--- a/bibliography/bibliography.bib	Tue May 05 17:27:21 2009 +0000
+++ b/bibliography/bibliography.bib	Sun May 24 20:30:45 2009 +0000
@@ -1,503 +1,523 @@
-@PREAMBLE{ "\newcommand{\noopsort}[1]{}" }
-
-@STRING{CUP = {Cambridge University Press}}
-@STRING{OUP = {Oxford University Press}}
-@STRING{PUP = {Princeton University Press}}
-@STRING{SV = {Springer-Verlag}}
-@STRING{AP = {Academic Press}}
-@STRING{AMS = {American Mathematical Society}}
-
-%S! fix this citation!
-@article{morrison-walker,
-    AUTHOR = {David Clark and Scott Morrison and Kevin Walker},
-    TITLE = {Fixing the functoriality of Khovanov homology},
-     note = {\arxiv{math.GT/0701339}},
-}
-
-@misc{kw:tqft,
-    AUTHOR = {Walker, Kevin},
-    TITLE  = {Topological Quantum Field Theories},
-    note    = {draft available at \url{http://canyon23.net/math/tc.pdf}},
-}
-
-@article {MR1854636,
-    AUTHOR = {Keller, Bernhard},
-     TITLE = {Introduction to {$A$}-infinity algebras and modules},
-   JOURNAL = {Homology Homotopy Appl.},
-  FJOURNAL = {Homology, Homotopy and Applications},
-    VOLUME = {3},
-      YEAR = {2001},
-    NUMBER = {1},
-     PAGES = {1--35 (electronic)},
-      ISSN = {1512-0139},
-   MRCLASS = {18E30 (18D50 18G40 55P43 55U35)},
-  MRNUMBER = {MR1854636 (2004a:18008a)},
-MRREVIEWER = {Ulrike Tillmann},
-      note = {\mathscinet{MR1854636} \arxiv{math.RA/9910179}},
-}
-
-@incollection {MR2061854,
-    AUTHOR = {McClure, James E. and Smith, Jeffrey H.},
-     TITLE = {Operads and cosimplicial objects: an introduction},
- BOOKTITLE = {Axiomatic, enriched and motivic homotopy theory},
-    SERIES = {NATO Sci. Ser. II Math. Phys. Chem.},
-    VOLUME = {131},
-     PAGES = {133--171},
- PUBLISHER = {Kluwer Acad. Publ.},
-   ADDRESS = {Dordrecht},
-      YEAR = {2004},
-   MRCLASS = {55P48 (18D50)},
-  MRNUMBER = {MR2061854 (2005b:55018)},
-MRREVIEWER = {David Chataur},
-      note = {\mathscinet{MR2061854} \arxiv{math.QA/0402117}},
-}
-
-@book {MR0420610,
-    AUTHOR = {May, J. P.},
-     TITLE = {The geometry of iterated loop spaces},
- PUBLISHER = {Springer-Verlag},
-   ADDRESS = {Berlin},
-      YEAR = {1972},
-     PAGES = {viii+175},
-   MRCLASS = {55D35},
-  MRNUMBER = {MR0420610 (54 \#8623b)},
-MRREVIEWER = {J. Stasheff},
-      note = {Lectures Notes in Mathematics, Vol. 271 \mathscinet{MR0420610} \href{http://www.math.uchicago.edu/~may/BOOKS/gils.pdf}{available online}},
-}
-
-@article {MR0236922,
-    AUTHOR = {Boardman, J. M. and Vogt, R. M.},
-     TITLE = {Homotopy-everything {$H$}-spaces},
-   JOURNAL = {Bull. Amer. Math. Soc.},
-  FJOURNAL = {Bulletin of the American Mathematical Society},
-    VOLUME = {74},
-      YEAR = {1968},
-     PAGES = {1117--1122},
-      ISSN = {0002-9904},
-   MRCLASS = {55.42},
-  MRNUMBER = {MR0236922 (38 \#5215)},
-MRREVIEWER = {R. J. Milgram},
-      note = {\mathscinet{MR0236922} \doi{10.1090/S0002-9904-1968-12070-1}},
-}
-
-@book {MR0420609,
-    AUTHOR = {Boardman, J. M. and Vogt, R. M.},
-     TITLE = {Homotopy invariant algebraic structures on topological spaces},
-    SERIES = {Lecture Notes in Mathematics, Vol. 347},
- PUBLISHER = {Springer-Verlag},
-   ADDRESS = {Berlin},
-      YEAR = {1973},
-     PAGES = {x+257},
-   MRCLASS = {55D35},
-  MRNUMBER = {MR0420609 (54 \#8623a)},
-MRREVIEWER = {J. Stasheff},
-      note = {\mathscinet{MR0420609}},
-}
-
-%The framed little discs operad:
-@article {MR1256989,
-    AUTHOR = {Getzler, E.},
-     TITLE = {Batalin-{V}ilkovisky algebras and two-dimensional topological
-              field theories},
-   JOURNAL = {Comm. Math. Phys.},
-  FJOURNAL = {Communications in Mathematical Physics},
-    VOLUME = {159},
-      YEAR = {1994},
-    NUMBER = {2},
-     PAGES = {265--285},
-      ISSN = {0010-3616},
-     CODEN = {CMPHAY},
-   MRCLASS = {81T70 (17B81 55Q99 58Z05 81T40)},
-  MRNUMBER = {MR1256989 (95h:81099)},
-MRREVIEWER = {J. Stasheff},
-      note = {\mathscinet{MR1256989} \euclid{1104254599}},
-}
-
-
-
-
-
-@article {MR1917056,
-    AUTHOR = {Bar-Natan, Dror},
-     TITLE = {On {K}hovanov's categorification of the {J}ones polynomial},
-   JOURNAL = {Algebr. Geom. Topol.},
-  FJOURNAL = {Algebraic \& Geometric Topology},
-    VOLUME = {2},
-      YEAR = {2002},
-     PAGES = {337--370 (electronic)},
-      ISSN = {1472-2747},
-   MRCLASS = {57M27},
-  MRNUMBER = {MR1917056 (2003h:57014)},
-MRREVIEWER = {Jacob Andrew Rasmussen},
-      note = {arXiv:\arxiv{math.QA/0201043}},
-}
-
-@incollection {MR2147420,
-    AUTHOR = {Bar-Natan, Dror},
-     TITLE = {Khovanov homology for knots and links with up to 11 crossings},
- BOOKTITLE = {Advances in topological quantum field theory},
-    SERIES = {NATO Sci. Ser. II Math. Phys. Chem.},
-    VOLUME = {179},
-     PAGES = {167--241},
- PUBLISHER = {Kluwer Acad. Publ.},
-   ADDRESS = {Dordrecht},
-      YEAR = {2004},
-   MRCLASS = {57M27},
-  MRNUMBER = {MR2147420 (2006c:57009)},
-MRREVIEWER = {Marta M. Asaeda},
-}
-
-@article {MR2174270,
-    AUTHOR = {Bar-Natan, Dror},
-     TITLE = {Khovanov's homology for tangles and cobordisms},
-   JOURNAL = {Geom. Topol.},
-  FJOURNAL = {Geometry and Topology},
-    VOLUME = {9},
-      YEAR = {2005},
-     PAGES = {1443--1499 (electronic)},
-      ISSN = {1465-3060},
-   MRCLASS = {57M27 (57M25)},
-  MRNUMBER = {MR2174270},
-      note = {arXiv:\arxiv{math.GT/0410495}},
-}
-
-@article {MR1680395,
-    AUTHOR = {Khovanov, Mikhail and Kuperberg, Greg},
-     TITLE = {Web bases for {${\rm sl}(3)$} are not dual canonical},
-   JOURNAL = {Pacific J. Math.},
-  FJOURNAL = {Pacific Journal of Mathematics},
-    VOLUME = {188},
-      YEAR = {1999},
-    NUMBER = {1},
-     PAGES = {129--153},
-      ISSN = {0030-8730},
-     CODEN = {PJMAAI},
-   MRCLASS = {17B37 (22E60 57M27 81R05)},
-  MRNUMBER = {MR1680395 (2000j:17023a)},
-MRREVIEWER = {Robert J. Marsh},
-      note = {arXiv:\arxiv{q-alg/9712046}},
-}
-
-@article {MR1740682,
-    AUTHOR = {Khovanov, Mikhail},
-     TITLE = {A categorification of the {J}ones polynomial},
-   JOURNAL = {Duke Math. J.},
-  FJOURNAL = {Duke Mathematical Journal},
-    VOLUME = {101},
-      YEAR = {2000},
-    NUMBER = {3},
-     PAGES = {359--426},
-      ISSN = {0012-7094},
-     CODEN = {DUMJAO},
-   MRCLASS = {57M27 (57R56)},
-  MRNUMBER = {MR1740682 (2002j:57025)},
-}
-
-@article {MR1928174,
-    AUTHOR = {Khovanov, Mikhail},
-     TITLE = {A functor-valued invariant of tangles},
-   JOURNAL = {Algebr. Geom. Topol.},
-  FJOURNAL = {Algebraic \& Geometric Topology},
-    VOLUME = {2},
-      YEAR = {2002},
-     PAGES = {665--741 (electronic)},
-      ISSN = {1472-2747},
-   MRCLASS = {57M27 (57R56)},
-  MRNUMBER = {MR1928174 (2004d:57016)},
-MRREVIEWER = {Jacob Andrew Rasmussen},
-    note   = {arXiv:\arxiv{math.GT/0103190}},
-}
-
-@article {MR2034399,
-    AUTHOR = {Khovanov, Mikhail},
-     TITLE = {Patterns in knot cohomology. {I}},
-   JOURNAL = {Experiment. Math.},
-  FJOURNAL = {Experimental Mathematics},
-    VOLUME = {12},
-      YEAR = {2003},
-    NUMBER = {3},
-     PAGES = {365--374},
-      ISSN = {1058-6458},
-   MRCLASS = {57M27 (18G60 57M25 57R56)},
-  MRNUMBER = {MR2034399 (2004m:57022)},
-MRREVIEWER = {Jacob Andrew Rasmussen},
-}
-
-@article {MR2100691,
-    AUTHOR = {Khovanov, Mikhail},
-     TITLE = {sl(3) link homology},
-   JOURNAL = {Algebr. Geom. Topol.},
-  FJOURNAL = {Algebraic \& Geometric Topology},
-    VOLUME = {4},
-      YEAR = {2004},
-     PAGES = {1045--1081 (electronic)},
-      ISSN = {1472-2747},
-   MRCLASS = {57M27 (18G60 57R56)},
-  MRNUMBER = {MR2100691 (2005g:57032)},
-MRREVIEWER = {Justin Sawon},
-      note = {arXiv:\arxiv{math.QA/0304375}},
-}
-
-@article {MR2124557,
-    AUTHOR = {Khovanov, Mikhail},
-     TITLE = {Categorifications of the colored {J}ones polynomial},
-   JOURNAL = {J. Knot Theory Ramifications},
-  FJOURNAL = {Journal of Knot Theory and its Ramifications},
-    VOLUME = {14},
-      YEAR = {2005},
-    NUMBER = {1},
-     PAGES = {111--130},
-      ISSN = {0218-2165},
-   MRCLASS = {57M27},
-  MRNUMBER = {MR2124557 (2006a:57016)},
-MRREVIEWER = {Marta M. Asaeda},
-}
-
-@article {MR2171235,
-    AUTHOR = {Khovanov, Mikhail},
-     TITLE = {An invariant of tangle cobordisms},
-   JOURNAL = {Trans. Amer. Math. Soc.},
-  FJOURNAL = {Transactions of the American Mathematical Society},
-    VOLUME = {358},
-      YEAR = {2006},
-    NUMBER = {1},
-     PAGES = {315--327 (electronic)},
-      ISSN = {0002-9947},
-     CODEN = {TAMTAM},
-   MRCLASS = {57Q45},
-  MRNUMBER = {MR2171235},
-      note = {arXiv:\arxiv{math.GT/0207264}},
-}
-
-@misc{math.QA/0401268,
-    title = {{Matrix factorizations and link homology}},
-    author = {Mikhail Khovanov and Lev Rozansky},
-      note = {arXiv:\arxiv{math.QA/0401268}}}
-
-@misc{math.QA/0505056,
-    title = {{Matrix factorizations and link homology II}},
-    author = {Mikhail Khovanov and Lev Rozansky},
-      note = {arXiv:\arxiv{math.QA/0505056}}}
-
-@incollection {MR2048108,
-    AUTHOR = {Thurston, Dylan P.},
-     TITLE = {The algebra of knotted trivalent graphs and {T}uraev's shadow
-              world},
- BOOKTITLE = {Invariants of knots and 3-manifolds (Kyoto, 2001)},
-    SERIES = {Geom. Topol. Monogr.},
-    VOLUME = {4},
-     PAGES = {337--362 (electronic)},
- PUBLISHER = {Geom. Topol. Publ., Coventry},
-      YEAR = {2002},
-   MRCLASS = {57M25 (57M15 57M20 57Q40)},
-  MRNUMBER = {MR2048108 (2005c:57010)},
-MRREVIEWER = {Marko Kranjc},
-      note = {arXiv:\arxiv{math.GT/0311458}},
-}
-
-@article {MR1403861,
-    AUTHOR = {Kuperberg, Greg},
-     TITLE = {Spiders for rank {$2$} {L}ie algebras},
-   JOURNAL = {Comm. Math. Phys.},
-  FJOURNAL = {Communications in Mathematical Physics},
-    VOLUME = {180},
-      YEAR = {1996},
-    NUMBER = {1},
-     PAGES = {109--151},
-      ISSN = {0010-3616},
-     CODEN = {CMPHAY},
-   MRCLASS = {17B10 (22E60 81R05)},
-  MRNUMBER = {MR1403861 (97f:17005)},
-MRREVIEWER = {Stefano Capparelli},
-      note = {arXiv:\arxiv{q-alg/9712003}},
-}
-
-@misc{math.GT/0606318,
-    title  = {{Fast Khovanov Homology Computations}},
-    author = {Dror Bar-Natan},
-    note   = {arXiv:\arxiv{math.GT/0606318}},
-    url    = {http://www.math.toronto.edu/~drorbn/papers/FastKh/}
-    }
-
-@misc{math.GT/0603347,
-    title = {{On the Algebraic Structure of Bar-Natan's Universal
-        Geometric Complex and the Geometric Structure of Khovanov
-        Link Homology Theories}},
-    author = {Gad Naot},
-    note   = {arXiv:\arxiv{math.GT/0603347}}}
-
-@misc{math.QA/9909027,
-    title = {{Planar algebras, I}},
-    author = {Vaughan F. R. Jones},
-    note   = {arXiv:\arxiv{math.QA/9909027}}}
-
-@article {MR1217386,
-    AUTHOR = {Turaev, V. and Wenzl, H.},
-     TITLE = {Quantum invariants of {$3$}-manifolds associated with
-              classical simple {L}ie algebras},
-   JOURNAL = {Internat. J. Math.},
-  FJOURNAL = {International Journal of Mathematics},
-    VOLUME = {4},
-      YEAR = {1993},
-    NUMBER = {2},
-     PAGES = {323--358},
-      ISSN = {0129-167X},
-   MRCLASS = {57M25 (17B37 57N10)},
-  MRNUMBER = {MR1217386 (94i:57019)},
-MRREVIEWER = {Toshitake Kohno},
-}
-
-@article {MR1470857,
-    AUTHOR = {Wenzl, Hans},
-     TITLE = {{$C\sp *$} tensor categories from quantum groups},
-   JOURNAL = {J. Amer. Math. Soc.},
-  FJOURNAL = {Journal of the American Mathematical Society},
-    VOLUME = {11},
-      YEAR = {1998},
-    NUMBER = {2},
-     PAGES = {261--282},
-      ISSN = {0894-0347},
-   MRCLASS = {46L89 (17B37 46L05)},
-  MRNUMBER = {MR1470857 (98k:46123)},
-MRREVIEWER = {Andrzej Sitarz},
-    eprint = {\url{http://ftp.ams.org/jams/1998-11-02/S0894-0347-98-00253-7/S0894-0347-98-00253-7.pdf}},
-}
-
-@misc{math.GT/0603307,
-    title  = {{The universal sl3-link homology}},
-    author = {Marco Mackaay and Pedro Vaz},
-    note  = {arXiv:\arxiv{math.GT/0603307}}}
-
- @misc{wiki:Grothendieck-group,
-   author = "Wikipedia",
-   title = "Grothendieck group --- Wikipedia{,} The Free Encyclopedia",
-   year = "2006",
-   note = "[\href{http://en.wikipedia.org/w/index.php?title=Grothendieck_group&oldid=52451663}{Online}; accessed 30-June-2006]"
- }
-
- @misc{wiki:Invariant-basis-number,
-   author = "Wikipedia",
-   title = "Invariant basis number --- Wikipedia{,} The Free Encyclopedia",
-   year = "2006",
-   note = "[\href{http://en.wikipedia.org/w/index.php?title=Invariant_basis_number&oldid=85220456}{Online}; accessed 24-December-2006]"
- }
-
-@article {MR1403351,
-    AUTHOR = {Etingof, Pavel and Kazhdan, David},
-     TITLE = {Quantization of {L}ie bialgebras. {I}},
-   JOURNAL = {Selecta Math. (N.S.)},
-  FJOURNAL = {Selecta Mathematica. New Series},
-    VOLUME = {2},
-      YEAR = {1996},
-    NUMBER = {1},
-     PAGES = {1--41},
-      ISSN = {1022-1824},
-     CODEN = {SMATF6},
-   MRCLASS = {17B37 (16W30 18D20 81R50)},
-  MRNUMBER = {MR1403351 (97f:17014)},
-MRREVIEWER = {Yu. N. Bespalov},
-}
-
-@article {MR1669953,
-    AUTHOR = {Etingof, Pavel and Kazhdan, David},
-     TITLE = {Quantization of {L}ie bialgebras. {II}},
-   JOURNAL = {Selecta Math. (N.S.)},
-  FJOURNAL = {Selecta Mathematica. New Series},
-    VOLUME = {4},
-      YEAR = {1998},
-    NUMBER = {2},
-     PAGES = {213--231, 233--269},
-      ISSN = {1022-1824},
-     CODEN = {SMATF6},
-   MRCLASS = {17B62 (16W35 17B37 18D20)},
-  MRNUMBER = {MR1669953 (2000i:17033)},
-MRREVIEWER = {Benjamin David Enriquez},
-      note = {arXiv:\arxiv{q-alg/9701038}},
-}
-
-@article {MR1771217,
-    AUTHOR = {Etingof, Pavel and Kazhdan, David},
-     TITLE = {Quantization of {L}ie bialgebras. {IV}. {T}he coinvariant
-              construction and the quantum {KZ} equations},
-   JOURNAL = {Selecta Math. (N.S.)},
-  FJOURNAL = {Selecta Mathematica. New Series},
-    VOLUME = {6},
-      YEAR = {2000},
-    NUMBER = {1},
-     PAGES = {79--104},
-      ISSN = {1022-1824},
-     CODEN = {SMATF6},
-   MRCLASS = {17B37 (32G34 81R50)},
-  MRNUMBER = {MR1771217 (2002i:17021)},
-}
-
-@article {MR1771218,
-    AUTHOR = {Etingof, Pavel and Kazhdan, David},
-     TITLE = {Quantization of {L}ie bialgebras. {V}. {Q}uantum vertex
-              operator algebras},
-   JOURNAL = {Selecta Math. (N.S.)},
-  FJOURNAL = {Selecta Mathematica. New Series},
-    VOLUME = {6},
-      YEAR = {2000},
-    NUMBER = {1},
-     PAGES = {105--130},
-      ISSN = {1022-1824},
-     CODEN = {SMATF6},
-   MRCLASS = {17B37 (17B69 81R50)},
-  MRNUMBER = {MR1771218 (2002i:17022)},
-}
-
-@article {MR2253455,
-    AUTHOR = {Bar-Natan, Dror and Morrison, Scott},
-     TITLE = {The {K}aroubi envelope and {L}ee's degeneration of {K}hovanov
-              homology},
-   JOURNAL = {Algebr. Geom. Topol.},
-  FJOURNAL = {Algebraic \& Geometric Topology},
-    VOLUME = {6},
-      YEAR = {2006},
-     PAGES = {1459--1469 (electronic)},
-      ISSN = {1472-2747},
-   MRCLASS = {57M27 (18E05 57M25)},
-  MRNUMBER = {MR2253455},
-      note = {arXiv:\arxiv{math.GT/0606542}},
-}
-
-
-@article{math.GT/0206303,
-    title = {{An invariant of link cobordisms from Khovanov homology}},
-    author = {Magnus Jacobsson},
-    journal = {Algebr. Geom. Topol.},
-    volume = 4,
-    year = 2004,
-    pages = {1211--1251},
-    note = {arXiv:\arxiv{math.GT/0206303}}}
-
-@misc{math.GT/0610650,
-    title = {{Khovanov-Rozansky homology via a canopolis formalism}},
-    author = {Ben Webster},
-      note = {arXiv:\arxiv{math.GT/0610650}}}
-
-@misc{green-implementation,
-    title = {{JavaKh}},
-    author= {Jeremey Green},
-    note = {\url{http://katlas.math.toronto.edu/wiki/Khovanov_Homology}},
-}
-
-@misc{ortiz-navarro,
-    title = {{Khovanov Homology and Reidemeister Torsion}},
-    author ={Juan Ariel Ortiz-Navarro and Chris Truman},
-    note = {a talk at the 2006 Toronto CMS meeting, slides at \url{http://www.math.uiowa.edu/~jortizna/Present-CMS-06.pdf}},
-}
-
-@book {MR1438306,
-    AUTHOR = {Gelfand, Sergei I. and Manin, Yuri I.},
-     TITLE = {Methods of homological algebra},
-      NOTE = {Translated from the 1988 Russian original},
- PUBLISHER = {Springer-Verlag},
-   ADDRESS = {Berlin},
-      YEAR = {1996},
-     PAGES = {xviii+372},
-      ISBN = {3-540-54746-0},
-   MRCLASS = {18-02 (18Exx 18Gxx 55U35)},
-  MRNUMBER = {MR1438306 (97j:18001)},
-}
+@PREAMBLE{ "\newcommand{\noopsort}[1]{}" }
+
+@STRING{CUP = {Cambridge University Press}}
+@STRING{OUP = {Oxford University Press}}
+@STRING{PUP = {Princeton University Press}}
+@STRING{SV = {Springer-Verlag}}
+@STRING{AP = {Academic Press}}
+@STRING{AMS = {American Mathematical Society}}
+
+%S! fix this citation!
+@article{morrison-walker,
+    AUTHOR = {David Clark and Scott Morrison and Kevin Walker},
+    TITLE = {Fixing the functoriality of Khovanov homology},
+     note = {\arxiv{math.GT/0701339}},
+}
+
+@misc{kw:tqft,
+    AUTHOR = {Walker, Kevin},
+    TITLE  = {Topological Quantum Field Theories},
+    note    = {draft available at \url{http://canyon23.net/math/tc.pdf}},
+}
+
+@article {MR1854636,
+    AUTHOR = {Keller, Bernhard},
+     TITLE = {Introduction to {$A$}-infinity algebras and modules},
+   JOURNAL = {Homology Homotopy Appl.},
+  FJOURNAL = {Homology, Homotopy and Applications},
+    VOLUME = {3},
+      YEAR = {2001},
+    NUMBER = {1},
+     PAGES = {1--35 (electronic)},
+      ISSN = {1512-0139},
+   MRCLASS = {18E30 (18D50 18G40 55P43 55U35)},
+  MRNUMBER = {MR1854636 (2004a:18008a)},
+MRREVIEWER = {Ulrike Tillmann},
+      note = {\mathscinet{MR1854636} \arxiv{math.RA/9910179}},
+}
+
+@incollection {MR2061854,
+    AUTHOR = {McClure, James E. and Smith, Jeffrey H.},
+     TITLE = {Operads and cosimplicial objects: an introduction},
+ BOOKTITLE = {Axiomatic, enriched and motivic homotopy theory},
+    SERIES = {NATO Sci. Ser. II Math. Phys. Chem.},
+    VOLUME = {131},
+     PAGES = {133--171},
+ PUBLISHER = {Kluwer Acad. Publ.},
+   ADDRESS = {Dordrecht},
+      YEAR = {2004},
+   MRCLASS = {55P48 (18D50)},
+  MRNUMBER = {MR2061854 (2005b:55018)},
+MRREVIEWER = {David Chataur},
+      note = {\mathscinet{MR2061854} \arxiv{math.QA/0402117}},
+}
+
+@book {MR0420610,
+    AUTHOR = {May, J. P.},
+     TITLE = {The geometry of iterated loop spaces},
+ PUBLISHER = {Springer-Verlag},
+   ADDRESS = {Berlin},
+      YEAR = {1972},
+     PAGES = {viii+175},
+   MRCLASS = {55D35},
+  MRNUMBER = {MR0420610 (54 \#8623b)},
+MRREVIEWER = {J. Stasheff},
+      note = {Lectures Notes in Mathematics, Vol. 271 \mathscinet{MR0420610} \href{http://www.math.uchicago.edu/~may/BOOKS/gils.pdf}{available online}},
+}
+
+@article {MR0236922,
+    AUTHOR = {Boardman, J. M. and Vogt, R. M.},
+     TITLE = {Homotopy-everything {$H$}-spaces},
+   JOURNAL = {Bull. Amer. Math. Soc.},
+  FJOURNAL = {Bulletin of the American Mathematical Society},
+    VOLUME = {74},
+      YEAR = {1968},
+     PAGES = {1117--1122},
+      ISSN = {0002-9904},
+   MRCLASS = {55.42},
+  MRNUMBER = {MR0236922 (38 \#5215)},
+MRREVIEWER = {R. J. Milgram},
+      note = {\mathscinet{MR0236922} \doi{10.1090/S0002-9904-1968-12070-1}},
+}
+
+@book {MR0420609,
+    AUTHOR = {Boardman, J. M. and Vogt, R. M.},
+     TITLE = {Homotopy invariant algebraic structures on topological spaces},
+    SERIES = {Lecture Notes in Mathematics, Vol. 347},
+ PUBLISHER = {Springer-Verlag},
+   ADDRESS = {Berlin},
+      YEAR = {1973},
+     PAGES = {x+257},
+   MRCLASS = {55D35},
+  MRNUMBER = {MR0420609 (54 \#8623a)},
+MRREVIEWER = {J. Stasheff},
+      note = {\mathscinet{MR0420609}},
+}
+
+%The framed little discs operad:
+@article {MR1256989,
+    AUTHOR = {Getzler, E.},
+     TITLE = {Batalin-{V}ilkovisky algebras and two-dimensional topological
+              field theories},
+   JOURNAL = {Comm. Math. Phys.},
+  FJOURNAL = {Communications in Mathematical Physics},
+    VOLUME = {159},
+      YEAR = {1994},
+    NUMBER = {2},
+     PAGES = {265--285},
+      ISSN = {0010-3616},
+     CODEN = {CMPHAY},
+   MRCLASS = {81T70 (17B81 55Q99 58Z05 81T40)},
+  MRNUMBER = {MR1256989 (95h:81099)},
+MRREVIEWER = {J. Stasheff},
+      note = {\mathscinet{MR1256989} \euclid{1104254599}},
+}
+
+
+
+
+
+@article {MR1917056,
+    AUTHOR = {Bar-Natan, Dror},
+     TITLE = {On {K}hovanov's categorification of the {J}ones polynomial},
+   JOURNAL = {Algebr. Geom. Topol.},
+  FJOURNAL = {Algebraic \& Geometric Topology},
+    VOLUME = {2},
+      YEAR = {2002},
+     PAGES = {337--370 (electronic)},
+      ISSN = {1472-2747},
+   MRCLASS = {57M27},
+  MRNUMBER = {MR1917056 (2003h:57014)},
+MRREVIEWER = {Jacob Andrew Rasmussen},
+      note = {arXiv:\arxiv{math.QA/0201043}},
+}
+
+@incollection {MR2147420,
+    AUTHOR = {Bar-Natan, Dror},
+     TITLE = {Khovanov homology for knots and links with up to 11 crossings},
+ BOOKTITLE = {Advances in topological quantum field theory},
+    SERIES = {NATO Sci. Ser. II Math. Phys. Chem.},
+    VOLUME = {179},
+     PAGES = {167--241},
+ PUBLISHER = {Kluwer Acad. Publ.},
+   ADDRESS = {Dordrecht},
+      YEAR = {2004},
+   MRCLASS = {57M27},
+  MRNUMBER = {MR2147420 (2006c:57009)},
+MRREVIEWER = {Marta M. Asaeda},
+}
+
+@article {MR2174270,
+    AUTHOR = {Bar-Natan, Dror},
+     TITLE = {Khovanov's homology for tangles and cobordisms},
+   JOURNAL = {Geom. Topol.},
+  FJOURNAL = {Geometry and Topology},
+    VOLUME = {9},
+      YEAR = {2005},
+     PAGES = {1443--1499 (electronic)},
+      ISSN = {1465-3060},
+   MRCLASS = {57M27 (57M25)},
+  MRNUMBER = {MR2174270},
+      note = {arXiv:\arxiv{math.GT/0410495}},
+}
+
+@article {MR1680395,
+    AUTHOR = {Khovanov, Mikhail and Kuperberg, Greg},
+     TITLE = {Web bases for {${\rm sl}(3)$} are not dual canonical},
+   JOURNAL = {Pacific J. Math.},
+  FJOURNAL = {Pacific Journal of Mathematics},
+    VOLUME = {188},
+      YEAR = {1999},
+    NUMBER = {1},
+     PAGES = {129--153},
+      ISSN = {0030-8730},
+     CODEN = {PJMAAI},
+   MRCLASS = {17B37 (22E60 57M27 81R05)},
+  MRNUMBER = {MR1680395 (2000j:17023a)},
+MRREVIEWER = {Robert J. Marsh},
+      note = {arXiv:\arxiv{q-alg/9712046}},
+}
+
+@article {MR1740682,
+    AUTHOR = {Khovanov, Mikhail},
+     TITLE = {A categorification of the {J}ones polynomial},
+   JOURNAL = {Duke Math. J.},
+  FJOURNAL = {Duke Mathematical Journal},
+    VOLUME = {101},
+      YEAR = {2000},
+    NUMBER = {3},
+     PAGES = {359--426},
+      ISSN = {0012-7094},
+     CODEN = {DUMJAO},
+   MRCLASS = {57M27 (57R56)},
+  MRNUMBER = {MR1740682 (2002j:57025)},
+}
+
+@article {MR1928174,
+    AUTHOR = {Khovanov, Mikhail},
+     TITLE = {A functor-valued invariant of tangles},
+   JOURNAL = {Algebr. Geom. Topol.},
+  FJOURNAL = {Algebraic \& Geometric Topology},
+    VOLUME = {2},
+      YEAR = {2002},
+     PAGES = {665--741 (electronic)},
+      ISSN = {1472-2747},
+   MRCLASS = {57M27 (57R56)},
+  MRNUMBER = {MR1928174 (2004d:57016)},
+MRREVIEWER = {Jacob Andrew Rasmussen},
+    note   = {arXiv:\arxiv{math.GT/0103190}},
+}
+
+@article {MR2034399,
+    AUTHOR = {Khovanov, Mikhail},
+     TITLE = {Patterns in knot cohomology. {I}},
+   JOURNAL = {Experiment. Math.},
+  FJOURNAL = {Experimental Mathematics},
+    VOLUME = {12},
+      YEAR = {2003},
+    NUMBER = {3},
+     PAGES = {365--374},
+      ISSN = {1058-6458},
+   MRCLASS = {57M27 (18G60 57M25 57R56)},
+  MRNUMBER = {MR2034399 (2004m:57022)},
+MRREVIEWER = {Jacob Andrew Rasmussen},
+}
+
+@article {MR2100691,
+    AUTHOR = {Khovanov, Mikhail},
+     TITLE = {sl(3) link homology},
+   JOURNAL = {Algebr. Geom. Topol.},
+  FJOURNAL = {Algebraic \& Geometric Topology},
+    VOLUME = {4},
+      YEAR = {2004},
+     PAGES = {1045--1081 (electronic)},
+      ISSN = {1472-2747},
+   MRCLASS = {57M27 (18G60 57R56)},
+  MRNUMBER = {MR2100691 (2005g:57032)},
+MRREVIEWER = {Justin Sawon},
+      note = {arXiv:\arxiv{math.QA/0304375}},
+}
+
+@article {MR2124557,
+    AUTHOR = {Khovanov, Mikhail},
+     TITLE = {Categorifications of the colored {J}ones polynomial},
+   JOURNAL = {J. Knot Theory Ramifications},
+  FJOURNAL = {Journal of Knot Theory and its Ramifications},
+    VOLUME = {14},
+      YEAR = {2005},
+    NUMBER = {1},
+     PAGES = {111--130},
+      ISSN = {0218-2165},
+   MRCLASS = {57M27},
+  MRNUMBER = {MR2124557 (2006a:57016)},
+MRREVIEWER = {Marta M. Asaeda},
+}
+
+@article {MR2171235,
+    AUTHOR = {Khovanov, Mikhail},
+     TITLE = {An invariant of tangle cobordisms},
+   JOURNAL = {Trans. Amer. Math. Soc.},
+  FJOURNAL = {Transactions of the American Mathematical Society},
+    VOLUME = {358},
+      YEAR = {2006},
+    NUMBER = {1},
+     PAGES = {315--327 (electronic)},
+      ISSN = {0002-9947},
+     CODEN = {TAMTAM},
+   MRCLASS = {57Q45},
+  MRNUMBER = {MR2171235},
+      note = {arXiv:\arxiv{math.GT/0207264}},
+}
+
+@misc{math.QA/0401268,
+    title = {{Matrix factorizations and link homology}},
+    author = {Mikhail Khovanov and Lev Rozansky},
+      note = {arXiv:\arxiv{math.QA/0401268}}}
+
+@misc{math.QA/0505056,
+    title = {{Matrix factorizations and link homology II}},
+    author = {Mikhail Khovanov and Lev Rozansky},
+      note = {arXiv:\arxiv{math.QA/0505056}}}
+
+@incollection {MR2048108,
+    AUTHOR = {Thurston, Dylan P.},
+     TITLE = {The algebra of knotted trivalent graphs and {T}uraev's shadow
+              world},
+ BOOKTITLE = {Invariants of knots and 3-manifolds (Kyoto, 2001)},
+    SERIES = {Geom. Topol. Monogr.},
+    VOLUME = {4},
+     PAGES = {337--362 (electronic)},
+ PUBLISHER = {Geom. Topol. Publ., Coventry},
+      YEAR = {2002},
+   MRCLASS = {57M25 (57M15 57M20 57Q40)},
+  MRNUMBER = {MR2048108 (2005c:57010)},
+MRREVIEWER = {Marko Kranjc},
+      note = {arXiv:\arxiv{math.GT/0311458}},
+}
+
+@article {MR1403861,
+    AUTHOR = {Kuperberg, Greg},
+     TITLE = {Spiders for rank {$2$} {L}ie algebras},
+   JOURNAL = {Comm. Math. Phys.},
+  FJOURNAL = {Communications in Mathematical Physics},
+    VOLUME = {180},
+      YEAR = {1996},
+    NUMBER = {1},
+     PAGES = {109--151},
+      ISSN = {0010-3616},
+     CODEN = {CMPHAY},
+   MRCLASS = {17B10 (22E60 81R05)},
+  MRNUMBER = {MR1403861 (97f:17005)},
+MRREVIEWER = {Stefano Capparelli},
+      note = {arXiv:\arxiv{q-alg/9712003}},
+}
+
+@misc{math.GT/0606318,
+    title  = {{Fast Khovanov Homology Computations}},
+    author = {Dror Bar-Natan},
+    note   = {arXiv:\arxiv{math.GT/0606318}},
+    url    = {http://www.math.toronto.edu/~drorbn/papers/FastKh/}
+    }
+
+@misc{math.GT/0603347,
+    title = {{On the Algebraic Structure of Bar-Natan's Universal
+        Geometric Complex and the Geometric Structure of Khovanov
+        Link Homology Theories}},
+    author = {Gad Naot},
+    note   = {arXiv:\arxiv{math.GT/0603347}}}
+
+@misc{math.QA/9909027,
+    title = {{Planar algebras, I}},
+    author = {Vaughan F. R. Jones},
+    note   = {arXiv:\arxiv{math.QA/9909027}}}
+
+@article {MR1217386,
+    AUTHOR = {Turaev, V. and Wenzl, H.},
+     TITLE = {Quantum invariants of {$3$}-manifolds associated with
+              classical simple {L}ie algebras},
+   JOURNAL = {Internat. J. Math.},
+  FJOURNAL = {International Journal of Mathematics},
+    VOLUME = {4},
+      YEAR = {1993},
+    NUMBER = {2},
+     PAGES = {323--358},
+      ISSN = {0129-167X},
+   MRCLASS = {57M25 (17B37 57N10)},
+  MRNUMBER = {MR1217386 (94i:57019)},
+MRREVIEWER = {Toshitake Kohno},
+}
+
+@article {MR1470857,
+    AUTHOR = {Wenzl, Hans},
+     TITLE = {{$C\sp *$} tensor categories from quantum groups},
+   JOURNAL = {J. Amer. Math. Soc.},
+  FJOURNAL = {Journal of the American Mathematical Society},
+    VOLUME = {11},
+      YEAR = {1998},
+    NUMBER = {2},
+     PAGES = {261--282},
+      ISSN = {0894-0347},
+   MRCLASS = {46L89 (17B37 46L05)},
+  MRNUMBER = {MR1470857 (98k:46123)},
+MRREVIEWER = {Andrzej Sitarz},
+    eprint = {\url{http://ftp.ams.org/jams/1998-11-02/S0894-0347-98-00253-7/S0894-0347-98-00253-7.pdf}},
+}
+
+@misc{math.GT/0603307,
+    title  = {{The universal sl3-link homology}},
+    author = {Marco Mackaay and Pedro Vaz},
+    note  = {arXiv:\arxiv{math.GT/0603307}}}
+
+ @misc{wiki:Grothendieck-group,
+   author = "Wikipedia",
+   title = "Grothendieck group --- Wikipedia{,} The Free Encyclopedia",
+   year = "2006",
+   note = "[\href{http://en.wikipedia.org/w/index.php?title=Grothendieck_group&oldid=52451663}{Online}; accessed 30-June-2006]"
+ }
+
+ @misc{wiki:Invariant-basis-number,
+   author = "Wikipedia",
+   title = "Invariant basis number --- Wikipedia{,} The Free Encyclopedia",
+   year = "2006",
+   note = "[\href{http://en.wikipedia.org/w/index.php?title=Invariant_basis_number&oldid=85220456}{Online}; accessed 24-December-2006]"
+ }
+
+@article {MR1403351,
+    AUTHOR = {Etingof, Pavel and Kazhdan, David},
+     TITLE = {Quantization of {L}ie bialgebras. {I}},
+   JOURNAL = {Selecta Math. (N.S.)},
+  FJOURNAL = {Selecta Mathematica. New Series},
+    VOLUME = {2},
+      YEAR = {1996},
+    NUMBER = {1},
+     PAGES = {1--41},
+      ISSN = {1022-1824},
+     CODEN = {SMATF6},
+   MRCLASS = {17B37 (16W30 18D20 81R50)},
+  MRNUMBER = {MR1403351 (97f:17014)},
+MRREVIEWER = {Yu. N. Bespalov},
+}
+
+@article {MR1669953,
+    AUTHOR = {Etingof, Pavel and Kazhdan, David},
+     TITLE = {Quantization of {L}ie bialgebras. {II}},
+   JOURNAL = {Selecta Math. (N.S.)},
+  FJOURNAL = {Selecta Mathematica. New Series},
+    VOLUME = {4},
+      YEAR = {1998},
+    NUMBER = {2},
+     PAGES = {213--231, 233--269},
+      ISSN = {1022-1824},
+     CODEN = {SMATF6},
+   MRCLASS = {17B62 (16W35 17B37 18D20)},
+  MRNUMBER = {MR1669953 (2000i:17033)},
+MRREVIEWER = {Benjamin David Enriquez},
+      note = {arXiv:\arxiv{q-alg/9701038}},
+}
+
+@article {MR1771217,
+    AUTHOR = {Etingof, Pavel and Kazhdan, David},
+     TITLE = {Quantization of {L}ie bialgebras. {IV}. {T}he coinvariant
+              construction and the quantum {KZ} equations},
+   JOURNAL = {Selecta Math. (N.S.)},
+  FJOURNAL = {Selecta Mathematica. New Series},
+    VOLUME = {6},
+      YEAR = {2000},
+    NUMBER = {1},
+     PAGES = {79--104},
+      ISSN = {1022-1824},
+     CODEN = {SMATF6},
+   MRCLASS = {17B37 (32G34 81R50)},
+  MRNUMBER = {MR1771217 (2002i:17021)},
+}
+
+@article {MR1771218,
+    AUTHOR = {Etingof, Pavel and Kazhdan, David},
+     TITLE = {Quantization of {L}ie bialgebras. {V}. {Q}uantum vertex
+              operator algebras},
+   JOURNAL = {Selecta Math. (N.S.)},
+  FJOURNAL = {Selecta Mathematica. New Series},
+    VOLUME = {6},
+      YEAR = {2000},
+    NUMBER = {1},
+     PAGES = {105--130},
+      ISSN = {1022-1824},
+     CODEN = {SMATF6},
+   MRCLASS = {17B37 (17B69 81R50)},
+  MRNUMBER = {MR1771218 (2002i:17022)},
+}
+
+@article {MR2253455,
+    AUTHOR = {Bar-Natan, Dror and Morrison, Scott},
+     TITLE = {The {K}aroubi envelope and {L}ee's degeneration of {K}hovanov
+              homology},
+   JOURNAL = {Algebr. Geom. Topol.},
+  FJOURNAL = {Algebraic \& Geometric Topology},
+    VOLUME = {6},
+      YEAR = {2006},
+     PAGES = {1459--1469 (electronic)},
+      ISSN = {1472-2747},
+   MRCLASS = {57M27 (18E05 57M25)},
+  MRNUMBER = {MR2253455},
+      note = {arXiv:\arxiv{math.GT/0606542}},
+}
+
+
+@article{math.GT/0206303,
+    title = {{An invariant of link cobordisms from Khovanov homology}},
+    author = {Magnus Jacobsson},
+    journal = {Algebr. Geom. Topol.},
+    volume = 4,
+    year = 2004,
+    pages = {1211--1251},
+    note = {arXiv:\arxiv{math.GT/0206303}}}
+
+@misc{math.GT/0610650,
+    title = {{Khovanov-Rozansky homology via a canopolis formalism}},
+    author = {Ben Webster},
+      note = {arXiv:\arxiv{math.GT/0610650}}}
+
+@misc{green-implementation,
+    title = {{JavaKh}},
+    author= {Jeremey Green},
+    note = {\url{http://katlas.math.toronto.edu/wiki/Khovanov_Homology}},
+}
+
+@misc{ortiz-navarro,
+    title = {{Khovanov Homology and Reidemeister Torsion}},
+    author ={Juan Ariel Ortiz-Navarro and Chris Truman},
+    note = {a talk at the 2006 Toronto CMS meeting, slides at \url{http://www.math.uiowa.edu/~jortizna/Present-CMS-06.pdf}},
+}
+
+@book {MR1438306,
+    AUTHOR = {Gelfand, Sergei I. and Manin, Yuri I.},
+     TITLE = {Methods of homological algebra},
+      NOTE = {Translated from the 1988 Russian original},
+ PUBLISHER = {Springer-Verlag},
+   ADDRESS = {Berlin},
+      YEAR = {1996},
+     PAGES = {xviii+372},
+      ISBN = {3-540-54746-0},
+   MRCLASS = {18-02 (18Exx 18Gxx 55U35)},
+  MRNUMBER = {MR1438306 (97j:18001)},
+}
+
+@article {MR1624157,
+    AUTHOR = {Bullock, Doug and Frohman, Charles and Kania-Bartoszy{\'n}ska,
+              Joanna},
+     TITLE = {Skein homology},
+   JOURNAL = {Canad. Math. Bull.},
+  FJOURNAL = {Canadian Mathematical Bulletin. Bulletin Canadien de
+              Math\'ematiques},
+    VOLUME = {41},
+      YEAR = {1998},
+    NUMBER = {2},
+     PAGES = {140--144},
+      ISSN = {0008-4395},
+     CODEN = {CMBUA3},
+   MRCLASS = {57M25 (55N35)},
+  MRNUMBER = {MR1624157 (99g:57006)},
+MRREVIEWER = {Daniel Ruberman},
+ note = {\arxiv{q-alg/9701019} \mathscinet{MR1624157}},
+}
+	

--- a/blob1.tex	Tue May 05 17:27:21 2009 +0000
+++ b/blob1.tex	Sun May 24 20:30:45 2009 +0000
@@ -112,6 +112,7 @@
 \begin{itemize}
 \item Derive Hochschild standard results from blob point of view?
 \item Kh
+\item Mention somewhere \cite{MR1624157} ``Skein homology''; it's not directly related, but has similar motivations.
 \end{itemize}
 \end{itemize}
 
@@ -293,11 +294,10 @@
 unoriented, topological, smooth, spin, etc. --- but for definiteness we
 will stick with oriented PL.)
 
-Fix a top dimension $n$.
+Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$.
 
-A {\it system of fields}
-is a collection of functors $\cC_k$, for $k \le n$, from $\cM_k$ to the
-category of sets,
+A $n$-dimensional {\it system of fields} in $\cS$
+is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$
 together with some additional data and satisfying some additional conditions, all specified below.
 
 \nn{refer somewhere to my TQFT notes \cite{kw:tqft}, and possibly also to paper with Chris}
@@ -322,11 +322,12 @@
 For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of 
 $\cC(X)$ which restricts to $c$.
 In this context, we will call $c$ a boundary condition.
+\item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields must be morphisms in $\cS$.
 \item There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps
 again comprise a natural transformation of functors.
 In addition, the orientation reversal maps are compatible with the boundary restriction maps.
 \item $\cC_k$ is compatible with the symmetric monoidal
-structures on $\cM_k$ and sets: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$,
+structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$,
 compatibly with homeomorphisms, restriction to boundary, and orientation reversal.
 We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$
 restriction maps.
@@ -400,16 +401,16 @@
 
 \nn{should also say something about pseudo-isotopy}
 
-\bigskip
-\hrule
-\bigskip
-
-\input{text/fields.tex}
-
-
-\bigskip
-\hrule
-\bigskip
+%\bigskip
+%\hrule
+%\bigskip
+%
+%\input{text/fields.tex}
+%
+%
+%\bigskip
+%\hrule
+%\bigskip
 
 \nn{note: probably will suppress from notation the distinction
 between fields and their (orientation-reversal) duals}
@@ -726,7 +727,7 @@
 \]
 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above.
 $\overline{c}$ runs over all boundary conditions, again as described above.
-$j$ runs over all indices of twig blobs.
+$j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are cuttable along all of the blobs in $\overline{B}$.
 
 The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows.
 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram.

--- a/build.xml	Tue May 05 17:27:21 2009 +0000
+++ b/build.xml	Sun May 24 20:30:45 2009 +0000
@@ -87,8 +87,8 @@
     </target>
     
     <target name="scott-copy-pdf" depends="pdf">
-        <copy file="blob1.pdf" tofile="../../hosts/tqft.net/papers/blobs.pdf"/>
-        <exec executable="svn" dir="../../hosts/tqft.net/papers/">
+        <copy file="blob1.pdf" tofile="../../Sites/tqft.net/papers/blobs.pdf"/>
+        <exec executable="svn" dir="../../Sites/tqft.net/papers/">
             <arg value="commit"/>
             <arg value="-m"/>
             <arg value="blob"/>

--- a/preamble.tex	Tue May 05 17:27:21 2009 +0000
+++ b/preamble.tex	Sun May 24 20:30:45 2009 +0000
@@ -146,6 +146,7 @@
 
 \newcommand{\Set}{\text{\textbf{Set}}}
 \newcommand{\Vect}{\text{\textbf{Vect}}}
+\newcommand{\Kom}{\text{\textbf{Kom}}}
 \newcommand{\Cat}{\mathcal{C}}
 
 \newcommand{\psmallmatrix}[1]{\left(\begin{smallmatrix} #1 \end{smallmatrix}\right)}

--- a/text/A-infty.tex	Tue May 05 17:27:21 2009 +0000
+++ b/text/A-infty.tex	Sun May 24 20:30:45 2009 +0000
@@ -1,25 +1,31 @@
 \section{Homological systems of fields}
+\label{sec:homological-fields}
 
 In this section, we extend the definition of blob homology to allow \emph{homological systems of fields}.
 
-We begin with a definition of a \emph{topological $A_\infty$ category}, and then introduce the notion of a homological system of fields. A topological $A_\infty$ category gives a $1$-dimensional homological system of fields. We'll suggest that any good definition of a topological $A_\infty$ $n$-category with duals should allow construction of an $n$-dimensional homological system of fields, but we won't propose any such definition here. Later, we extend the definition of blob homology to allow homological fields as input. These definitions allow us to state and prove a theorem about the blob homology of a product manifold, and an intermediate theorem about gluing, in preparation for the proof of \ref{thm:gluing}.
+We begin with a definition of a \emph{topological $A_\infty$ category}, and then introduce the notion of a homological system of fields. A topological $A_\infty$ category gives a $1$-dimensional homological system of fields. We'll suggest that any good definition of a topological $A_\infty$ $n$-category with duals should allow construction of an $n$-dimensional homological system of fields, but we won't propose any such definition here. Later, we extend the definition of blob homology to allow homological fields as input. These definitions allow us to state and prove a theorem about the blob homology of a product manifold, and an intermediate theorem about gluing, in preparation for the proof of Property \ref{property:gluing}.
 
 \subsection{Topological $A_\infty$ categories}
 
-First recall the \emph{coloured little intervals operad}. Given a set of labels $\cL$, the operations are indexed by \emph{decompositions of the interval}, each of which is a collection of disjoint subintervals $\{(a_i,b_i)\}_{i=1}^k$ of $[0,1]$, along with a labeling of the complementary regions by $\cL$, $\{l_0, \ldots, l_k\}$.  Given two decompositions $\cJ_1$ and $\cJ_2$, and an index $m$ such that $l^{(1)}_{m-1} = l^{(2)}_0$ and $l^{(1)}_{m} = l^{(2)}_{k^{(2)}}$, we can form a new decomposition by inserting the intervals of $\cJ_2$ linearly inside the $m$-th interval of $\cJ_1$. We call the resulting decomposition $\cJ_1 \circ_m \cJ_2$.
+First recall the \emph{coloured little intervals operad}. Given a set of labels $\cL$, the operations are indexed by \emph{decompositions of the interval}, each of which is a collection of disjoint subintervals $\{(a_i,b_i)\}_{i=1}^k$ of $[0,1]$, along with a labeling of the complementary regions by $\cL$, $\{l_0, \ldots, l_k\}$.  Given two decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$ such that $l^{(1)}_{m-1} = l^{(2)}_0$ and $l^{(1)}_{m} = l^{(2)}_{k^{(2)}}$, we can form a new decomposition by inserting the intervals of $\cJ^{(2)}$ linearly inside the $m$-th interval of $\cJ^{(1)}$. We call the resulting decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$.
 
 \begin{defn}
-A \emph{topological $A_\infty$ category} $\cC$ has a set of objects $\Obj(\cC)$ and for each $a,b \in \Obj(\cC)$ a chain complex $\cC_{a,b}$, along with a compatible `composition map' and `action of families of diffeomorphisms'.
+A \emph{topological $A_\infty$ category} $\cC$ has a set of objects $\Obj(\cC)$ and for each $a,b \in \Obj(\cC)$ a chain complex $\cC_{a,b}$, along with a compatible `composition map' and an `action of families of diffeomorphisms'.
 
 A \emph{composition map} $f$ is a family of chain maps, one for each decomposition of the interval, $f_\cJ : A^{\tensor k} \to A$, making $\cC$ into a category over the coloured little intervals operad, with labels $\cL = \Obj(\cC)$. Thus the chain maps satisfy the identity 
 \begin{equation*}
-f_{\cJ_1 \circ_m \cJ_2} = f_{\cJ_1} \circ (\id^{\tensor m-1} \tensor f_{\cJ_2} \tensor \id^{\tensor k^{(1)} - m}).
+f_{\cJ^{(1)} \circ_m \cJ^{(2)}} = f_{\cJ^{(1)}} \circ (\id^{\tensor m-1} \tensor f_{\cJ^{(2)}} \tensor \id^{\tensor k^{(1)} - m}).
 \end{equation*}
 
 An \emph{action of families of diffeomorphisms} is a chain map $ev: \CD{[0,1]} \tensor A \to A$, such that \todo{What goes here, if anything?} 
 \begin{enumerate}
 \item The diagram 
-\todo{}
+\begin{equation*}
+\xymatrix{
+\CD{[0,1]} \tensor \CD{[0,1]} \tensor A \ar[r]^{\id \tensor ev} \ar[d]^{\circ \tensor \id} & \CD{[0,1]} \tensor A \ar[d]^{ev} \\
+\CD{[0,1]} \tensor A \ar[r]^{ev} & A
+}
+\end{equation*}
 commutes up to weakly unique \todo{???} homotopy.
 \item If $\phi \in \Diff([0,1])$ and $\cJ$ is a decomposition of the interval, we obtain a new decomposition $\phi(\cJ)$ and a collection $\phi_m \in \Diff([0,1])$ of diffeomorphisms obtained by taking the restrictions $\restrict{\phi}{[a_m,b_m]} : [a_m,b_m] \to [\phi(a_m),\phi(b_m)]$ and pre- and post-composing these with the linear diffeomorphisms $[0,1] \to [a_m,b_m]$ and $[\phi(a_m),\phi(b_m)] \to [0,1]$. We require that
 \begin{equation*}
@@ -41,12 +47,12 @@
 \intertext{and thus that}
 m_1 \circ m_3 & =  m_2 \circ (\id \tensor m_2) - m_2 \circ (m_2 \tensor \id) - \\ & \qquad - m_3 \circ (m_1 \tensor \id \tensor \id) - m_3 \circ (\id \tensor m_1 \tensor \id) - m_3 \circ (\id \tensor \id \tensor m_1)
 \end{align*}
-as required (c.f. \cite[p. 6]{MR1854636}.
+as required (c.f. \cite[p. 6]{MR1854636}).
 \todo{then the general case.}
 We won't describe a reverse construction (producing a topological $A_\infty$ category from a `conventional' $A_\infty$ category), but we presume that this will be easy for the experts.
 
 \subsection{Homological systems of fields}
-\todo{Describe homological fields}
+A homological system of fields $\cF$ is nothing more than a system of fields in the category $\Kom$ of complexes of vector spaces; that is, the set of top level fields with given boundary conditions is always a complex.
 
 A topological $A_\infty$ category $\cC$ gives rise to a one dimensional homological system of fields. The functor $\cF_0$ simply assigns the set of objects of $\cC$ to a point. 
 For a $1$-manifold $X$, define a \emph{decomposition of $X$} with labels in $\cL$ as a (possibly empty) set of disjoint closed intervals $\{J\}$ in $X$, and a labeling of the complementary regions by elements of $\cL$.
@@ -79,13 +85,15 @@
 \cF^{\times F}(M) = \cB_*(M \times F, \cF).
 \end{equation*}
 \end{thm}
+We might suggestively write $\cF^{\times F}$ as  $\cB_*(F \times [0,1]^b, \cF)$, interpreting this as an (undefined!) $A_\infty$ $b$-category, and then as the resulting homological system of fields, following a recipe analogous to that given above for $A_\infty$ $1$-categories.
+
 
 In later sections, we'll prove the following two unsurprising theorems, about the (as-yet-undefined) blob homology of these homological systems of fields.
 
 
 \begin{thm}
 \begin{equation*}
-\cB_*(M, \CM{-}{X}) \iso \CM{M}{X}))
+\cB_*(M, \Xi) \iso \Xi(M)
 \end{equation*}
 \end{thm}
 
@@ -93,34 +101,41 @@
 Given a $b$-manifold $B$, an $f$-manifold $F$ and a $b+f$ dimensional system of fields,
 there is a quasi-isomorphism
 \begin{align*}
-\cB_*(B \times F, \cF) & \quismto \cB_*(B, \cF^{\times F}) \\
-\intertext{or suggestively}
-\cB_*(B \times F, \cF) & \quismto  \cB_*(B, \cB_*(F \times [0,1]^b, \cF)).
+\cB_*(B \times F, \cF) & \quismto \cB_*(B, \cF^{\times F})
 \end{align*}
-where on the right we intend $\cB_*(F \times [0,1]^b, \cF)$ to be interpreted as the homological system of fields coming from an (undefined) $A_\infty$ $b$-category.
 \end{thm}
 
 \begin{question}
 Is it possible to compute the blob homology of a non-trivial bundle in terms of the blob homology of its fiber?
 \end{question}
 
-
 \subsection{Blob homology}
 The definition of blob homology for $(\cF, \cU)$ a homological system of fields and local relations is essentially the same as that given before in \S \ref{???}, except now there are some extra terms in the differential accounting for the `internal' differential acting on the fields.
-The blob complex $\cB_*^{\cF,\cU}(M)$ is a doubly-graded vector space, with a `blob degree' and an `internal degree'. 
 
-We'll write $\cT$ for the set of finite rooted trees. We'll think of each such a rooted tree as a category, with vertices as objects  and each morphism set either empty or a singleton, with $v \to w$ if $w$ is closer to a root of the tree than $v$. We'll write $\hat{v}$ for the `parent' of a vertex $v$ if $v$ is not a root (that is, $\hat{v}$ is the unique vertex such that $v \to \hat{v}$ but there is no $w$ with $v \to w \to \hat{v}$. If $v$ is a root, we'll write $\hat{v}=\star$. Further, for each tree $t$, let's arbitrarily choose an orientation $\lambda_t$, that is, an alternating $\pm1$-valued function on orderings of the vertices.
+As before
+\begin{equation*}
+	\cB_*^{\cF,\cU}(M) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}}
+		\left( \otimes_j \cU(B_j; c_j)\right) \otimes \cF(M \setmin B^t; c^t)
+\end{equation*}
+with $\overline{B}$ running over configurations of blobs satisfying the usual conditions, and $\overline{c}$ running over all boundary conditions. This is a doubly-graded vector space, graded by blob degree (the number of blobs) and internal degree (the sum of the homological degrees of the tensor factor fields). It becomes a complex by taking the homological degree to the be the sum of the blob and internal degrees, and defining $d$ by
 
-Given $v \in t$ there's a functor $\partial_v : t \to t \setminus \{v\}$ which removes the vertex $v$. Notice that removing a vertex naturally produces an orientation on $t \setminus \{v\}$ from the orientation on $t$, by $(\partial_v \lambda_t)(o) = \lambda_t(vo)$. This orientation may or may not agree with the chosen orientation of $t \setminus \{v\}$. We'll define $\sigma(v \in t) = \pm 1$ according to whether or not they agree. Notice that $$\sigma(v \in t) \sigma(w \in \partial_v t) = - \sigma(w \in t) \sigma(v \in \partial_w t).$$
+\begin{equation*}
+d f = \sum_{v \in t} \partial_v f + \sum_{v' \in t \cup \{\star\}} d_{v'} f,
+\end{equation*}
+
 
-Let $\operatorname{balls}(M)$ denote the category of open balls in $M$ with inclusions. Given a tree $t \in \cT$ we'll call a functor $b : t \to \operatorname{balls}(M)$ such that if $b(v) \cap b(v') \neq \emptyset$) then either $v \to v'$ or $v' \to v$, \emph{non-intersecting}.\footnote{Equivalently, if $b(v)$ and $b(v')$ are spanned in $\operatorname{balls}(M)$, then $v$ and $v'$ are spanned in $t$. That is, if there exists some ball $B \subset M$ so $B \subset b(v)$ and $B \subset b(v')$, then there must exist some $v'' \in t$ so $v'' \to v$ and $v'' \to v'$. Because $t$ is a tree, this implies either $v \to v'$ or $v' \to v$} For each non-intersecting functor $b$ define  
-\begin{equation*}
-\cF(t,b) = \cF\left(M \setminus b(t)\right) \tensor \left(\Tensor_{\substack{v \in t \\ \text{$v$ not a leaf}}} \cF\left(b(v) \setminus b(v' \to v)\right)\right) \tensor \left(\Tensor_{\substack{v \in t \\ \text{$v$ a leaf}}} \cU\left(b(v)\right)\right)
-\end{equation*}
-and then the vector space
-\begin{equation*}
-\cB_*^{\cF,\cU}(M) = \DirectSum_{t \in \cT} \DirectSum_{\substack{\text{non-intersecting}\\\text{functors} \\ b: t \to \operatorname{balls}(M)}} \cF(t,b)
-\end{equation*}
+%We'll write $\cT$ for the set of finite rooted trees. We'll think of each such a rooted tree as a category, with vertices as objects  and each morphism set either empty or a singleton, with $v \to w$ if $w$ is closer to a root of the tree than $v$. We'll write $\hat{v}$ for the `parent' of a vertex $v$ if $v$ is not a root (that is, $\hat{v}$ is the unique vertex such that $v \to \hat{v}$ but there is no $w$ with $v \to w \to \hat{v}$. If $v$ is a root, we'll write $\hat{v}=\star$. Further, for each tree $t$, let's arbitrarily choose an orientation $\lambda_t$, that is, an alternating $\pm1$-valued function on orderings of the vertices.
+
+%Given $v \in t$ there's a functor $\partial_v : t \to t \setminus \{v\}$ which removes the vertex $v$. Notice that removing a vertex naturally produces an orientation on $t \setminus \{v\}$ from the orientation on $t$, by $(\partial_v \lambda_t)(o) = \lambda_t(vo)$. This orientation may or may not agree with the chosen orientation of $t \setminus \{v\}$. We'll define $\sigma(v \in t) = \pm 1$ according to whether or not they agree. Notice that $$\sigma(v \in t) \sigma(w \in \partial_v t) = - \sigma(w \in t) \sigma(v \in \partial_w t).$$
+
+%Let $\operatorname{balls}(M)$ denote the category of open balls in $M$ with inclusions. Given a tree $t \in \cT$ we'll call a functor $b : t \to \operatorname{balls}(M)$ such that if $b(v) \cap b(v') \neq \emptyset$) then either $v \to v'$ or $v' \to v$, \emph{non-intersecting}.\footnote{Equivalently, if $b(v)$ and $b(v')$ are spanned in $\operatorname{balls}(M)$, then $v$ and $v'$ are spanned in $t$. That is, if there exists some ball $B \subset M$ so $B \subset b(v)$ and $B \subset b(v')$, then there must exist some $v'' \in t$ so $v'' \to v$ and $v'' \to v'$. Because $t$ is a tree, this implies either $v \to v'$ or $v' \to v$} For each non-intersecting functor $b$ define  
+%\begin{equation*}
+%\cF(t,b) = \cF\left(M \setminus b(t)\right) \tensor \left(\Tensor_{\substack{v \in t \\ \text{$v$ not a leaf}}} \cF\left(b(v) \setminus b(v' \to v)\right)\right) \tensor \left(\Tensor_{\substack{v \in t \\ \text{$v$ a leaf}}} \cU\left(b(v)\right)\right)
+%\end{equation*}
+%and then the vector space
+%\begin{equation*}
+%\cB_*^{\cF,\cU}(M) = \DirectSum_{t \in \cT} \DirectSum_{\substack{\text{non-intersecting}\\\text{functors} \\ b: t \to \operatorname{balls}(M)}} \cF(t,b)
+%\end{equation*}
 
 The blob degree of an element of $\cF(t,b)$ is the number of vertices in $t$, and the internal degree is the sum of the homological degrees in the tensor factors.
 The vector space $\cB_*^{\cF,\cU}(M)$ becomes a chain complex by taking the homological degree to be the sum of the blob and internal degrees, and defining $d$ on $\cF(t,b)$ by

--- a/text/fields.tex	Tue May 05 17:27:21 2009 +0000
+++ b/text/fields.tex	Sun May 24 20:30:45 2009 +0000
@@ -1,3 +1,5 @@
+\nn{This file is obsolete.}
+
 \todo{beginning of scott's attempt to write down what fields are...}
 
 \newcommand{\manifolds}[1]{\cM_{#1}}