is a $1$-blob (here $\beta$ is the ball in $b$, and this is the action of a one parameter family of homeomorphisms on a $1$-blob, so a $2$-blob).
+
+\begin{align*}
+(\bdy h+h \bdy)(b) & = \bdy (\phi_{\beta}(b) - \phi_{\eset \prec \beta}{\bdy b}) + \phi_\eset(\bdy b) \\
+ & = b - \phi_\beta(1,0)(b) - \phi_\beta(\bdy b) - (\bdy \phi_{\eset \prec \beta})(\bdy b) + \phi_\eset(\bdy b) \\
+ & = b - \phi_\beta(1,0)(b) - \phi_\beta(\bdy b) - \phi_\eset(\bdy b) + \phi_\beta(\bdy b) - \restrict{\phi_{\eset \prec \beta}}{u=0}(\bdy b) + \phi_\eset(\bdy b) \\
+ & = b - \phi_\beta(1,0)(b) - \restrict{\phi_{\eset \prec \beta}}{u=0}(\bdy b) \\
+ & = (\id - i \circ s)(b)
+\end{align*}
+
+
+Given a blob diagram $b \in \bc_k(M)$, denote by $b_\cS$ for $\cS \subset \{1, \ldots, k\}$ the blob diagram obtained by erasing the corresponding blobs. In particular, $b_\eset = b$, $b_{\{1,\ldots,k\}} \in \bc_0(M)$, and $d b_\cS = \sum_{\cS' = \cS'\sqcup\{i\}} \pm b_{\cS'}$.
+Similarly, for a disjoint embedding of $k$ balls $\beta$ (that is, a blob diagram but without the labels on regions), $\beta_\cS$ denotes the result of erasing a subset of blobs. We'll write $\beta' \prec \beta$ if $\beta' = \beta_\cS$ for some $\cS$. Finally, for finite sequences, we'll write $i \prec i'$ if $i$ is subsequence of $i'$, and $i \prec_1 i$ if the lengths differ by exactly 1.
Next, we'll choose a `shrinking system' for $\cU$, namely for each increasing sequence of blob configurations
$\beta_0 \prec \beta_1 \prec \cdots \prec \beta_m$, an $m$ parameter family of diffeomorphisms
@@ -22,13 +47,20 @@
\phi_{\beta_0 \prec \cdots \prec \beta_m}(x_0, \ldots, x_{i-1},0,x_{i+1},\ldots,x_m) & = \phi_{\beta_0 \prec \cdots \beta_{i-1} \prec \beta_{i+1} \prec \beta_m}(x_0,\ldots, x_{i-1},x_{i+1},\ldots,x_m).
\end{align*}
\end{itemize}
-It's not immediately obvious that it's possible to make such choices, but it follows quickly from
-\begin{claim}
-If $\beta$ is a collection of disjointly embedded balls in $M$, and $\varphi: B^k \to \Diff{M}$ is a map into diffeomorphisms such that for every $x\in \bdy B^k$, $\varphi(x)(\beta)$ is subordinate to $\cU$, then we can extend $\varphi$ to $\varphi:B^{k+1} \to \Diff{M}$, with the original $B^k$ as $\bdy^{\text{north}}(B^{k+1})$, and $\varphi(x)(\beta)$ subordinate to $\cU$ for every $x \in \bdy^{\text{south}}(B^{k+1})$.
+It's not immediately obvious that it's possible to make such choices, but it follows readily from the following Lemma.
+
+When $\beta$ is a collection of disjoint embedded balls in $M$, we say that a homeomorphism of $M$ `makes $\beta$ small' if the image of each ball in $\beta$ under the homeomorphism is contained in some open set of $\cU$.
-In fact, for a fixed $\beta$, $\Diff{M}$ retracts onto the subset $\setc{\varphi \in \Diff{M}}{\text{$\varphi(\beta)$ is subordinate to $\cU$}}$.
-\end{claim}
-\nn{need to check that this is true.}
+\begin{lem}
+\label{lem:extend-small-homeomorphisms}
+Fix a collection of disjoint embedded balls $\beta$ in $M$. Suppose we have a map $f : X \to \Homeo(M)$ on some compact $X$ such that for each $x \in \bdy X$, $f(x)$ makes $\beta$ small. Then we can extend $f$ to a map $\tilde{f} : X \times [0,1] \to \Homeo(M)$ so that $\tilde{f}(x,0) = f(x)$ and for every $x \in \bdy X \times [0,1] \cup X \times \{1\}$, $\tilde{f}(x)$ makes $\beta$ small.
+\end{lem}
+\begin{proof}
+Fix a metric on $M$, and pick $\epsilon > 0$ so every $\epsilon$ ball in $M$ is contained in some open set of $\cU$. First construct a family of homeomorphisms $g_s : M \to M$, $s \in [1,\infty)$ so $g_1$ is the identity, and $g_s(\beta_i) \subset \beta_i$ and $\rad g_s(\beta_i) \leq \frac{1}{s} \rad \beta_i$ for each ball $\beta_i$.
+There is some $K$ which uniformly bounds the expansion factors of all the homeomorphisms $f(x)$, that is $d(f(x)(a), f(x)(b)) < K d(a,b)$ for all $x \in X, a,b \in M$. Write $S=\epsilon^{-1} K \max_i \{\rad \beta_i\}$ (note that is $S<1$, we can just take $S=1$, as already $f(x)$ makes $\beta$ small for all $x$). Now define $\tilde{f}(t, x) = f(x) \compose g_{(S-1)t+1}$.
+
+If $x \in \bdy X$, then $g_{(S-1)t+1}(\beta_i) \subset \beta_i$, and by hypothesis $f(x)$ makes $\beta_i$ small, so $\tilde{f}(t, x)$ makes $\beta$ small for all $t \in [0,1]$. Alternatively, $\rad g_S(\beta_i) \leq \frac{1}{S} \rad \beta_i \leq \frac{\epsilon}{K}$, so $\rad \tilde{f}(1,x)(\beta_i) \leq \epsilon$, and so $\tilde{f}(1,x)$ makes $\beta$ small for all $x \in X$.
+\end{proof}
We'll need a stronger version of Property \ref{property:evaluation}; while the evaluation map $ev: \CD{M} \tensor \bc_*(M) \to \bc_*(M)$ is not unique, it has an up-to-homotopy representative (satisfying the usual conditions) which restricts to become a chain map $ev: \CD{M} \tensor \bc^{\cU}_*(M) \to \bc^{\cU}_*(M)$. The proof is straightforward: when deforming the family of diffeomorphisms to shrink its supports to a union of open sets, do so such that those open sets are subordinate to the cover.
--- a/text/tqftreview.tex Thu Mar 18 19:40:46 2010 +0000
+++ b/text/tqftreview.tex Sat Mar 27 03:07:45 2010 +0000
@@ -290,6 +290,7 @@
\nn{maybe examples of local relations before general def?}
\subsection{Constructing a TQFT}
+\label{sec:constructing-a-tqft}
In this subsection we briefly review the construction of a TQFT from a system of fields and local relations.
(For more details, see \cite{kw:tqft}.)