author | Kevin Walker <kevin@canyon23.net> |
Mon, 29 Nov 2010 10:01:34 -0700 | |
changeset 668 | a064476a3265 |
parent 667 | 0f45668726dd |
child 669 | c21da249a015 |
permissions | -rw-r--r-- |
566 | 1 |
%% PNAStmpl.tex |
2 |
%% Template file to use for PNAS articles prepared in LaTeX |
|
3 |
%% Version: Apr 14, 2008 |
|
4 |
||
5 |
||
6 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
7 |
%% BASIC CLASS FILE |
|
8 |
%% PNAStwo for two column articles is called by default. |
|
9 |
%% Uncomment PNASone for single column articles. One column class |
|
10 |
%% and style files are available upon request from pnas@nas.edu. |
|
11 |
%% (uncomment means get rid of the '%' in front of the command) |
|
12 |
||
13 |
%\documentclass{pnasone} |
|
14 |
\documentclass{pnastwo} |
|
15 |
||
16 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
17 |
%% Changing position of text on physical page: |
|
18 |
%% Since not all printers position |
|
19 |
%% the printed page in the same place on the physical page, |
|
20 |
%% you can change the position yourself here, if you need to: |
|
21 |
||
22 |
% \advance\voffset -.5in % Minus dimension will raise the printed page on the |
|
23 |
% physical page; positive dimension will lower it. |
|
24 |
||
25 |
%% You may set the dimension to the size that you need. |
|
26 |
||
27 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
28 |
%% OPTIONAL GRAPHICS STYLE FILE |
|
29 |
||
30 |
%% Requires graphics style file (graphicx.sty), used for inserting |
|
31 |
%% .eps files into LaTeX articles. |
|
32 |
%% Note that inclusion of .eps files is for your reference only; |
|
33 |
%% when submitting to PNAS please submit figures separately. |
|
34 |
||
35 |
%% Type into the square brackets the name of the driver program |
|
36 |
%% that you are using. If you don't know, try dvips, which is the |
|
37 |
%% most common PC driver, or textures for the Mac. These are the options: |
|
38 |
||
39 |
% [dvips], [xdvi], [dvipdf], [dvipdfm], [dvipdfmx], [pdftex], [dvipsone], |
|
40 |
% [dviwindo], [emtex], [dviwin], [pctexps], [pctexwin], [pctexhp], [pctex32], |
|
41 |
% [truetex], [tcidvi], [vtex], [oztex], [textures], [xetex] |
|
42 |
||
43 |
%\usepackage[dvips]{graphicx} |
|
44 |
||
45 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
46 |
%% OPTIONAL POSTSCRIPT FONT FILES |
|
47 |
||
48 |
%% PostScript font files: You may need to edit the PNASoneF.sty |
|
49 |
%% or PNAStwoF.sty file to make the font names match those on your system. |
|
50 |
%% Alternatively, you can leave the font style file commands commented out |
|
51 |
%% and typeset your article using the default Computer Modern |
|
52 |
%% fonts (recommended). If accepted, your article will be typeset |
|
53 |
%% at PNAS using PostScript fonts. |
|
54 |
||
55 |
||
56 |
% Choose PNASoneF for one column; PNAStwoF for two column: |
|
57 |
%\usepackage{PNASoneF} |
|
58 |
%\usepackage{PNAStwoF} |
|
59 |
||
60 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
61 |
%% ADDITIONAL OPTIONAL STYLE FILES |
|
62 |
||
63 |
%% The AMS math files are commonly used to gain access to useful features |
|
64 |
%% like extended math fonts and math commands. |
|
65 |
||
571
f958e0ea62f8
compilable PNAS file\! The blob intro typesets (poorly) onto 3 2-column pages
Scott Morrison <scott@tqft.net>
parents:
566
diff
changeset
|
66 |
\usepackage{amssymb,amsfonts,amsmath,amsthm} |
566 | 67 |
|
68 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
69 |
%% OPTIONAL MACRO FILES |
|
70 |
%% Insert self-defined macros here. |
|
71 |
%% \newcommand definitions are recommended; \def definitions are supported |
|
72 |
||
73 |
%\newcommand{\mfrac}[2]{\frac{\displaystyle #1}{\displaystyle #2}} |
|
74 |
%\def\s{\sigma} |
|
75 |
||
571
f958e0ea62f8
compilable PNAS file\! The blob intro typesets (poorly) onto 3 2-column pages
Scott Morrison <scott@tqft.net>
parents:
566
diff
changeset
|
76 |
\input{preamble} |
566 | 77 |
|
78 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
79 |
%% Don't type in anything in the following section: |
|
80 |
%%%%%%%%%%%% |
|
81 |
%% For PNAS Only: |
|
82 |
\contributor{Submitted to Proceedings |
|
83 |
of the National Academy of Sciences of the United States of America} |
|
571
f958e0ea62f8
compilable PNAS file\! The blob intro typesets (poorly) onto 3 2-column pages
Scott Morrison <scott@tqft.net>
parents:
566
diff
changeset
|
84 |
%\url{www.pnas.org/cgi/doi/10.1073/pnas.0709640104} |
566 | 85 |
\copyrightyear{2008} |
86 |
\issuedate{Issue Date} |
|
87 |
\volume{Volume} |
|
88 |
\issuenumber{Issue Number} |
|
89 |
%%%%%%%%%%%% |
|
90 |
||
91 |
\begin{document} |
|
92 |
||
93 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
94 |
||
95 |
||
96 |
%% For titles, only capitalize the first letter |
|
97 |
%% \title{Almost sharp fronts for the surface quasi-geostrophic equation} |
|
98 |
||
653 | 99 |
\title{Higher categories, colimits, and the blob complex} |
566 | 100 |
|
101 |
||
102 |
%% Enter authors via the \author command. |
|
103 |
%% Use \affil to define affiliations. |
|
104 |
%% (Leave no spaces between author name and \affil command) |
|
105 |
||
106 |
%% Note that the \thanks{} command has been disabled in favor of |
|
107 |
%% a generic, reserved space for PNAS publication footnotes. |
|
108 |
||
109 |
%% \author{<author name> |
|
110 |
%% \affil{<number>}{<Institution>}} One number for each institution. |
|
111 |
%% The same number should be used for authors that |
|
112 |
%% are affiliated with the same institution, after the first time |
|
113 |
%% only the number is needed, ie, \affil{number}{text}, \affil{number}{} |
|
114 |
%% Then, before last author ... |
|
115 |
%% \and |
|
116 |
%% \author{<author name> |
|
117 |
%% \affil{<number>}{}} |
|
118 |
||
119 |
%% For example, assuming Garcia and Sonnery are both affiliated with |
|
120 |
%% Universidad de Murcia: |
|
121 |
%% \author{Roberta Graff\affil{1}{University of Cambridge, Cambridge, |
|
122 |
%% United Kingdom}, |
|
123 |
%% Javier de Ruiz Garcia\affil{2}{Universidad de Murcia, Bioquimica y Biologia |
|
124 |
%% Molecular, Murcia, Spain}, \and Franklin Sonnery\affil{2}{}} |
|
125 |
||
651 | 126 |
\author{Scott Morrison\affil{1}{Miller Institute for Basic Research, UC Berkeley, CA 94704, USA} |
127 |
\and Kevin Walker\affil{2}{Microsoft Station Q, 2243 CNSI Building, UC Santa Barbara, CA 93106, USA}} |
|
566 | 128 |
|
129 |
\contributor{Submitted to Proceedings of the National Academy of Sciences |
|
130 |
of the United States of America} |
|
131 |
||
132 |
%% The \maketitle command is necessary to build the title page. |
|
133 |
\maketitle |
|
134 |
||
135 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
136 |
\begin{article} |
|
137 |
||
650 | 138 |
\begin{abstract} |
662
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
139 |
\nn{needs revision} |
651 | 140 |
We explain the need for new axioms for topological quantum field theories that include ideas from derived |
657
9fbd8e63ab2e
fixing single quotes and long lines
Kevin Walker <kevin@canyon23.net>
parents:
656
diff
changeset
|
141 |
categories and homotopy theory. We summarize our axioms for higher categories, and describe the ``blob complex". |
651 | 142 |
Fixing an $n$-category $\cC$, the blob complex associates a chain complex $\bc_*(W;\cC)$ to any $n$-manifold $W$. |
143 |
The $0$-th homology of this chain complex recovers the usual TQFT invariants of $W$. |
|
144 |
The higher homology groups should be viewed as generalizations of Hochschild homology. |
|
145 |
The blob complex has a very natural definition in terms of homotopy colimits along decompositions of the manifold $W$. |
|
146 |
We outline the important properties of the blob complex, and sketch the proof of a generalization of |
|
147 |
Deligne's conjecture on Hochschild cohomology and the little discs operad to higher dimensions. |
|
650 | 148 |
\end{abstract} |
566 | 149 |
|
150 |
||
151 |
%% When adding keywords, separate each term with a straight line: | |
|
578 | 152 |
\keywords{n-categories | topological quantum field theory | hochschild homology} |
566 | 153 |
|
154 |
%% Optional for entering abbreviations, separate the abbreviation from |
|
155 |
%% its definition with a comma, separate each pair with a semicolon: |
|
156 |
%% for example: |
|
157 |
%% \abbreviations{SAM, self-assembled monolayer; OTS, |
|
158 |
%% octadecyltrichlorosilane} |
|
159 |
||
602
109ecc26c50d
writing intro; just an expanded version of the existing notes, feel free to savage
Scott Morrison <scott@tqft.net>
parents:
601
diff
changeset
|
160 |
% \abbreviations{TQFT, topological quantum field theory} |
566 | 161 |
|
162 |
%% The first letter of the article should be drop cap: \dropcap{} |
|
163 |
%\dropcap{I}n this article we study the evolution of ''almost-sharp'' fronts |
|
164 |
||
165 |
%% Enter the text of your article beginning here and ending before |
|
166 |
%% \begin{acknowledgements} |
|
167 |
%% Section head commands for your reference: |
|
168 |
%% \section{} |
|
169 |
%% \subsection{} |
|
170 |
%% \subsubsection{} |
|
171 |
||
637
c1cf892a4ab7
minor changes to rewritten intro
Scott Morrison <scott@tqft.net>
parents:
636
diff
changeset
|
172 |
\dropcap{T}he aim of this paper is to describe a derived category analogue of topological quantum field theories. |
630
e0093da0d39f
silly commit to move stuff from office to home
Kevin Walker <kevin@canyon23.net>
parents:
627
diff
changeset
|
173 |
|
e0093da0d39f
silly commit to move stuff from office to home
Kevin Walker <kevin@canyon23.net>
parents:
627
diff
changeset
|
174 |
For our purposes, an $n{+}1$-dimensional TQFT is a locally defined system of |
653 | 175 |
invariants of manifolds of dimensions 0 through $n{+}1$. In particular, |
637
c1cf892a4ab7
minor changes to rewritten intro
Scott Morrison <scott@tqft.net>
parents:
636
diff
changeset
|
176 |
the TQFT invariant $A(Y)$ of a closed $k$-manifold $Y$ is a linear $(n{-}k)$-category. |
632 | 177 |
If $Y$ has boundary then $A(Y)$ is a collection of $(n{-}k)$-categories which afford |
178 |
a representation of the $(n{-}k{+}1)$-category $A(\bd Y)$. |
|
179 |
(See \cite{1009.5025} and \cite{kw:tqft}; |
|
180 |
for a more homotopy-theoretic point of view see \cite{0905.0465}.) |
|
602
109ecc26c50d
writing intro; just an expanded version of the existing notes, feel free to savage
Scott Morrison <scott@tqft.net>
parents:
601
diff
changeset
|
181 |
|
632 | 182 |
We now comment on some particular values of $k$ above. |
637
c1cf892a4ab7
minor changes to rewritten intro
Scott Morrison <scott@tqft.net>
parents:
636
diff
changeset
|
183 |
A linear 0-category is a vector space, and a representation |
632 | 184 |
of a vector space is an element of the dual space. |
637
c1cf892a4ab7
minor changes to rewritten intro
Scott Morrison <scott@tqft.net>
parents:
636
diff
changeset
|
185 |
Thus a TQFT assigns to each closed $n$-manifold $Y$ a vector space $A(Y)$, |
632 | 186 |
and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$. |
637
c1cf892a4ab7
minor changes to rewritten intro
Scott Morrison <scott@tqft.net>
parents:
636
diff
changeset
|
187 |
For the remainder of this paper we will in fact be interested in so-called $(n{+}\epsilon)$-dimensional |
658
c56a3fe75d1e
changes from proof-read, 1st installment
Kevin Walker <kevin@canyon23.net>
parents:
657
diff
changeset
|
188 |
TQFTs, which are slightly weaker structures in that they assign |
c56a3fe75d1e
changes from proof-read, 1st installment
Kevin Walker <kevin@canyon23.net>
parents:
657
diff
changeset
|
189 |
invariants to mapping cylinders of homeomorphisms between $n$-manifolds, but not to general $(n{+}1)$-manifolds. |
602
109ecc26c50d
writing intro; just an expanded version of the existing notes, feel free to savage
Scott Morrison <scott@tqft.net>
parents:
601
diff
changeset
|
190 |
|
660 | 191 |
When $k=n{-}1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$, |
632 | 192 |
and a representation of $A(\bd Y)$ for each $n$-manifold $Y$. |
637
c1cf892a4ab7
minor changes to rewritten intro
Scott Morrison <scott@tqft.net>
parents:
636
diff
changeset
|
193 |
The TQFT gluing rule in dimension $n$ states that |
632 | 194 |
$A(Y_1\cup_S Y_2) \cong A(Y_1) \ot_{A(S)} A(Y_2)$, |
658
c56a3fe75d1e
changes from proof-read, 1st installment
Kevin Walker <kevin@canyon23.net>
parents:
657
diff
changeset
|
195 |
where $Y_1$ and $Y_2$ are $n$-manifolds with common boundary $S$. |
632 | 196 |
|
197 |
When $k=0$ we have an $n$-category $A(pt)$. |
|
637
c1cf892a4ab7
minor changes to rewritten intro
Scott Morrison <scott@tqft.net>
parents:
636
diff
changeset
|
198 |
This can be thought of as the local part of the TQFT, and the full TQFT can be reconstructed from $A(pt)$ |
632 | 199 |
via colimits (see below). |
602
109ecc26c50d
writing intro; just an expanded version of the existing notes, feel free to savage
Scott Morrison <scott@tqft.net>
parents:
601
diff
changeset
|
200 |
|
632 | 201 |
We call a TQFT semisimple if $A(S)$ is a semisimple 1-category for all $(n{-}1)$-manifolds $S$ |
202 |
and $A(Y)$ is a finite-dimensional vector space for all $n$-manifolds $Y$. |
|
637
c1cf892a4ab7
minor changes to rewritten intro
Scott Morrison <scott@tqft.net>
parents:
636
diff
changeset
|
203 |
Examples of semisimple TQFTs include Witten-Reshetikhin-Turaev theories, |
632 | 204 |
Turaev-Viro theories, and Dijkgraaf-Witten theories. |
205 |
These can all be given satisfactory accounts in the framework outlined above. |
|
651 | 206 |
(The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories with only a weak |
207 |
dependence on interiors in order to be |
|
637
c1cf892a4ab7
minor changes to rewritten intro
Scott Morrison <scott@tqft.net>
parents:
636
diff
changeset
|
208 |
extended all the way down to dimension 0.) |
602
109ecc26c50d
writing intro; just an expanded version of the existing notes, feel free to savage
Scott Morrison <scott@tqft.net>
parents:
601
diff
changeset
|
209 |
|
639
11f8331ea7c4
maybe if I commit now merging will be easier?
Kevin Walker <kevin@canyon23.net>
parents:
632
diff
changeset
|
210 |
For other non-semisimple TQFT-like invariants, however, the above framework seems to be inadequate. |
658
c56a3fe75d1e
changes from proof-read, 1st installment
Kevin Walker <kevin@canyon23.net>
parents:
657
diff
changeset
|
211 |
For example, the gluing rule for 3-manifolds in Ozsv\'ath-Szab\'o/Seiberg-Witten theory |
639
11f8331ea7c4
maybe if I commit now merging will be easier?
Kevin Walker <kevin@canyon23.net>
parents:
632
diff
changeset
|
212 |
involves a tensor product over an $A_\infty$ 1-category associated to 2-manifolds \cite{1003.0598,1005.1248}. |
11f8331ea7c4
maybe if I commit now merging will be easier?
Kevin Walker <kevin@canyon23.net>
parents:
632
diff
changeset
|
213 |
Long exact sequences are important computational tools in these theories, |
11f8331ea7c4
maybe if I commit now merging will be easier?
Kevin Walker <kevin@canyon23.net>
parents:
632
diff
changeset
|
214 |
and also in Khovanov homology, but the colimit construction breaks exactness. |
11f8331ea7c4
maybe if I commit now merging will be easier?
Kevin Walker <kevin@canyon23.net>
parents:
632
diff
changeset
|
215 |
For these reasons and others, it is desirable to |
11f8331ea7c4
maybe if I commit now merging will be easier?
Kevin Walker <kevin@canyon23.net>
parents:
632
diff
changeset
|
216 |
extend to above framework to incorporate ideas from derived categories. |
602
109ecc26c50d
writing intro; just an expanded version of the existing notes, feel free to savage
Scott Morrison <scott@tqft.net>
parents:
601
diff
changeset
|
217 |
|
639
11f8331ea7c4
maybe if I commit now merging will be easier?
Kevin Walker <kevin@canyon23.net>
parents:
632
diff
changeset
|
218 |
One approach to such a generalization might be to simply define a |
11f8331ea7c4
maybe if I commit now merging will be easier?
Kevin Walker <kevin@canyon23.net>
parents:
632
diff
changeset
|
219 |
TQFT via its gluing formulas, replacing tensor products with |
643
212991f176d1
citing rozansky for s2 x s1: is there actually a paper by khovanov about this?
Scott Morrison <scott@tqft.net>
parents:
642
diff
changeset
|
220 |
derived tensor products (c.f. \cite{1011.1958}). |
639
11f8331ea7c4
maybe if I commit now merging will be easier?
Kevin Walker <kevin@canyon23.net>
parents:
632
diff
changeset
|
221 |
However, it is probably difficult to prove |
602
109ecc26c50d
writing intro; just an expanded version of the existing notes, feel free to savage
Scott Morrison <scott@tqft.net>
parents:
601
diff
changeset
|
222 |
the invariance of such a definition, as the object associated to a manifold |
109ecc26c50d
writing intro; just an expanded version of the existing notes, feel free to savage
Scott Morrison <scott@tqft.net>
parents:
601
diff
changeset
|
223 |
will a priori depend on the explicit presentation used to apply the gluing formulas. |
109ecc26c50d
writing intro; just an expanded version of the existing notes, feel free to savage
Scott Morrison <scott@tqft.net>
parents:
601
diff
changeset
|
224 |
We instead give a manifestly invariant construction, and |
639
11f8331ea7c4
maybe if I commit now merging will be easier?
Kevin Walker <kevin@canyon23.net>
parents:
632
diff
changeset
|
225 |
deduce from it the gluing formulas based on $A_\infty$ tensor products. |
602
109ecc26c50d
writing intro; just an expanded version of the existing notes, feel free to savage
Scott Morrison <scott@tqft.net>
parents:
601
diff
changeset
|
226 |
|
639
11f8331ea7c4
maybe if I commit now merging will be easier?
Kevin Walker <kevin@canyon23.net>
parents:
632
diff
changeset
|
227 |
This paper is organized as follows. |
11f8331ea7c4
maybe if I commit now merging will be easier?
Kevin Walker <kevin@canyon23.net>
parents:
632
diff
changeset
|
228 |
We first give an account of our version of $n$-categories. |
11f8331ea7c4
maybe if I commit now merging will be easier?
Kevin Walker <kevin@canyon23.net>
parents:
632
diff
changeset
|
229 |
According to our definition, $n$-categories are, among other things, |
11f8331ea7c4
maybe if I commit now merging will be easier?
Kevin Walker <kevin@canyon23.net>
parents:
632
diff
changeset
|
230 |
functorial invariants of $k$-balls, $0\le k \le n$, which behave well with respect to gluing. |
644
975c807661ca
minor changes in introduction
Scott Morrison <scott@tqft.net>
parents:
643
diff
changeset
|
231 |
We then show how to extend an $n$-category from balls to arbitrary $k$-manifolds, |
975c807661ca
minor changes in introduction
Scott Morrison <scott@tqft.net>
parents:
643
diff
changeset
|
232 |
using colimits and homotopy colimits. |
975c807661ca
minor changes in introduction
Scott Morrison <scott@tqft.net>
parents:
643
diff
changeset
|
233 |
This extension, which we call the blob complex, has as $0$-th homology the usual TQFT invariant. |
639
11f8331ea7c4
maybe if I commit now merging will be easier?
Kevin Walker <kevin@canyon23.net>
parents:
632
diff
changeset
|
234 |
(The name comes from the ``blobs" which feature prominently |
11f8331ea7c4
maybe if I commit now merging will be easier?
Kevin Walker <kevin@canyon23.net>
parents:
632
diff
changeset
|
235 |
in a concrete version of the homotopy colimit.) |
641 | 236 |
We then review some basic properties of the blob complex, and finish by showing how it |
237 |
yields a higher categorical and higher dimensional generalization of Deligne's |
|
238 |
conjecture on Hochschild cochains and the little 2-disks operad. |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
239 |
|
651 | 240 |
At several points we only sketch an argument briefly; full details can be found in \cite{1009.5025}. |
241 |
In this paper we attempt to give a clear view of the big picture without getting |
|
242 |
bogged down in technical details. |
|
641 | 243 |
|
632 | 244 |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
245 |
\section{Definitions} |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
246 |
\subsection{$n$-categories} \mbox{} |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
247 |
|
642
61287354218c
short version of cat sect intro; longer intro desirable?
Kevin Walker <kevin@canyon23.net>
parents:
641
diff
changeset
|
248 |
In this section we give a definition of $n$-categories designed to work well with TQFTs. |
61287354218c
short version of cat sect intro; longer intro desirable?
Kevin Walker <kevin@canyon23.net>
parents:
641
diff
changeset
|
249 |
The main idea is to base the definition on actual balls, rather combinatorial models of them. |
61287354218c
short version of cat sect intro; longer intro desirable?
Kevin Walker <kevin@canyon23.net>
parents:
641
diff
changeset
|
250 |
This has the advantages of avoiding a proliferation of coherency axioms and building in a strong |
61287354218c
short version of cat sect intro; longer intro desirable?
Kevin Walker <kevin@canyon23.net>
parents:
641
diff
changeset
|
251 |
version of duality from the start. |
581 | 252 |
|
253 |
||
642
61287354218c
short version of cat sect intro; longer intro desirable?
Kevin Walker <kevin@canyon23.net>
parents:
641
diff
changeset
|
254 |
%\nn{maybe say something about goals: well-suited to TQFTs; avoid proliferation of coherency axioms; |
61287354218c
short version of cat sect intro; longer intro desirable?
Kevin Walker <kevin@canyon23.net>
parents:
641
diff
changeset
|
255 |
%non-recursive (n-cats not defined n terms of (n-1)-cats; easy to show that the motivating |
61287354218c
short version of cat sect intro; longer intro desirable?
Kevin Walker <kevin@canyon23.net>
parents:
641
diff
changeset
|
256 |
%examples satisfy the axioms; strong duality; both plain and infty case; |
61287354218c
short version of cat sect intro; longer intro desirable?
Kevin Walker <kevin@canyon23.net>
parents:
641
diff
changeset
|
257 |
%(?) easy to see that axioms are correct, in the sense of nothing missing (need |
61287354218c
short version of cat sect intro; longer intro desirable?
Kevin Walker <kevin@canyon23.net>
parents:
641
diff
changeset
|
258 |
%to say this better if we keep it)} |
61287354218c
short version of cat sect intro; longer intro desirable?
Kevin Walker <kevin@canyon23.net>
parents:
641
diff
changeset
|
259 |
% |
61287354218c
short version of cat sect intro; longer intro desirable?
Kevin Walker <kevin@canyon23.net>
parents:
641
diff
changeset
|
260 |
%\nn{maybe: the typical n-cat definition tries to do two things at once: (1) give a list of basic properties |
61287354218c
short version of cat sect intro; longer intro desirable?
Kevin Walker <kevin@canyon23.net>
parents:
641
diff
changeset
|
261 |
%which are weak enough to include the basic examples and strong enough to support the proofs |
61287354218c
short version of cat sect intro; longer intro desirable?
Kevin Walker <kevin@canyon23.net>
parents:
641
diff
changeset
|
262 |
%of the main theorems; and (2) specify a minimal set of generators and/or axioms. |
61287354218c
short version of cat sect intro; longer intro desirable?
Kevin Walker <kevin@canyon23.net>
parents:
641
diff
changeset
|
263 |
%We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second. |
61287354218c
short version of cat sect intro; longer intro desirable?
Kevin Walker <kevin@canyon23.net>
parents:
641
diff
changeset
|
264 |
%More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
581 | 265 |
|
666
6b6c565bd76e
move Lurie-comparison paragraph to n-cat section
Kevin Walker <kevin@canyon23.net>
parents:
665
diff
changeset
|
266 |
Of course, there are currently many interesting alternative notions of $n$-category. |
6b6c565bd76e
move Lurie-comparison paragraph to n-cat section
Kevin Walker <kevin@canyon23.net>
parents:
665
diff
changeset
|
267 |
We note that our $n$-categories are both more and less general |
6b6c565bd76e
move Lurie-comparison paragraph to n-cat section
Kevin Walker <kevin@canyon23.net>
parents:
665
diff
changeset
|
268 |
than the ``fully dualizable" ones which play a prominent role in \cite{0905.0465}. |
6b6c565bd76e
move Lurie-comparison paragraph to n-cat section
Kevin Walker <kevin@canyon23.net>
parents:
665
diff
changeset
|
269 |
They are more general in that we make no duality assumptions in the top dimension $n{+}1$. |
6b6c565bd76e
move Lurie-comparison paragraph to n-cat section
Kevin Walker <kevin@canyon23.net>
parents:
665
diff
changeset
|
270 |
They are less general in that we impose stronger duality requirements in dimensions 0 through $n$. |
6b6c565bd76e
move Lurie-comparison paragraph to n-cat section
Kevin Walker <kevin@canyon23.net>
parents:
665
diff
changeset
|
271 |
Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional {\it unoriented} or {\it oriented} TQFTs, while |
6b6c565bd76e
move Lurie-comparison paragraph to n-cat section
Kevin Walker <kevin@canyon23.net>
parents:
665
diff
changeset
|
272 |
Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional {\it framed} TQFTs. |
6b6c565bd76e
move Lurie-comparison paragraph to n-cat section
Kevin Walker <kevin@canyon23.net>
parents:
665
diff
changeset
|
273 |
|
655
71eb442b8500
trying out 'isotopy n-category', and explaining the difference better
Scott Morrison <scott@tqft.net>
parents:
654
diff
changeset
|
274 |
We will define two variations simultaneously, as all but one of the axioms are identical in the two cases. |
662
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
275 |
These variations are ``plain $n$-categories", where homeomorphisms fixing the boundary |
655
71eb442b8500
trying out 'isotopy n-category', and explaining the difference better
Scott Morrison <scott@tqft.net>
parents:
654
diff
changeset
|
276 |
act trivially on the sets associated to $n$-balls |
71eb442b8500
trying out 'isotopy n-category', and explaining the difference better
Scott Morrison <scott@tqft.net>
parents:
654
diff
changeset
|
277 |
(and these sets are usually vector spaces or more generally modules over a commutative ring) |
657
9fbd8e63ab2e
fixing single quotes and long lines
Kevin Walker <kevin@canyon23.net>
parents:
656
diff
changeset
|
278 |
and ``$A_\infty$ $n$-categories", where there is a homotopy action of |
655
71eb442b8500
trying out 'isotopy n-category', and explaining the difference better
Scott Morrison <scott@tqft.net>
parents:
654
diff
changeset
|
279 |
$k$-parameter families of homeomorphisms on these sets |
71eb442b8500
trying out 'isotopy n-category', and explaining the difference better
Scott Morrison <scott@tqft.net>
parents:
654
diff
changeset
|
280 |
(which are usually chain complexes or topological spaces). |
581 | 281 |
|
610 | 282 |
There are five basic ingredients |
615
222da6df3edc
various minor, and moving mapping spaces to 'specializations'
Scott Morrison <scott@tqft.net>
parents:
614
diff
changeset
|
283 |
\cite{life-of-brian} of an $n$-category definition: |
581 | 284 |
$k$-morphisms (for $0\le k \le n$), domain and range, composition, |
285 |
identity morphisms, and special behavior in dimension $n$ (e.g. enrichment |
|
286 |
in some auxiliary category, or strict associativity instead of weak associativity). |
|
584 | 287 |
We will treat each of these in turn. |
581 | 288 |
|
289 |
To motivate our morphism axiom, consider the venerable notion of the Moore loop space |
|
599 | 290 |
\cite[\S 2.2]{MR505692}. |
581 | 291 |
In the standard definition of a loop space, loops are always parameterized by the unit interval $I = [0,1]$, |
292 |
so composition of loops requires a reparameterization $I\cup I \cong I$, and this leads to a proliferation |
|
293 |
of higher associativity relations. |
|
294 |
While this proliferation is manageable for 1-categories (and indeed leads to an elegant theory |
|
295 |
of Stasheff polyhedra and $A_\infty$ categories), it becomes undesirably complex for higher categories. |
|
296 |
In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a |
|
297 |
{\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. |
|
298 |
Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
|
299 |
We wish to imitate this strategy in higher categories. |
|
300 |
Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
|
664 | 301 |
a product of $k$ intervals (c.f. \cite{ulrike-tillmann-2008,0909.2212}) but rather with any $k$-ball, that is, |
651 | 302 |
any $k$-manifold which is homeomorphic |
582 | 303 |
to the standard $k$-ball $B^k$. |
583 | 304 |
|
645 | 305 |
By default our balls are unoriented, |
644
975c807661ca
minor changes in introduction
Scott Morrison <scott@tqft.net>
parents:
643
diff
changeset
|
306 |
but it is useful at times to vary this, |
645 | 307 |
for example by considering oriented or Spin balls. |
644
975c807661ca
minor changes in introduction
Scott Morrison <scott@tqft.net>
parents:
643
diff
changeset
|
308 |
We can also consider more exotic structures, such as balls with a map to some target space, |
600
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
309 |
or equipped with $m$ independent vector fields. |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
310 |
(The latter structure would model $n$-categories with less duality than we usually assume.) |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
311 |
|
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
312 |
\begin{axiom}[Morphisms] |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
313 |
\label{axiom:morphisms} |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
314 |
For each $0 \le k \le n$, we have a functor $\cC_k$ from |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
315 |
the category of $k$-balls and |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
316 |
homeomorphisms to the category of sets and bijections. |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
317 |
\end{axiom} |
582 | 318 |
|
586
0510346848ed
restore and complete the fragment
Kevin Walker <kevin@canyon23.net>
parents:
585
diff
changeset
|
319 |
Note that the functoriality in the above axiom allows us to operate via |
0510346848ed
restore and complete the fragment
Kevin Walker <kevin@canyon23.net>
parents:
585
diff
changeset
|
320 |
homeomorphisms which are not the identity on the boundary of the $k$-ball. |
0510346848ed
restore and complete the fragment
Kevin Walker <kevin@canyon23.net>
parents:
585
diff
changeset
|
321 |
The action of these homeomorphisms gives the ``strong duality" structure. |
658
c56a3fe75d1e
changes from proof-read, 1st installment
Kevin Walker <kevin@canyon23.net>
parents:
657
diff
changeset
|
322 |
For this reason we don't subdivide the boundary of a morphism |
c56a3fe75d1e
changes from proof-read, 1st installment
Kevin Walker <kevin@canyon23.net>
parents:
657
diff
changeset
|
323 |
into domain and range in the next axiom --- the duality operations can convert between domain and range. |
582 | 324 |
|
651 | 325 |
Later we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ |
658
c56a3fe75d1e
changes from proof-read, 1st installment
Kevin Walker <kevin@canyon23.net>
parents:
657
diff
changeset
|
326 |
defined on arbitrary manifolds. |
c56a3fe75d1e
changes from proof-read, 1st installment
Kevin Walker <kevin@canyon23.net>
parents:
657
diff
changeset
|
327 |
We need these functors for $k$-spheres, for $k<n$, for the next axiom. |
582 | 328 |
|
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
329 |
\begin{axiom}[Boundaries]\label{nca-boundary} |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
330 |
For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
331 |
These maps, for various $X$, comprise a natural transformation of functors. |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
332 |
\end{axiom} |
582 | 333 |
|
594
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
334 |
For $c\in \cl{\cC}_{k-1}(\bd X)$ we define $\cC_k(X; c) = \bd^{-1}(c)$. |
587
38ec3d05d0d8
enrichment; decompositions (meta)
Kevin Walker <kevin@canyon23.net>
parents:
586
diff
changeset
|
335 |
|
38ec3d05d0d8
enrichment; decompositions (meta)
Kevin Walker <kevin@canyon23.net>
parents:
586
diff
changeset
|
336 |
Many of the examples we are interested in are enriched in some auxiliary category $\cS$ |
597 | 337 |
(e.g. vector spaces or rings, or, in the $A_\infty$ case, chain complexes or topological spaces). |
595
9c708975b61b
making pinched products axioms terser, and writing a short proof of the higher deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
594
diff
changeset
|
338 |
This means that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure |
587
38ec3d05d0d8
enrichment; decompositions (meta)
Kevin Walker <kevin@canyon23.net>
parents:
586
diff
changeset
|
339 |
of an object of $\cS$, and all of the structure maps of the category (above and below) are |
38ec3d05d0d8
enrichment; decompositions (meta)
Kevin Walker <kevin@canyon23.net>
parents:
586
diff
changeset
|
340 |
compatible with the $\cS$ structure on $\cC_n(X; c)$. |
38ec3d05d0d8
enrichment; decompositions (meta)
Kevin Walker <kevin@canyon23.net>
parents:
586
diff
changeset
|
341 |
|
38ec3d05d0d8
enrichment; decompositions (meta)
Kevin Walker <kevin@canyon23.net>
parents:
586
diff
changeset
|
342 |
|
657
9fbd8e63ab2e
fixing single quotes and long lines
Kevin Walker <kevin@canyon23.net>
parents:
656
diff
changeset
|
343 |
Given two hemispheres (a ``domain" and ``range") that agree on the equator, we need to be able to |
651 | 344 |
assemble them into a boundary value of the entire sphere. |
585
e2996d7b4e6c
various, mostly working on axioms
Scott Morrison <scott@tqft.net>
parents:
584
diff
changeset
|
345 |
|
e2996d7b4e6c
various, mostly working on axioms
Scott Morrison <scott@tqft.net>
parents:
584
diff
changeset
|
346 |
\begin{lem} |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
347 |
\label{lem:domain-and-range} |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
348 |
Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
349 |
$B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
350 |
Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
351 |
two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$. |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
352 |
Then we have an injective map |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
353 |
\[ |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
354 |
\gl_E : \cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2) \into \cl{\cC}(S) |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
355 |
\] |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
356 |
which is natural with respect to the actions of homeomorphisms. |
585
e2996d7b4e6c
various, mostly working on axioms
Scott Morrison <scott@tqft.net>
parents:
584
diff
changeset
|
357 |
%(When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product |
e2996d7b4e6c
various, mostly working on axioms
Scott Morrison <scott@tqft.net>
parents:
584
diff
changeset
|
358 |
%becomes a normal product.) |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
359 |
\end{lem} |
582 | 360 |
|
585
e2996d7b4e6c
various, mostly working on axioms
Scott Morrison <scott@tqft.net>
parents:
584
diff
changeset
|
361 |
If $\bdy B = S$, we denote $\bdy^{-1}(\im(\gl_E))$ by $\cC(B)_E$. |
e2996d7b4e6c
various, mostly working on axioms
Scott Morrison <scott@tqft.net>
parents:
584
diff
changeset
|
362 |
|
e2996d7b4e6c
various, mostly working on axioms
Scott Morrison <scott@tqft.net>
parents:
584
diff
changeset
|
363 |
\begin{axiom}[Gluing] |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
364 |
\label{axiom:composition} |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
365 |
Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
366 |
and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
367 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
585
e2996d7b4e6c
various, mostly working on axioms
Scott Morrison <scott@tqft.net>
parents:
584
diff
changeset
|
368 |
%Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
e2996d7b4e6c
various, mostly working on axioms
Scott Morrison <scott@tqft.net>
parents:
584
diff
changeset
|
369 |
We have restriction maps $\cC(B_i)_E \to \cC(Y)$. |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
370 |
Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
371 |
We have a map |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
372 |
\[ |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
373 |
\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
374 |
\] |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
375 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
376 |
to the intersection of the boundaries of $B$ and $B_i$. |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
377 |
If $k < n$, |
658
c56a3fe75d1e
changes from proof-read, 1st installment
Kevin Walker <kevin@canyon23.net>
parents:
657
diff
changeset
|
378 |
or if $k=n$ and we are in the $A_\infty$ case, |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
379 |
we require that $\gl_Y$ is injective. |
662
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
380 |
(For $k=n$ in the plain $n$-category case, see Axiom \ref{axiom:extended-isotopies}.) |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
381 |
\end{axiom} |
582 | 382 |
|
646 | 383 |
\begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity} |
585
e2996d7b4e6c
various, mostly working on axioms
Scott Morrison <scott@tqft.net>
parents:
584
diff
changeset
|
384 |
The gluing maps above are strictly associative. |
584 | 385 |
Given any decomposition of a ball $B$ into smaller balls |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
386 |
$$\bigsqcup B_i \to B,$$ |
584 | 387 |
any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result. |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
388 |
\end{axiom} |
658
c56a3fe75d1e
changes from proof-read, 1st installment
Kevin Walker <kevin@canyon23.net>
parents:
657
diff
changeset
|
389 |
%This axiom is only reasonable because the definition assigns a set to every ball; |
c56a3fe75d1e
changes from proof-read, 1st installment
Kevin Walker <kevin@canyon23.net>
parents:
657
diff
changeset
|
390 |
%any identifications would limit the extent to which we can demand associativity. |
c56a3fe75d1e
changes from proof-read, 1st installment
Kevin Walker <kevin@canyon23.net>
parents:
657
diff
changeset
|
391 |
%%%% KW: It took me quite a while figure out what you [or I??] meant by the above, so I'm attempting a rewrite. |
c56a3fe75d1e
changes from proof-read, 1st installment
Kevin Walker <kevin@canyon23.net>
parents:
657
diff
changeset
|
392 |
Note that even though our $n$-categories are ``weak" in the traditional sense, we can require |
c56a3fe75d1e
changes from proof-read, 1st installment
Kevin Walker <kevin@canyon23.net>
parents:
657
diff
changeset
|
393 |
strict associativity because we have more morphisms (cf.\ discussion of Moore loops above). |
c56a3fe75d1e
changes from proof-read, 1st installment
Kevin Walker <kevin@canyon23.net>
parents:
657
diff
changeset
|
394 |
|
585
e2996d7b4e6c
various, mostly working on axioms
Scott Morrison <scott@tqft.net>
parents:
584
diff
changeset
|
395 |
For the next axiom, a \emph{pinched product} is a map locally modeled on a degeneracy map between simplices. |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
396 |
\begin{axiom}[Product (identity) morphisms] |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
397 |
\label{axiom:product} |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
398 |
For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
399 |
there is a map $\pi^*:\cC(X)\to \cC(E)$. |
595
9c708975b61b
making pinched products axioms terser, and writing a short proof of the higher deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
594
diff
changeset
|
400 |
These maps must be |
9c708975b61b
making pinched products axioms terser, and writing a short proof of the higher deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
594
diff
changeset
|
401 |
\begin{enumerate} |
9c708975b61b
making pinched products axioms terser, and writing a short proof of the higher deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
594
diff
changeset
|
402 |
\item natural with respect to maps of pinched products, |
9c708975b61b
making pinched products axioms terser, and writing a short proof of the higher deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
594
diff
changeset
|
403 |
\item functorial with respect to composition of pinched products, |
9c708975b61b
making pinched products axioms terser, and writing a short proof of the higher deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
594
diff
changeset
|
404 |
\item compatible with gluing and restriction of pinched products. |
9c708975b61b
making pinched products axioms terser, and writing a short proof of the higher deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
594
diff
changeset
|
405 |
\end{enumerate} |
9c708975b61b
making pinched products axioms terser, and writing a short proof of the higher deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
594
diff
changeset
|
406 |
|
9c708975b61b
making pinched products axioms terser, and writing a short proof of the higher deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
594
diff
changeset
|
407 |
%%% begin noop %%% |
9c708975b61b
making pinched products axioms terser, and writing a short proof of the higher deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
594
diff
changeset
|
408 |
% this was the original list of conditions, which I've replaced with the much terser list above -S |
9c708975b61b
making pinched products axioms terser, and writing a short proof of the higher deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
594
diff
changeset
|
409 |
\noop{ |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
410 |
These maps must satisfy the following conditions. |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
411 |
\begin{enumerate} |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
412 |
\item |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
413 |
If $\pi:E\to X$ and $\pi':E'\to X'$ are pinched products, and |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
414 |
if $f:X\to X'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
415 |
\[ \xymatrix{ |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
416 |
E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\ |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
417 |
X \ar[r]^{f} & X' |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
418 |
} \] |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
419 |
commutes, then we have |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
420 |
\[ |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
421 |
\pi'^*\circ f = \tilde{f}\circ \pi^*. |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
422 |
\] |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
423 |
\item |
585
e2996d7b4e6c
various, mostly working on axioms
Scott Morrison <scott@tqft.net>
parents:
584
diff
changeset
|
424 |
Product morphisms are compatible with gluing. |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
425 |
Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
426 |
be pinched products with $E = E_1\cup E_2$. |
611 | 427 |
Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\subset X$. |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
428 |
Then |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
429 |
\[ |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
430 |
\pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
431 |
\] |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
432 |
\item |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
433 |
Product morphisms are associative. |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
434 |
If $\pi:E\to X$ and $\rho:D\to E$ are pinched products then |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
435 |
\[ |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
436 |
\rho^*\circ\pi^* = (\pi\circ\rho)^* . |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
437 |
\] |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
438 |
\item |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
439 |
Product morphisms are compatible with restriction. |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
440 |
If we have a commutative diagram |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
441 |
\[ \xymatrix{ |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
442 |
D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\ |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
443 |
Y \ar@{^(->}[r] & X |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
444 |
} \] |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
445 |
such that $\rho$ and $\pi$ are pinched products, then |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
446 |
\[ |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
447 |
\res_D\circ\pi^* = \rho^*\circ\res_Y . |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
448 |
\] |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
449 |
\end{enumerate} |
595
9c708975b61b
making pinched products axioms terser, and writing a short proof of the higher deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
594
diff
changeset
|
450 |
} %%% end \noop %%% |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
451 |
\end{axiom} |
604
f0dff7f0f337
definition of collar maps
Kevin Walker <kevin@canyon23.net>
parents:
603
diff
changeset
|
452 |
|
f0dff7f0f337
definition of collar maps
Kevin Walker <kevin@canyon23.net>
parents:
603
diff
changeset
|
453 |
To state the next axiom we need the notion of {\it collar maps} on $k$-morphisms. |
611 | 454 |
Let $X$ be a $k$-ball and $Y\subset\bd X$ be a $(k{-}1)$-ball. |
604
f0dff7f0f337
definition of collar maps
Kevin Walker <kevin@canyon23.net>
parents:
603
diff
changeset
|
455 |
Let $J$ be a 1-ball. |
f0dff7f0f337
definition of collar maps
Kevin Walker <kevin@canyon23.net>
parents:
603
diff
changeset
|
456 |
Let $Y\times_p J$ denote $Y\times J$ pinched along $(\bd Y)\times J$. |
f0dff7f0f337
definition of collar maps
Kevin Walker <kevin@canyon23.net>
parents:
603
diff
changeset
|
457 |
A collar map is an instance of the composition |
f0dff7f0f337
definition of collar maps
Kevin Walker <kevin@canyon23.net>
parents:
603
diff
changeset
|
458 |
\[ |
f0dff7f0f337
definition of collar maps
Kevin Walker <kevin@canyon23.net>
parents:
603
diff
changeset
|
459 |
\cC(X) \to \cC(X\cup_Y (Y\times_p J)) \to \cC(X) , |
f0dff7f0f337
definition of collar maps
Kevin Walker <kevin@canyon23.net>
parents:
603
diff
changeset
|
460 |
\] |
f0dff7f0f337
definition of collar maps
Kevin Walker <kevin@canyon23.net>
parents:
603
diff
changeset
|
461 |
where the first arrow is gluing with a product morphism on $Y\times_p J$ and |
f0dff7f0f337
definition of collar maps
Kevin Walker <kevin@canyon23.net>
parents:
603
diff
changeset
|
462 |
the second is induced by a homeomorphism from $X\cup_Y (Y\times_p J)$ to $X$ which restricts |
f0dff7f0f337
definition of collar maps
Kevin Walker <kevin@canyon23.net>
parents:
603
diff
changeset
|
463 |
to the identity on the boundary. |
f0dff7f0f337
definition of collar maps
Kevin Walker <kevin@canyon23.net>
parents:
603
diff
changeset
|
464 |
|
f0dff7f0f337
definition of collar maps
Kevin Walker <kevin@canyon23.net>
parents:
603
diff
changeset
|
465 |
|
662
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
466 |
\begin{axiom}[\textup{\textbf{[for plain $n$-categories]}} Extended isotopy invariance in dimension $n$.] |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
467 |
\label{axiom:extended-isotopies} |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
468 |
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
469 |
to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
470 |
Then $f$ acts trivially on $\cC(X)$. |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
471 |
In addition, collar maps act trivially on $\cC(X)$. |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
472 |
\end{axiom} |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
473 |
|
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
474 |
\smallskip |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
475 |
|
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
476 |
For $A_\infty$ $n$-categories, we replace |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
477 |
isotopy invariance with the requirement that families of homeomorphisms act. |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
478 |
For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
479 |
Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
480 |
$C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
481 |
|
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
482 |
|
655
71eb442b8500
trying out 'isotopy n-category', and explaining the difference better
Scott Morrison <scott@tqft.net>
parents:
654
diff
changeset
|
483 |
\begin{axiom}[\textup{\textbf{[for $A_\infty$ $n$-categories]}} Families of homeomorphisms act in dimension $n$.] |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
484 |
\label{axiom:families} |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
485 |
For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
486 |
\[ |
611 | 487 |
C_*(\Homeo_\bd(X))\tensor \cC(X; c) \to \cC(X; c) . |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
488 |
\] |
652
821d79885bfe
minor changes from proofreading
Scott Morrison <scott@tqft.net>
parents:
651
diff
changeset
|
489 |
These action maps are required to restrict to the usual action of homeomorphisms on $C_0$, be associative up to homotopy, |
821d79885bfe
minor changes from proofreading
Scott Morrison <scott@tqft.net>
parents:
651
diff
changeset
|
490 |
and also be compatible with composition (gluing) in the sense that |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
491 |
a diagram like the one in Theorem \ref{thm:CH} commutes. |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
492 |
\end{axiom} |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
493 |
|
658
c56a3fe75d1e
changes from proof-read, 1st installment
Kevin Walker <kevin@canyon23.net>
parents:
657
diff
changeset
|
494 |
\subsection{Example (the fundamental $n$-groupoid)} \mbox{} |
601
6bfa35fb758a
minor changes to cone-product polyhedra discussion
Scott Morrison <scott@tqft.net>
parents:
600
diff
changeset
|
495 |
We will define $\pi_{\le n}(T)$, the fundamental $n$-groupoid of a topological space $T$. |
6bfa35fb758a
minor changes to cone-product polyhedra discussion
Scott Morrison <scott@tqft.net>
parents:
600
diff
changeset
|
496 |
When $X$ is a $k$-ball with $k<n$, define $\pi_{\le n}(T)(X)$ |
600
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
497 |
to be the set of continuous maps from $X$ to $T$. |
601
6bfa35fb758a
minor changes to cone-product polyhedra discussion
Scott Morrison <scott@tqft.net>
parents:
600
diff
changeset
|
498 |
When $X$ is an $n$-ball, define $\pi_{\le n}(T)(X)$ to be homotopy classes (rel boundary) of such maps. |
600
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
499 |
Define boundary restrictions and gluing in the obvious way. |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
500 |
If $\rho:E\to X$ is a pinched product and $f:X\to T$ is a $k$-morphism, |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
501 |
define the product morphism $\rho^*(f)$ to be $f\circ\rho$. |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
502 |
|
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
503 |
We can also define an $A_\infty$ version $\pi_{\le n}^\infty(T)$ of the fundamental $n$-groupoid. |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
504 |
For $X$ an $n$-ball define $\pi_{\le n}^\infty(T)(X)$ to be the space of all maps from $X$ to $T$ |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
505 |
(if we are enriching over spaces) or the singular chains on that space (if we are enriching over chain complexes). |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
506 |
|
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
507 |
|
658
c56a3fe75d1e
changes from proof-read, 1st installment
Kevin Walker <kevin@canyon23.net>
parents:
657
diff
changeset
|
508 |
\subsection{Example (string diagrams)} \mbox{} |
657
9fbd8e63ab2e
fixing single quotes and long lines
Kevin Walker <kevin@canyon23.net>
parents:
656
diff
changeset
|
509 |
Fix a ``traditional" $n$-category $C$ with strong duality (e.g.\ a pivotal 2-category). |
600
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
510 |
Let $X$ be a $k$-ball and define $\cS_C(X)$ to be the set of $C$ string diagrams drawn on $X$; |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
511 |
that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$. |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
512 |
If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
513 |
Boundary restrictions and gluing are again straightforward to define. |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
514 |
Define product morphisms via product cell decompositions. |
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
515 |
|
658
c56a3fe75d1e
changes from proof-read, 1st installment
Kevin Walker <kevin@canyon23.net>
parents:
657
diff
changeset
|
516 |
\subsection{Example (bordism)} \mbox{} |
612 | 517 |
When $X$ is a $k$-ball with $k<n$, $\Bord^n(X)$ is the set of all $k$-dimensional |
518 |
submanifolds $W$ in $X\times \bbR^\infty$ which project to $X$ transversely |
|
519 |
to $\bd X$. |
|
520 |
For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes rel boundary of such $n$-dimensional submanifolds. |
|
600
e9032f8dee24
Examples and misc.; quality of writing perhaps not so great.
Kevin Walker <kevin@canyon23.net>
parents:
599
diff
changeset
|
521 |
|
651 | 522 |
There is an $A_\infty$ analogue enriched in topological spaces, where at the top level we take |
523 |
all such submanifolds, rather than homeomorphism classes. |
|
652
821d79885bfe
minor changes from proofreading
Scott Morrison <scott@tqft.net>
parents:
651
diff
changeset
|
524 |
For each fixed $\bdy W \subset \bdy X \times \bbR^\infty$, we |
651 | 525 |
topologize the set of submanifolds by ambient isotopy rel boundary. |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
526 |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
527 |
\subsection{The blob complex} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
528 |
\subsubsection{Decompositions of manifolds} |
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
529 |
|
646 | 530 |
A \emph{ball decomposition} of a $k$-manifold $W$ is a |
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
531 |
sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
574 | 532 |
$\du_a X_a$ and each $M_i$ is a manifold. |
533 |
If $X_a$ is some component of $M_0$, its image in $W$ need not be a ball; $\bd X_a$ may have been glued to itself. |
|
534 |
A {\it permissible decomposition} of $W$ is a map |
|
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
535 |
\[ |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
536 |
\coprod_a X_a \to W, |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
537 |
\] |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
538 |
which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. |
574 | 539 |
A permissible decomposition is weaker than a ball decomposition; we forget the order in which the balls |
540 |
are glued up to yield $W$, and just require that there is some non-pathological way to do this. |
|
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
541 |
|
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
542 |
Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
543 |
of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
544 |
with $\du_b Y_b = M_i$ for some $i$. |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
545 |
|
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
546 |
\begin{defn} |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
547 |
The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
548 |
and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
549 |
See Figure \ref{partofJfig} for an example. |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
550 |
\end{defn} |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
551 |
|
651 | 552 |
This poset in fact has more structure, since we can glue together permissible decompositions of |
553 |
$W_1$ and $W_2$ to obtain a permissible decomposition of $W_1 \sqcup W_2$. |
|
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
554 |
|
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
555 |
An $n$-category $\cC$ determines |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
556 |
a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
557 |
(possibly with additional structure if $k=n$). |
653 | 558 |
Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-manifolds, |
611 | 559 |
and there is a subset $\cC(X)\spl \subset \cC(X)$ of morphisms whose boundaries |
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
560 |
are splittable along this decomposition. |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
561 |
|
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
562 |
\begin{defn} |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
563 |
Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
564 |
For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
574 | 565 |
\begin{equation*} |
566 |
%\label{eq:psi-C} |
|
611 | 567 |
\psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl |
574 | 568 |
\end{equation*} |
653 | 569 |
where the restrictions to the various pieces of shared boundaries amongst the balls |
662
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
570 |
$X_a$ all agree (similar to a fibered product). |
657
9fbd8e63ab2e
fixing single quotes and long lines
Kevin Walker <kevin@canyon23.net>
parents:
656
diff
changeset
|
571 |
When $k=n$, the ``subset" and ``product" in the above formula should be |
651 | 572 |
interpreted in the appropriate enriching category. |
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
573 |
If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
574 |
\end{defn} |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
575 |
|
662
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
576 |
%We will use the term ``field on $W$" to refer to a point of this functor, |
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
577 |
%that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
578 |
|
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
579 |
|
632 | 580 |
\subsubsection{Colimits} |
655
71eb442b8500
trying out 'isotopy n-category', and explaining the difference better
Scott Morrison <scott@tqft.net>
parents:
654
diff
changeset
|
581 |
Recall that our definition of an $n$-category is essentially a collection of functors |
659
cc0c2dfe61f3
2nd installment of changes from proof-read
Kevin Walker <kevin@canyon23.net>
parents:
658
diff
changeset
|
582 |
defined on the categories of homeomorphisms of $k$-balls |
651 | 583 |
for $k \leq n$ satisfying certain axioms. |
655
71eb442b8500
trying out 'isotopy n-category', and explaining the difference better
Scott Morrison <scott@tqft.net>
parents:
654
diff
changeset
|
584 |
It is natural to hope to extend such functors to the |
651 | 585 |
larger categories of all $k$-manifolds (again, with homeomorphisms). |
655
71eb442b8500
trying out 'isotopy n-category', and explaining the difference better
Scott Morrison <scott@tqft.net>
parents:
654
diff
changeset
|
586 |
In fact, the axioms stated above already require such an extension to $k$-spheres for $k<n$. |
638
6a7f2a6295d1
very paltry start on colimits, out of time for now
Scott Morrison <scott@tqft.net>
parents:
637
diff
changeset
|
587 |
|
655
71eb442b8500
trying out 'isotopy n-category', and explaining the difference better
Scott Morrison <scott@tqft.net>
parents:
654
diff
changeset
|
588 |
The natural construction achieving this is a colimit along the poset of permissible decompositions. |
662
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
589 |
Given a plain $n$-category $\cC$, |
659
cc0c2dfe61f3
2nd installment of changes from proof-read
Kevin Walker <kevin@canyon23.net>
parents:
658
diff
changeset
|
590 |
we will denote its extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$, |
656
28592849a474
some more fixes in the colimit section
Scott Morrison <scott@tqft.net>
parents:
655
diff
changeset
|
591 |
this is defined to be the colimit along $\cell(W)$ of the functor $\psi_{\cC;W}$. |
651 | 592 |
Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity} |
593 |
imply that $\cl{\cC}(X) \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$. |
|
660 | 594 |
Suppose that $\cC$ is enriched in vector spaces: this means that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball, |
595 |
the set $\cC(X;c)$ is a vector space. |
|
596 |
In this case, for $W$ an arbitrary $n$-manifold and $c \in \cl{\cC}(\bdy W)$, |
|
597 |
the set $\cl{\cC}(W;c) = \bdy^{-1} (c)$ inherits the structure of a vector space. |
|
651 | 598 |
These are the usual TQFT skein module invariants on $n$-manifolds. |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
599 |
|
598
20de3d710f77
writing inconclusively about homotopy colimits, but have to run
Scott Morrison <scott@tqft.net>
parents:
597
diff
changeset
|
600 |
We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
659
cc0c2dfe61f3
2nd installment of changes from proof-read
Kevin Walker <kevin@canyon23.net>
parents:
658
diff
changeset
|
601 |
with coefficients in the $n$-category $\cC$ as the {\it homotopy} colimit along $\cell(W)$ |
598
20de3d710f77
writing inconclusively about homotopy colimits, but have to run
Scott Morrison <scott@tqft.net>
parents:
597
diff
changeset
|
602 |
of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
20de3d710f77
writing inconclusively about homotopy colimits, but have to run
Scott Morrison <scott@tqft.net>
parents:
597
diff
changeset
|
603 |
|
651 | 604 |
An explicit realization of the homotopy colimit is provided by the simplices of the |
605 |
functor $\psi_{\cC; W}$. That is, $$\clh{\cC}(W) = \DirectSum_{\bar{x}} \psi_{\cC; W}(x_0)[m],$$ |
|
606 |
where $\bar{x} = x_0 \leq \cdots \leq x_m$ is a simplex in $\cell(W)$. |
|
607 |
The differential acts on $(\bar{x},a)$ (here $a \in \psi_{\cC; W}(x_0)$) as |
|
599 | 608 |
$$\bdy (\bar{x},a) = (\bar{x}, \bdy a) + (-1)^{\deg a} \left( (d_0 \bar{x}, g(a)) + \sum_{i=1}^m (-1)^i (d_i \bar{x}, a) \right)$$ |
609 |
where $g$ is the gluing map from $x_0$ to $x_1$, and $d_i \bar{x}$ denotes the $i$-th face of the simplex $\bar{x}$. |
|
598
20de3d710f77
writing inconclusively about homotopy colimits, but have to run
Scott Morrison <scott@tqft.net>
parents:
597
diff
changeset
|
610 |
|
651 | 611 |
Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the |
612 |
homotopy colimit via the cone-product polyhedra in $\cell(W)$. |
|
613 |
A cone-product polyhedra is obtained from a point by successively taking the cone or taking the |
|
614 |
product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, |
|
615 |
cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, |
|
616 |
and taking product identifies the roots of several trees. |
|
657
9fbd8e63ab2e
fixing single quotes and long lines
Kevin Walker <kevin@canyon23.net>
parents:
656
diff
changeset
|
617 |
The ``local homotopy colimit" is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$. |
661 | 618 |
We further require that all (compositions of) morphisms in a directed tree are not expressible as a product. |
651 | 619 |
The differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required. |
657
9fbd8e63ab2e
fixing single quotes and long lines
Kevin Walker <kevin@canyon23.net>
parents:
656
diff
changeset
|
620 |
A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
621 |
|
605
78db9976b145
intro to more concrete \bc_* definition and misc
Kevin Walker <kevin@canyon23.net>
parents:
604
diff
changeset
|
622 |
%When $\cC$ is a topological $n$-category, |
78db9976b145
intro to more concrete \bc_* definition and misc
Kevin Walker <kevin@canyon23.net>
parents:
604
diff
changeset
|
623 |
%the flexibility available in the construction of a homotopy colimit allows |
78db9976b145
intro to more concrete \bc_* definition and misc
Kevin Walker <kevin@canyon23.net>
parents:
604
diff
changeset
|
624 |
%us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
78db9976b145
intro to more concrete \bc_* definition and misc
Kevin Walker <kevin@canyon23.net>
parents:
604
diff
changeset
|
625 |
%\todo{either need to explain why this is the same, or significantly rewrite this section} |
662
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
626 |
When $\cC$ is the plain $n$-category based on string diagrams for a traditional |
605
78db9976b145
intro to more concrete \bc_* definition and misc
Kevin Walker <kevin@canyon23.net>
parents:
604
diff
changeset
|
627 |
$n$-category $C$, |
610 | 628 |
one can show \cite{1009.5025} that the above two constructions of the homotopy colimit |
606 | 629 |
are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; \cC)$. |
630 |
Roughly speaking, the generators of $\bc_k(W; \cC)$ are string diagrams on $W$ together with |
|
605
78db9976b145
intro to more concrete \bc_* definition and misc
Kevin Walker <kevin@canyon23.net>
parents:
604
diff
changeset
|
631 |
a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested. |
78db9976b145
intro to more concrete \bc_* definition and misc
Kevin Walker <kevin@canyon23.net>
parents:
604
diff
changeset
|
632 |
The restriction of the string diagram to innermost blobs is required to be ``null" in the sense that |
78db9976b145
intro to more concrete \bc_* definition and misc
Kevin Walker <kevin@canyon23.net>
parents:
604
diff
changeset
|
633 |
it evaluates to a zero $n$-morphism of $C$. |
78db9976b145
intro to more concrete \bc_* definition and misc
Kevin Walker <kevin@canyon23.net>
parents:
604
diff
changeset
|
634 |
The next few paragraphs describe this in more detail. |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
635 |
|
667
0f45668726dd
more string diagram / field nonsense
Kevin Walker <kevin@canyon23.net>
parents:
666
diff
changeset
|
636 |
We will call a string diagram on a manifold a ``field". |
0f45668726dd
more string diagram / field nonsense
Kevin Walker <kevin@canyon23.net>
parents:
666
diff
changeset
|
637 |
(See \cite{1009.5025} for a more general notion of field.) |
0f45668726dd
more string diagram / field nonsense
Kevin Walker <kevin@canyon23.net>
parents:
666
diff
changeset
|
638 |
|
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
639 |
We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
640 |
if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
651 | 641 |
each $B_i$ appears as a connected component of one of the $M_j$. |
659
cc0c2dfe61f3
2nd installment of changes from proof-read
Kevin Walker <kevin@canyon23.net>
parents:
658
diff
changeset
|
642 |
Note that this forces the balls to be pairwise either disjoint or nested. |
651 | 643 |
Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. |
663 | 644 |
These pieces need not be manifolds, |
645 |
but they can be further subdivided into pieces which are manifolds |
|
646 |
and which fit into a permissible decomposition of $W$. |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
647 |
|
662
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
648 |
The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. |
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
649 |
A $k$-blob diagram consists of |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
650 |
\begin{itemize} |
663 | 651 |
\item a permissible collection of $k$ embedded balls, and |
652 |
\item a linear combination $s$ of string diagrams on $W$, |
|
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
653 |
\end{itemize} |
663 | 654 |
such that |
655 |
\begin{itemize} |
|
656 |
\item there is a permissible decomposition of $W$, compatible with the $k$ blobs, such that |
|
667
0f45668726dd
more string diagram / field nonsense
Kevin Walker <kevin@canyon23.net>
parents:
666
diff
changeset
|
657 |
$s$ is the product of linear combinations of fields $s_i$ on the initial pieces $X_i$ of the decomposition |
663 | 658 |
(for fixed restrictions to the boundaries of the pieces), |
659 |
\item the $s_i$'s corresponding to innermost blobs evaluate to zero in $\cC$, and |
|
667
0f45668726dd
more string diagram / field nonsense
Kevin Walker <kevin@canyon23.net>
parents:
666
diff
changeset
|
660 |
\item the $s_i$'s corresponding to the other pieces are single fields (linear combinations with only one term). |
663 | 661 |
\end{itemize} |
662 |
%that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. |
|
663 |
We call such linear combinations which evaluate to zero on a blob $B$ a ``null field on $B$". |
|
668 | 664 |
% could maybe say something here like "if blobs have nice complements then this is just...." |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
665 |
|
608 | 666 |
The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs. |
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
667 |
|
662
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
668 |
We now spell this out for some small values of $k$. |
667
0f45668726dd
more string diagram / field nonsense
Kevin Walker <kevin@canyon23.net>
parents:
666
diff
changeset
|
669 |
For $k=0$, the $0$-blob group is simply linear combinations of fields (string diagrams) on $W$. |
651 | 670 |
For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field to that ball is a null field. |
671 |
The differential simply forgets the ball. |
|
672 |
Thus we see that $H_0$ of the blob complex is the quotient of fields by fields which are null on some ball. |
|
580
99611dfed1f3
k-blobs for small k, and blob cochains
Scott Morrison <scott@tqft.net>
parents:
579
diff
changeset
|
673 |
|
651 | 674 |
For $k=2$, we have a two types of generators; they each consists of a field $f$ on $W$, and two balls $B_1$ and $B_2$. |
675 |
In the first case, the balls are disjoint, and $f$ restricted to either of the $B_i$ is a null field. |
|
676 |
In the second case, the balls are properly nested, say $B_1 \subset B_2$, and $f$ restricted to $B_1$ is null. |
|
677 |
Note that this implies that $f$ restricted to $B_2$ is also null, by the associativity of the gluing operation. |
|
678 |
This ensures that the differential is well-defined. |
|
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
679 |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
680 |
\section{Properties of the blob complex} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
681 |
\subsection{Formal properties} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
682 |
\label{sec:properties} |
602
109ecc26c50d
writing intro; just an expanded version of the existing notes, feel free to savage
Scott Morrison <scott@tqft.net>
parents:
601
diff
changeset
|
683 |
The blob complex enjoys the following list of formal properties. The first three are immediate from the definitions. |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
684 |
|
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
685 |
\begin{property}[Functoriality] |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
686 |
\label{property:functoriality}% |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
687 |
The blob complex is functorial with respect to homeomorphisms. |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
688 |
That is, |
574 | 689 |
for a fixed $n$-category $\cC$, the association |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
690 |
\begin{equation*} |
574 | 691 |
X \mapsto \bc_*(X; \cC) |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
692 |
\end{equation*} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
693 |
is a functor from $n$-manifolds and homeomorphisms between them to chain |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
694 |
complexes and isomorphisms between them. |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
695 |
\end{property} |
574 | 696 |
As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cC)$; |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
697 |
this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:CH} below. |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
698 |
|
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
699 |
\begin{property}[Disjoint union] |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
700 |
\label{property:disjoint-union} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
701 |
The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes. |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
702 |
\begin{equation*} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
703 |
\bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
704 |
\end{equation*} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
705 |
\end{property} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
706 |
|
574 | 707 |
If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ (we allow $Y = \eset$) as a codimension $0$ submanifold of its boundary, |
708 |
write $X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
709 |
\begin{property}[Gluing map] |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
710 |
\label{property:gluing-map}% |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
711 |
%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 |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
712 |
%\begin{equation*} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
713 |
%\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
714 |
%\end{equation*} |
607 | 715 |
Given a gluing $X \to X \bigcup_{Y}\selfarrow$, there is |
574 | 716 |
a map |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
717 |
\[ |
574 | 718 |
\bc_*(X) \to \bc_*(X \bigcup_{Y}\selfarrow), |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
719 |
\] |
574 | 720 |
natural with respect to homeomorphisms, and associative with respect to iterated gluings. |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
721 |
\end{property} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
722 |
|
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
723 |
\begin{property}[Contractibility] |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
724 |
\label{property:contractibility}% |
589
14b7d867e423
a few changes, maybe bad ones...
Scott Morrison <scott@tqft.net>
parents:
577
diff
changeset
|
725 |
The blob complex on an $n$-ball is contractible in the sense |
14b7d867e423
a few changes, maybe bad ones...
Scott Morrison <scott@tqft.net>
parents:
577
diff
changeset
|
726 |
that it is homotopic to its $0$-th homology, and this is just the vector space associated to the ball by the $n$-category. |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
727 |
\begin{equation*} |
649 | 728 |
\xymatrix{\bc_*(B^n;\cC) \ar[r]^(0.4){\htpy} & H_0(\bc_*(B^n;\cC)) \ar[r]^(0.6)\iso & \cC(B^n)} |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
729 |
\end{equation*} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
730 |
\end{property} |
627
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
731 |
%\nn{maybe should say something about the $A_\infty$ case} |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
732 |
|
583 | 733 |
\begin{proof}(Sketch) |
734 |
For $k\ge 1$, the contracting homotopy sends a $k$-blob diagram to the $(k{+}1)$-blob diagram |
|
735 |
obtained by adding an outer $(k{+}1)$-st blob consisting of all $B^n$. |
|
736 |
For $k=0$ we choose a splitting $s: H_0(\bc_*(B^n)) \to \bc_0(B^n)$ and send |
|
737 |
$x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$. |
|
738 |
\end{proof} |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
739 |
|
649 | 740 |
If $\cC$ is an $A_\infty$ $n$-category then $\bc_*(B^n;\cC)$ is still homotopy equivalent to $\cC(B^n)$, |
627
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
741 |
but this is no longer concentrated in degree zero. |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
742 |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
743 |
\subsection{Specializations} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
744 |
\label{sec:specializations} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
745 |
|
615
222da6df3edc
various minor, and moving mapping spaces to 'specializations'
Scott Morrison <scott@tqft.net>
parents:
614
diff
changeset
|
746 |
The blob complex has several important special cases. |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
747 |
|
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
748 |
\begin{thm}[Skein modules] |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
749 |
\label{thm:skein-modules} |
662
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
750 |
Suppose $\cC$ is a plain $n$-category. |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
751 |
The $0$-th blob homology of $X$ is the usual |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
752 |
(dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
589
14b7d867e423
a few changes, maybe bad ones...
Scott Morrison <scott@tqft.net>
parents:
577
diff
changeset
|
753 |
by $\cC$. |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
754 |
\begin{equation*} |
649 | 755 |
H_0(\bc_*(X;\cC)) \iso \cl{\cC}(X) |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
756 |
\end{equation*} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
757 |
\end{thm} |
651 | 758 |
This follows from the fact that the $0$-th homology of a homotopy colimit is the usual colimit, |
759 |
or directly from the explicit description of the blob complex. |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
760 |
|
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
761 |
\begin{thm}[Hochschild homology when $X=S^1$] |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
762 |
\label{thm:hochschild} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
763 |
The blob complex for a $1$-category $\cC$ on the circle is |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
764 |
quasi-isomorphic to the Hochschild complex. |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
765 |
\begin{equation*} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
766 |
\xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
767 |
\end{equation*} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
768 |
\end{thm} |
651 | 769 |
This theorem is established by extending the statement to bimodules as well as categories, |
770 |
then verifying that the universal properties of Hochschild homology also hold for $\bc_*(S^1; -)$. |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
771 |
|
615
222da6df3edc
various minor, and moving mapping spaces to 'specializations'
Scott Morrison <scott@tqft.net>
parents:
614
diff
changeset
|
772 |
\begin{thm}[Mapping spaces] |
222da6df3edc
various minor, and moving mapping spaces to 'specializations'
Scott Morrison <scott@tqft.net>
parents:
614
diff
changeset
|
773 |
\label{thm:map-recon} |
222da6df3edc
various minor, and moving mapping spaces to 'specializations'
Scott Morrison <scott@tqft.net>
parents:
614
diff
changeset
|
774 |
Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
222da6df3edc
various minor, and moving mapping spaces to 'specializations'
Scott Morrison <scott@tqft.net>
parents:
614
diff
changeset
|
775 |
$B^n \to T$. |
222da6df3edc
various minor, and moving mapping spaces to 'specializations'
Scott Morrison <scott@tqft.net>
parents:
614
diff
changeset
|
776 |
(The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
222da6df3edc
various minor, and moving mapping spaces to 'specializations'
Scott Morrison <scott@tqft.net>
parents:
614
diff
changeset
|
777 |
Then |
222da6df3edc
various minor, and moving mapping spaces to 'specializations'
Scott Morrison <scott@tqft.net>
parents:
614
diff
changeset
|
778 |
$$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
222da6df3edc
various minor, and moving mapping spaces to 'specializations'
Scott Morrison <scott@tqft.net>
parents:
614
diff
changeset
|
779 |
\end{thm} |
222da6df3edc
various minor, and moving mapping spaces to 'specializations'
Scott Morrison <scott@tqft.net>
parents:
614
diff
changeset
|
780 |
|
222da6df3edc
various minor, and moving mapping spaces to 'specializations'
Scott Morrison <scott@tqft.net>
parents:
614
diff
changeset
|
781 |
This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
651 | 782 |
Note that there is no restriction on the connectivity of $T$ as there is for |
783 |
the corresponding result in topological chiral homology \cite[Theorem 3.8.6]{0911.0018}. |
|
659
cc0c2dfe61f3
2nd installment of changes from proof-read
Kevin Walker <kevin@canyon23.net>
parents:
658
diff
changeset
|
784 |
The result is proved in \cite[\S 7.3]{1009.5025}. |
615
222da6df3edc
various minor, and moving mapping spaces to 'specializations'
Scott Morrison <scott@tqft.net>
parents:
614
diff
changeset
|
785 |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
786 |
\subsection{Structure of the blob complex} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
787 |
\label{sec:structure} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
788 |
|
651 | 789 |
In the following $\CH{X} = C_*(\Homeo(X))$ is the singular chain complex of the space |
790 |
of homeomorphisms of $X$, fixed on $\bdy X$. |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
791 |
|
585
e2996d7b4e6c
various, mostly working on axioms
Scott Morrison <scott@tqft.net>
parents:
584
diff
changeset
|
792 |
\begin{thm} |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
793 |
\label{thm:CH}\label{thm:evaluation} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
794 |
There is a chain map |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
795 |
\begin{equation*} |
585
e2996d7b4e6c
various, mostly working on axioms
Scott Morrison <scott@tqft.net>
parents:
584
diff
changeset
|
796 |
e_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X) |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
797 |
\end{equation*} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
798 |
such that |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
799 |
\begin{enumerate} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
800 |
\item Restricted to $CH_0(X)$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
801 |
|
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
802 |
\item For |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
803 |
any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
804 |
(using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
805 |
\begin{equation*} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
806 |
\xymatrix@C+0.3cm{ |
611 | 807 |
\CH{X} \tensor \bc_*(X) |
808 |
\ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \tensor \gl_Y} & |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
809 |
\bc_*(X) \ar[d]_{\gl_Y} \\ |
611 | 810 |
\CH{X \bigcup_Y \selfarrow} \tensor \bc_*(X \bigcup_Y \selfarrow) \ar[r]_<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
811 |
} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
812 |
\end{equation*} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
813 |
\end{enumerate} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
814 |
|
609 | 815 |
Further, this map is associative, in the sense that the following diagram commutes (up to homotopy). |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
816 |
\begin{equation*} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
817 |
\xymatrix{ |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
818 |
\CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor e_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{e_X} \\ |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
819 |
\CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X) |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
820 |
} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
821 |
\end{equation*} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
822 |
\end{thm} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
823 |
|
609 | 824 |
\begin{proof}(Sketch.) |
622
dda6d3a00b09
minor tweaks in sketch proofs
Scott Morrison <scott@tqft.net>
parents:
620
diff
changeset
|
825 |
We introduce yet another homotopy equivalent version of |
609 | 826 |
the blob complex, $\cB\cT_*(X)$. |
827 |
Blob diagrams have a natural topology, which is ignored by $\bc_*(X)$. |
|
828 |
In $\cB\cT_*(X)$ we take this topology into account, treating the blob diagrams as something |
|
829 |
analogous to a simplicial space (but with cone-product polyhedra replacing simplices). |
|
651 | 830 |
More specifically, a generator of $\cB\cT_k(X)$ is an $i$-parameter family of $j$-blob diagrams, with $i+j=k$. |
831 |
An essential step in the proof of this equivalence is a result to the effect that a $k$-parameter |
|
652
821d79885bfe
minor changes from proofreading
Scott Morrison <scott@tqft.net>
parents:
651
diff
changeset
|
832 |
family of homeomorphisms can be localized to at most $k$ small sets. |
609 | 833 |
|
652
821d79885bfe
minor changes from proofreading
Scott Morrison <scott@tqft.net>
parents:
651
diff
changeset
|
834 |
With this alternate version in hand, the theorem is straightforward. |
657
9fbd8e63ab2e
fixing single quotes and long lines
Kevin Walker <kevin@canyon23.net>
parents:
656
diff
changeset
|
835 |
By functoriality (Property \ref{property:functoriality}) $\Homeo(X)$ acts on the set $BD_j(X)$ of $j$-blob diagrams, and this |
614 | 836 |
induces a chain map $\CH{X}\tensor C_*(BD_j(X))\to C_*(BD_j(X))$ |
837 |
and hence a map $e_X: \CH{X} \tensor \cB\cT_*(X) \to \cB\cT_*(X)$. |
|
609 | 838 |
It is easy to check that $e_X$ thus defined has the desired properties. |
839 |
\end{proof} |
|
575
4e6f00784bd3
writing on the plane to kyoto: the blob complex as homotopy colimit and explicitly (but not why these are the same), and copy and paste of statements of axioms
Scott Morrison <scott@tqft.net>
parents:
574
diff
changeset
|
840 |
|
585
e2996d7b4e6c
various, mostly working on axioms
Scott Morrison <scott@tqft.net>
parents:
584
diff
changeset
|
841 |
\begin{thm} |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
842 |
\label{thm:blobs-ainfty} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
843 |
Let $\cC$ be a topological $n$-category. |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
844 |
Let $Y$ be an $n{-}k$-manifold. |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
845 |
There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, |
610 | 846 |
to be the set $$\bc_*(Y;\cC)(D) = \cl{\cC}(Y \times D)$$ and on $k$-balls $D$ to be the set |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
847 |
$$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
848 |
(When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
849 |
These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
850 |
Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
851 |
\end{thm} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
852 |
\begin{rem} |
610 | 853 |
When $Y$ is a point this produces an $A_\infty$ $n$-category from a topological $n$-category, |
854 |
which can be thought of as a free resolution. |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
855 |
\end{rem} |
610 | 856 |
This result is described in more detail as Example 6.2.8 of \cite{1009.5025}. |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
857 |
|
618 | 858 |
Fix a topological $n$-category $\cC$, which we'll now omit from notation. |
661 | 859 |
From the above, associated to any $(n{-}1)$-manifold $Y$ is an $A_\infty$ category $\bc_*(Y)$. |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
860 |
|
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
861 |
\begin{thm}[Gluing formula] |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
862 |
\label{thm:gluing} |
652
821d79885bfe
minor changes from proofreading
Scott Morrison <scott@tqft.net>
parents:
651
diff
changeset
|
863 |
\mbox{}\vspace{-0.2cm}% <-- gets the indenting right |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
864 |
\begin{itemize} |
651 | 865 |
\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, |
866 |
the blob complex of $X$ is naturally an |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
867 |
$A_\infty$ module for $\bc_*(Y)$. |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
868 |
|
585
e2996d7b4e6c
various, mostly working on axioms
Scott Morrison <scott@tqft.net>
parents:
584
diff
changeset
|
869 |
\item The blob complex of a glued manifold $X\bigcup_Y \selfarrow$ is the $A_\infty$ self-tensor product of |
659
cc0c2dfe61f3
2nd installment of changes from proof-read
Kevin Walker <kevin@canyon23.net>
parents:
658
diff
changeset
|
870 |
$\bc_*(X)$ as a $\bc_*(Y)$-bimodule: |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
871 |
\begin{equation*} |
585
e2996d7b4e6c
various, mostly working on axioms
Scott Morrison <scott@tqft.net>
parents:
584
diff
changeset
|
872 |
\bc_*(X\bigcup_Y \selfarrow) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
873 |
\end{equation*} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
874 |
\end{itemize} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
875 |
\end{thm} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
876 |
|
618 | 877 |
\begin{proof} (Sketch.) |
620
28b016b716b1
adding some proof sketches
Kevin Walker <kevin@canyon23.net>
parents:
619
diff
changeset
|
878 |
The $A_\infty$ action of $\bc_*(Y)$ follows from the naturality of the blob complex with respect to gluing |
28b016b716b1
adding some proof sketches
Kevin Walker <kevin@canyon23.net>
parents:
619
diff
changeset
|
879 |
and the $C_*(\Homeo(-))$ action of Theorem \ref{thm:evaluation}. |
618 | 880 |
|
620
28b016b716b1
adding some proof sketches
Kevin Walker <kevin@canyon23.net>
parents:
619
diff
changeset
|
881 |
Let $T_*$ denote the self tensor product of $\bc_*(X)$, which is a homotopy colimit. |
622
dda6d3a00b09
minor tweaks in sketch proofs
Scott Morrison <scott@tqft.net>
parents:
620
diff
changeset
|
882 |
There is a tautological map from the 0-simplices of $T_*$ to $\bc_*(X\bigcup_Y \selfarrow)$, |
620
28b016b716b1
adding some proof sketches
Kevin Walker <kevin@canyon23.net>
parents:
619
diff
changeset
|
883 |
and this map can be extended to a chain map on all of $T_*$ by sending the higher simplices to zero. |
652
821d79885bfe
minor changes from proofreading
Scott Morrison <scott@tqft.net>
parents:
651
diff
changeset
|
884 |
Constructing a homotopy inverse to this natural map involves making various choices, but one can show that the |
659
cc0c2dfe61f3
2nd installment of changes from proof-read
Kevin Walker <kevin@canyon23.net>
parents:
658
diff
changeset
|
885 |
choices form contractible subcomplexes and apply the acyclic models theorem. |
618 | 886 |
\end{proof} |
610 | 887 |
|
652
821d79885bfe
minor changes from proofreading
Scott Morrison <scott@tqft.net>
parents:
651
diff
changeset
|
888 |
We next describe the blob complex for product manifolds, in terms of the |
821d79885bfe
minor changes from proofreading
Scott Morrison <scott@tqft.net>
parents:
651
diff
changeset
|
889 |
blob complexes for the $A_\infty$ $n$-categories constructed as above. |
610 | 890 |
|
891 |
\begin{thm}[Product formula] |
|
892 |
\label{thm:product} |
|
652
821d79885bfe
minor changes from proofreading
Scott Morrison <scott@tqft.net>
parents:
651
diff
changeset
|
893 |
Let $W$ be a $k$-manifold and $Y$ be an $n{-}k$ manifold. |
662
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
894 |
Let $\cC$ be a plain $n$-category. |
610 | 895 |
Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above. |
896 |
Then |
|
897 |
\[ |
|
898 |
\bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). |
|
899 |
\] |
|
659
cc0c2dfe61f3
2nd installment of changes from proof-read
Kevin Walker <kevin@canyon23.net>
parents:
658
diff
changeset
|
900 |
That is, the blob complex of $Y\times W$ with coefficients in $\cC$ is homotopy equivalent |
cc0c2dfe61f3
2nd installment of changes from proof-read
Kevin Walker <kevin@canyon23.net>
parents:
658
diff
changeset
|
901 |
to the blob complex of $W$ with coefficients in $\bc_*(Y;\cC)$. |
610 | 902 |
\end{thm} |
903 |
The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
|
904 |
(see \cite[\S7.1]{1009.5025}). |
|
905 |
||
620
28b016b716b1
adding some proof sketches
Kevin Walker <kevin@canyon23.net>
parents:
619
diff
changeset
|
906 |
\begin{proof} (Sketch.) |
623 | 907 |
The proof is similar to that of the second part of Theorem \ref{thm:gluing}. |
908 |
There is a natural map from the 0-simplices of $\clh{\bc_*(Y;\cC)}(W)$ to $\bc_*(Y\times W; \cC)$, |
|
909 |
given by reinterpreting a decomposition of $W$ labeled by $(n{-}k)$-morphisms of $\bc_*(Y; \cC)$ as a blob |
|
910 |
diagram on $W\times Y$. |
|
911 |
This map can be extended to all of $\clh{\bc_*(Y;\cC)}(W)$ by sending higher simplices to zero. |
|
620
28b016b716b1
adding some proof sketches
Kevin Walker <kevin@canyon23.net>
parents:
619
diff
changeset
|
912 |
|
623 | 913 |
To construct the homotopy inverse of the above map one first shows that |
914 |
$\bc_*(Y\times W; \cC)$ is homotopy equivalent to the subcomplex generated by blob diagrams which |
|
652
821d79885bfe
minor changes from proofreading
Scott Morrison <scott@tqft.net>
parents:
651
diff
changeset
|
915 |
are small with respect to any fixed open cover of $Y\times W$. |
623 | 916 |
For a sufficiently fine open cover the generators of this ``small" blob complex are in the image of the map |
917 |
of the previous paragraph, and furthermore the preimage in $\clh{\bc_*(Y;\cC)}(W)$ of such small diagrams |
|
918 |
lie in contractible subcomplexes. |
|
919 |
A standard acyclic models argument now constructs the homotopy inverse. |
|
620
28b016b716b1
adding some proof sketches
Kevin Walker <kevin@canyon23.net>
parents:
619
diff
changeset
|
920 |
\end{proof} |
28b016b716b1
adding some proof sketches
Kevin Walker <kevin@canyon23.net>
parents:
619
diff
changeset
|
921 |
|
28b016b716b1
adding some proof sketches
Kevin Walker <kevin@canyon23.net>
parents:
619
diff
changeset
|
922 |
%\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
923 |
|
662
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
924 |
\section{Extending Deligne's conjecture to $n$-categories} |
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
925 |
\label{sec:applications} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
926 |
|
625
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
927 |
Let $M$ and $N$ be $n$-manifolds with common boundary $E$. |
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
928 |
Recall (Theorem \ref{thm:gluing}) that the $A_\infty$ category $A = \bc_*(E)$ |
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
929 |
acts on $\bc_*(M)$ and $\bc_*(N)$. |
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
930 |
Let $\hom_A(\bc_*(M), \bc_*(N))$ denote the chain complex of $A_\infty$ module maps |
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
931 |
from $\bc_*(M)$ to $\bc_*(N)$. |
652
821d79885bfe
minor changes from proofreading
Scott Morrison <scott@tqft.net>
parents:
651
diff
changeset
|
932 |
Let $R$ be another $n$-manifold with boundary $E^\text{op}$. |
625
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
933 |
There is a chain map |
661 | 934 |
\begin{equation*} |
625
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
935 |
\hom_A(\bc_*(M), \bc_*(N)) \ot \bc_*(M) \ot_A \bc_*(R) \to \bc_*(N) \ot_A \bc_*(R) . |
661 | 936 |
\end{equation*} |
625
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
937 |
We think of this map as being associated to a surgery which cuts $M$ out of $M\cup_E R$ and |
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
938 |
replaces it with $N$, yielding $N\cup_E R$. |
661 | 939 |
(This is a more general notion of surgery that usual: $M$ and $N$ can be any manifolds |
626 | 940 |
which share a common boundary.) |
627
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
941 |
In analogy to Hochschild cochains, we will call elements of $\hom_A(\bc_*(M), \bc_*(N))$ ``blob cochains". |
625
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
942 |
|
626 | 943 |
Recall (Theorem \ref{thm:evaluation}) that chains on the space of mapping cylinders also act on the |
625
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
944 |
blob complex. |
626 | 945 |
An $n$-dimensional surgery cylinder is |
946 |
defined to be a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), |
|
947 |
modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. |
|
659
cc0c2dfe61f3
2nd installment of changes from proof-read
Kevin Walker <kevin@canyon23.net>
parents:
658
diff
changeset
|
948 |
One can associate to this data an $(n{+}1)$-manifold with a foliation by intervals, |
626 | 949 |
and the relations we impose correspond to homeomorphisms of the $(n{+}1)$-manifolds |
950 |
which preserve the foliation. |
|
625
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
951 |
|
626 | 952 |
Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another. |
625
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
953 |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
954 |
\begin{thm}[Higher dimensional Deligne conjecture] |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
955 |
\label{thm:deligne} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
956 |
The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
957 |
\end{thm} |
577
9a60488cd2fc
out of battery. writing a little about the deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
575
diff
changeset
|
958 |
|
627
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
959 |
More specifically, let $M_0, N_0, \ldots, M_k, N_k$ be $n$-manifolds and let $SC^n_{\overline{M}, \overline{N}}$ |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
960 |
denote the component of the operad with outer boundary $M_0\cup N_0$ and inner boundaries |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
961 |
$M_1\cup N_1,\ldots, M_k\cup N_k$. |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
962 |
Then there is a collection of chain maps |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
963 |
\begin{multline*} |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
964 |
C_*(SC^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots \\ |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
965 |
\otimes \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
966 |
\end{multline*} |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
967 |
which satisfy the operad compatibility conditions. |
577
9a60488cd2fc
out of battery. writing a little about the deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
575
diff
changeset
|
968 |
|
661 | 969 |
\begin{proof} (Sketch.) |
610 | 970 |
We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, |
971 |
and the action of surgeries is just composition of maps of $A_\infty$-modules. |
|
662
57bd9fab3827
part way done with post phone call edits
Kevin Walker <kevin@canyon23.net>
parents:
661
diff
changeset
|
972 |
We only need to check that the relations of the surgery cylinder operad are satisfied. |
610 | 973 |
This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity. |
595
9c708975b61b
making pinched products axioms terser, and writing a short proof of the higher deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
594
diff
changeset
|
974 |
\end{proof} |
577
9a60488cd2fc
out of battery. writing a little about the deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
575
diff
changeset
|
975 |
|
652
821d79885bfe
minor changes from proofreading
Scott Morrison <scott@tqft.net>
parents:
651
diff
changeset
|
976 |
Consider the special case where $n=1$ and all of the $M_i$'s and $N_i$'s are intervals. |
627
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
977 |
We have that $SC^1_{\overline{M}, \overline{N}}$ is homotopy equivalent to the little |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
978 |
disks operad and $\hom(\bc_*(M_i), \bc_*(N_i))$ is homotopy equivalent to Hochschild cochains. |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
979 |
This special case is just the usual Deligne conjecture |
628
4cce595ae1d3
adding Gerstenhaber-Voronov, explicitly not proving the mapping spaces result, and slight tweaks
Scott Morrison <scott@tqft.net>
parents:
627
diff
changeset
|
980 |
(see \cite{hep-th/9403055, MR1328534, MR1805894, MR1805923, MR2064592}). |
627
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
981 |
|
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
982 |
The general case when $n=1$ goes beyond the original Deligne conjecture, as the $M_i$'s and $N_i$'s |
652
821d79885bfe
minor changes from proofreading
Scott Morrison <scott@tqft.net>
parents:
651
diff
changeset
|
983 |
could be disjoint unions of intervals and circles, and the surgery cylinders could be high genus surfaces. |
627
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
984 |
|
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
985 |
If all of the $M_i$'s and $N_i$'s are $n$-balls, then $SC^n_{\overline{M}, \overline{N}}$ |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
986 |
contains a copy of the little $(n{+}1)$-balls operad. |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
987 |
Thus the little $(n{+}1)$-balls operad acts on blob cochains of the $n$-ball. |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
988 |
|
577
9a60488cd2fc
out of battery. writing a little about the deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
575
diff
changeset
|
989 |
|
566 | 990 |
|
991 |
%% == end of paper: |
|
992 |
||
993 |
%% Optional Materials and Methods Section |
|
994 |
%% The Materials and Methods section header will be added automatically. |
|
995 |
||
996 |
%% Enter any subheads and the Materials and Methods text below. |
|
997 |
%\begin{materials} |
|
998 |
% Materials text |
|
999 |
%\end{materials} |
|
1000 |
||
1001 |
||
1002 |
%% Optional Appendix or Appendices |
|
1003 |
%% \appendix Appendix text... |
|
1004 |
%% or, for appendix with title, use square brackets: |
|
1005 |
%% \appendix[Appendix Title] |
|
1006 |
||
1007 |
\begin{acknowledgments} |
|
610 | 1008 |
It is a pleasure to acknowledge helpful conversations with |
1009 |
Kevin Costello, |
|
625
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
1010 |
Michael Freedman, |
610 | 1011 |
Justin Roberts, |
1012 |
and |
|
1013 |
Peter Teichner. |
|
1014 |
We also thank the Aspen Center for Physics for providing a pleasant and productive |
|
1015 |
environment during the last stages of this project. |
|
566 | 1016 |
\end{acknowledgments} |
1017 |
||
1018 |
%% PNAS does not support submission of supporting .tex files such as BibTeX. |
|
1019 |
%% Instead all references must be included in the article .tex document. |
|
1020 |
%% If you currently use BibTeX, your bibliography is formed because the |
|
1021 |
%% command \verb+\bibliography{}+ brings the <filename>.bbl file into your |
|
1022 |
%% .tex document. To conform to PNAS requirements, copy the reference listings |
|
1023 |
%% from your .bbl file and add them to the article .tex file, using the |
|
1024 |
%% bibliography environment described above. |
|
1025 |
||
1026 |
%% Contact pnas@nas.edu if you need assistance with your |
|
1027 |
%% bibliography. |
|
1028 |
||
1029 |
% Sample bibliography item in PNAS format: |
|
1030 |
%% \bibitem{in-text reference} comma-separated author names up to 5, |
|
1031 |
%% for more than 5 authors use first author last name et al. (year published) |
|
1032 |
%% article title {\it Journal Name} volume #: start page-end page. |
|
1033 |
%% ie, |
|
1034 |
% \bibitem{Neuhaus} Neuhaus J-M, Sitcher L, Meins F, Jr, Boller T (1991) |
|
1035 |
% A short C-terminal sequence is necessary and sufficient for the |
|
1036 |
% targeting of chitinases to the plant vacuole. |
|
1037 |
% {\it Proc Natl Acad Sci USA} 88:10362-10366. |
|
1038 |
||
1039 |
||
1040 |
%% Enter the largest bibliography number in the facing curly brackets |
|
1041 |
%% following \begin{thebibliography} |
|
1042 |
||
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
1043 |
%%%% BIBTEX |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
1044 |
\bibliographystyle{alpha} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
1045 |
\bibliography{../bibliography/bibliography} |
566 | 1046 |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
1047 |
%%%% non-BIBTEX |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
1048 |
%\begin{thebibliography}{} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
1049 |
% |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
1050 |
%\end{thebibliography} |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
1051 |
|
566 | 1052 |
|
1053 |
\end{article} |
|
1054 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1055 |
||
1056 |
%% Adding Figure and Table References |
|
1057 |
%% Be sure to add figures and tables after \end{article} |
|
1058 |
%% and before \end{document} |
|
1059 |
||
1060 |
%% For figures, put the caption below the illustration. |
|
1061 |
%% |
|
1062 |
%% \begin{figure} |
|
1063 |
%% \caption{Almost Sharp Front}\label{afoto} |
|
1064 |
%% \end{figure} |
|
1065 |
||
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
1066 |
|
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
1067 |
\begin{figure} |
594
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1068 |
\centering |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1069 |
\begin{tikzpicture}[%every label/.style={green} |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1070 |
] |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1071 |
\node[fill=black, circle, label=below:$E$, inner sep=1.5pt](S) at (0,0) {}; |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1072 |
\node[fill=black, circle, label=above:$E$, inner sep=1.5pt](N) at (0,2) {}; |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1073 |
\draw (S) arc (-90:90:1); |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1074 |
\draw (N) arc (90:270:1); |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1075 |
\node[left] at (-1,1) {$B_1$}; |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1076 |
\node[right] at (1,1) {$B_2$}; |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1077 |
\end{tikzpicture} |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1078 |
\caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1079 |
|
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1080 |
\begin{figure} |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1081 |
\centering |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1082 |
\begin{tikzpicture}[%every label/.style={green}, |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1083 |
x=1.5cm,y=1.5cm] |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1084 |
\node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {}; |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1085 |
\node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {}; |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1086 |
\draw (S) arc (-90:90:1); |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1087 |
\draw (N) arc (90:270:1); |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1088 |
\draw (N) -- (S); |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1089 |
\node[left] at (-1/4,1) {$B_1$}; |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1090 |
\node[right] at (1/4,1) {$B_2$}; |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1091 |
\node at (1/6,3/2) {$Y$}; |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1092 |
\end{tikzpicture} |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1093 |
\caption{From two balls to one ball.}\label{blah5}\end{figure} |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1094 |
|
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1095 |
\begin{figure} |
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
1096 |
\begin{equation*} |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
1097 |
\mathfig{.23}{ncat/zz2} |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
1098 |
\end{equation*} |
594
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1099 |
\caption{A small part of $\cell(W)$.} |
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
1100 |
\label{partofJfig} |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
1101 |
\end{figure} |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
1102 |
|
577
9a60488cd2fc
out of battery. writing a little about the deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
575
diff
changeset
|
1103 |
\begin{figure} |
626 | 1104 |
%$$\mathfig{.4}{deligne/manifolds}$$ |
1105 |
$$\mathfig{.4}{deligne/mapping-cylinders}$$ |
|
594
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
1106 |
\caption{An $n$-dimensional surgery cylinder.}\label{delfig2} |
577
9a60488cd2fc
out of battery. writing a little about the deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
575
diff
changeset
|
1107 |
\end{figure} |
9a60488cd2fc
out of battery. writing a little about the deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
575
diff
changeset
|
1108 |
|
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
1109 |
|
566 | 1110 |
%% For Tables, put caption above table |
1111 |
%% |
|
1112 |
%% Table caption should start with a capital letter, continue with lower case |
|
1113 |
%% and not have a period at the end |
|
1114 |
%% Using @{\vrule height ?? depth ?? width0pt} in the tabular preamble will |
|
1115 |
%% keep that much space between every line in the table. |
|
1116 |
||
1117 |
%% \begin{table} |
|
1118 |
%% \caption{Repeat length of longer allele by age of onset class} |
|
1119 |
%% \begin{tabular}{@{\vrule height 10.5pt depth4pt width0pt}lrcccc} |
|
1120 |
%% table text |
|
1121 |
%% \end{tabular} |
|
1122 |
%% \end{table} |
|
1123 |
||
1124 |
%% For two column figures and tables, use the following: |
|
1125 |
||
1126 |
%% \begin{figure*} |
|
1127 |
%% \caption{Almost Sharp Front}\label{afoto} |
|
1128 |
%% \end{figure*} |
|
1129 |
||
1130 |
%% \begin{table*} |
|
1131 |
%% \caption{Repeat length of longer allele by age of onset class} |
|
1132 |
%% \begin{tabular}{ccc} |
|
1133 |
%% table text |
|
1134 |
%% \end{tabular} |
|
1135 |
%% \end{table*} |
|
1136 |
||
1137 |
\end{document} |
|
1138 |