author | Scott Morrison <scott@tqft.net> |
Fri, 16 Jul 2010 17:24:20 -0600 | |
changeset 444 | c3fb6e8a7136 |
parent 437 | 93ce0ba3d2d7 |
child 454 | 3377d4db80d9 |
permissions | -rw-r--r-- |
98 | 1 |
%!TEX root = ../blob1.tex |
2 |
||
3 |
\section{Introduction} |
|
4 |
||
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
5 |
We construct the ``blob complex'' $\bc_*(M; \cC)$ associated to an $n$-manifold $M$ and a ``linear $n$-category with strong duality'' $\cC$. |
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
6 |
This blob complex provides a simultaneous generalization of several well-understood constructions: |
147 | 7 |
\begin{itemize} |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
8 |
\item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. |
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
9 |
(See Theorem \ref{thm:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
10 |
\item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
11 |
the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. |
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
12 |
(See Theorem \ref{thm:hochschild} and \S \ref{sec:hochschild}.) |
437 | 13 |
\item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category, we have |
155 | 14 |
that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
437 | 15 |
on the configuration space of unlabeled points in $M$. (See \S \ref{sec:comm_alg}.) |
155 | 16 |
%$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ |
147 | 17 |
\end{itemize} |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
18 |
The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
19 |
(replacing quotient of fields by local relations with some sort of resolution), |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
20 |
and for a generalization of Hochschild homology to higher $n$-categories. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
21 |
We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
22 |
The blob complex gives us all of these! More detailed motivations are described in \S \ref{sec:motivations}. |
147 | 23 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
24 |
The blob complex has good formal properties, summarized in \S \ref{sec:properties}. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
25 |
These include an action of $\CH{M}$, |
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
26 |
extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Theorem \ref{thm:evaluation}) and a gluing |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
27 |
formula allowing calculations by cutting manifolds into smaller parts (see Theorem \ref{thm:gluing}). |
147 | 28 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
29 |
We expect applications of the blob complex to contact topology and Khovanov homology but do not address these in this paper. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
30 |
See \S \ref{sec:future} for slightly more detail. |
147 | 31 |
|
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
32 |
\subsection{Structure of the paper} |
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
33 |
The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
34 |
summarise the formal properties of the blob complex (see \S \ref{sec:properties}), describe known specializations (see \S \ref{sec:specializations}), outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications}) |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
35 |
and outline anticipated future directions (see \S \ref{sec:future}). |
151 | 36 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
37 |
The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex, |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
38 |
and establishes some of its properties. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
39 |
There are many alternative definitions of $n$-categories, and part of our difficulty defining the blob complex is |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
40 |
simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. |
417
d3b05641e7ca
making quotation marks consistently "American style"
Kevin Walker <kevin@canyon23.net>
parents:
415
diff
changeset
|
41 |
At first we entirely avoid this problem by introducing the notion of a ``system of fields", and define the blob complex |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
42 |
associated to an $n$-manifold and an $n$-dimensional system of fields. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
43 |
We sketch the construction of a system of fields from a $1$-category or from a pivotal $2$-category. |
147 | 44 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
45 |
Nevertheless, when we attempt to establish all of the observed properties of the blob complex, |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
46 |
we find this situation unsatisfactory. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
47 |
Thus, in the second part of the paper (\S\S \ref{sec:ncats}-\ref{sec:ainfblob}) we give yet another |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
48 |
definition of an $n$-category, or rather a definition of an $n$-category with strong duality. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
49 |
(It appears that removing the duality conditions from our definition would make it more complicated rather than less.) |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
50 |
We call these ``topological $n$-categories'', to differentiate them from previous versions. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
51 |
Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms. |
147 | 52 |
|
417
d3b05641e7ca
making quotation marks consistently "American style"
Kevin Walker <kevin@canyon23.net>
parents:
415
diff
changeset
|
53 |
The basic idea is that each potential definition of an $n$-category makes a choice about the ``shape" of morphisms. |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
54 |
We try to be as lax as possible: a topological $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
55 |
These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
56 |
For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to each such $B$ and ask that the action of |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
57 |
homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
58 |
The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
59 |
topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
147 | 60 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
61 |
In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
62 |
(using a colimit along cellulations of a manifold), and in \S \ref{sec:ainfblob} give an alternative definition |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
63 |
of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). |
417
d3b05641e7ca
making quotation marks consistently "American style"
Kevin Walker <kevin@canyon23.net>
parents:
415
diff
changeset
|
64 |
Using these definitions, we show how to use the blob complex to ``resolve" any topological $n$-category as an |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
65 |
$A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
66 |
We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), |
417
d3b05641e7ca
making quotation marks consistently "American style"
Kevin Walker <kevin@canyon23.net>
parents:
415
diff
changeset
|
67 |
in particular the ``gluing formula" of Theorem \ref{thm:gluing} below. |
313 | 68 |
|
69 |
The relationship between all these ideas is sketched in Figure \ref{fig:outline}. |
|
150 | 70 |
|
307 | 71 |
\tikzstyle{box} = [rectangle, rounded corners, draw,outer sep = 5pt, inner sep = 5pt, line width=0.5pt] |
72 |
||
313 | 73 |
\begin{figure}[!ht] |
307 | 74 |
{\center |
75 |
||
76 |
\begin{tikzpicture}[align=center,line width = 1.5pt] |
|
77 |
\newcommand{\xa}{2} |
|
78 |
\newcommand{\xb}{10} |
|
79 |
\newcommand{\ya}{14} |
|
80 |
\newcommand{\yb}{10} |
|
81 |
\newcommand{\yc}{6} |
|
82 |
||
83 |
\node[box] at (\xa,\ya) (C) {$\cC$ \\ a topological \\ $n$-category}; |
|
319
121c580d5ef7
editting all over the place
Scott Morrison <scott@tqft.net>
parents:
314
diff
changeset
|
84 |
\node[box] at (\xb,\ya) (A) {$\underrightarrow{\cC}(M)$ \\ the (dual) TQFT \\ Hilbert space}; |
307 | 85 |
\node[box] at (\xa,\yb) (FU) {$(\cF, \cU)$ \\ fields and\\ local relations}; |
86 |
\node[box] at (\xb,\yb) (BC) {$\bc_*(M; \cC)$ \\ the blob complex}; |
|
87 |
\node[box] at (\xa,\yc) (Cs) {$\cC_*$ \\ an $A_\infty$ \\$n$-category}; |
|
319
121c580d5ef7
editting all over the place
Scott Morrison <scott@tqft.net>
parents:
314
diff
changeset
|
88 |
\node[box] at (\xb,\yc) (BCs) {$\underrightarrow{\cC_*}(M)$}; |
307 | 89 |
|
90 |
||
91 |
||
314 | 92 |
\draw[->] (C) -- node[above] {$\displaystyle \colim_{\cell(M)} \cC$} node[below] {\S\S \ref{sec:constructing-a-tqft} \& \ref{ss:ncat_fields}} (A); |
307 | 93 |
\draw[->] (FU) -- node[below] {blob complex \\ for $M$} (BC); |
314 | 94 |
\draw[->] (Cs) -- node[above] {$\displaystyle \hocolim_{\cell(M)} \cC_*$} node[below] {\S \ref{ss:ncat_fields}} (BCs); |
307 | 95 |
|
96 |
\draw[->] (FU) -- node[right=10pt] {$\cF(M)/\cU$} (A); |
|
97 |
||
314 | 98 |
\draw[->] (C) -- node[left=10pt] { |
99 |
Example \ref{ex:traditional-n-categories(fields)} \\ and \S \ref{ss:ncat_fields} |
|
307 | 100 |
%$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle \cU(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$ |
101 |
} (FU); |
|
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
102 |
\draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Theorem \ref{thm:skein-modules}} (A); |
307 | 103 |
|
104 |
\draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); |
|
105 |
\draw (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); |
|
106 |
\end{tikzpicture} |
|
107 |
||
108 |
} |
|
313 | 109 |
\caption{The main gadgets and constructions of the paper.} |
110 |
\label{fig:outline} |
|
111 |
\end{figure} |
|
307 | 112 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
113 |
Finally, later sections address other topics. |
437 | 114 |
Section \S \ref{sec:deligne} gives |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
115 |
a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. |
417
d3b05641e7ca
making quotation marks consistently "American style"
Kevin Walker <kevin@canyon23.net>
parents:
415
diff
changeset
|
116 |
The appendixes prove technical results about $\CH{M}$ and the ``small blob complex", |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
117 |
and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
437 | 118 |
as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
119 |
thought of as a topological $n$-category, in terms of the topology of $M$. |
|
150 | 120 |
|
155 | 121 |
|
150 | 122 |
\nn{some more things to cover in the intro} |
131 | 123 |
\begin{itemize} |
124 |
\item related: we are being unsophisticated from a homotopy theory point of |
|
160 | 125 |
view and using chain complexes in many places where we could get by with spaces |
131 | 126 |
\item ? one of the points we make (far) below is that there is not really much |
127 |
difference between (a) systems of fields and local relations and (b) $n$-cats; |
|
128 |
thus we tend to switch between talking in terms of one or the other |
|
129 |
\end{itemize} |
|
130 |
||
131 |
\medskip\hrule\medskip |
|
132 |
||
151 | 133 |
\subsection{Motivations} |
134 |
\label{sec:motivations} |
|
135 |
||
160 | 136 |
We will briefly sketch our original motivation for defining the blob complex. |
137 |
\nn{this is adapted from an old draft of the intro; it needs further modification |
|
138 |
in order to better integrate it into the current intro.} |
|
139 |
||
140 |
As a starting point, consider TQFTs constructed via fields and local relations. |
|
426
8aca80203f9d
search & replace: s/((sub?)section|appendix)\s+\\ref/\S\ref/
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
141 |
(See \S\ref{sec:tqftsviafields} or \cite{kw:tqft}.) |
98 | 142 |
This gives a satisfactory treatment for semisimple TQFTs |
143 |
(i.e.\ TQFTs for which the cylinder 1-category associated to an |
|
144 |
$n{-}1$-manifold $Y$ is semisimple for all $Y$). |
|
160 | 145 |
|
166 | 146 |
For non-semi-simple TQFTs, this approach is less satisfactory. |
98 | 147 |
Our main motivating example (though we will not develop it in this paper) |
160 | 148 |
is the (decapitated) $4{+}1$-dimensional TQFT associated to Khovanov homology. |
98 | 149 |
It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
150 |
with a link $L \subset \bd W$. |
|
151 |
The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
|
437 | 152 |
\todo{I'm tempted to replace $A_{Kh}$ with $\cl{Kh}$ throughout this page -S} |
160 | 153 |
|
154 |
How would we go about computing $A_{Kh}(W^4, L)$? |
|
437 | 155 |
For the Khovanov homology of a link in $S^3$ the main tool is the exact triangle (long exact sequence) |
166 | 156 |
relating resolutions of a crossing. |
98 | 157 |
Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt |
158 |
to compute $A_{Kh}(S^1\times B^3, L)$. |
|
437 | 159 |
According to the gluing theorem for TQFTs, gluing along $B^3 \subset \bd B^4$ |
98 | 160 |
corresponds to taking a coend (self tensor product) over the cylinder category |
161 |
associated to $B^3$ (with appropriate boundary conditions). |
|
162 |
The coend is not an exact functor, so the exactness of the triangle breaks. |
|
160 | 163 |
|
164 |
||
165 |
The obvious solution to this problem is to replace the coend with its derived counterpart. |
|
98 | 166 |
This presumably works fine for $S^1\times B^3$ (the answer being the Hochschild homology |
167 |
of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired. |
|
168 |
If we build our manifold up via a handle decomposition, the computation |
|
169 |
would be a sequence of derived coends. |
|
170 |
A different handle decomposition of the same manifold would yield a different |
|
171 |
sequence of derived coends. |
|
172 |
To show that our definition in terms of derived coends is well-defined, we |
|
173 |
would need to show that the above two sequences of derived coends yield the same answer. |
|
174 |
This is probably not easy to do. |
|
160 | 175 |
|
176 |
Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$ |
|
98 | 177 |
which is manifestly invariant. |
178 |
(That is, a definition that does not |
|
179 |
involve choosing a decomposition of $W$. |
|
180 |
After all, one of the virtues of our starting point --- TQFTs via field and local relations --- |
|
181 |
is that it has just this sort of manifest invariance.) |
|
160 | 182 |
|
183 |
The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient |
|
98 | 184 |
\[ |
185 |
\text{linear combinations of fields} \;\big/\; \text{local relations} , |
|
186 |
\] |
|
187 |
with an appropriately free resolution (the ``blob complex") |
|
188 |
\[ |
|
189 |
\cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) . |
|
190 |
\] |
|
191 |
Here $\bc_0$ is linear combinations of fields on $W$, |
|
192 |
$\bc_1$ is linear combinations of local relations on $W$, |
|
193 |
$\bc_2$ is linear combinations of relations amongst relations on $W$, |
|
194 |
and so on. |
|
160 | 195 |
|
196 |
None of the above ideas depend on the details of the Khovanov homology example, |
|
166 | 197 |
so we develop the general theory in this paper and postpone specific applications |
98 | 198 |
to later papers. |
160 | 199 |
|
200 |
||
98 | 201 |
|
147 | 202 |
\subsection{Formal properties} |
203 |
\label{sec:properties} |
|
437 | 204 |
The blob complex enjoys the following list of formal properties. |
98 | 205 |
|
206 |
\begin{property}[Functoriality] |
|
207 |
\label{property:functoriality}% |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
208 |
The blob complex is functorial with respect to homeomorphisms. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
209 |
That is, |
187 | 210 |
for a fixed $n$-dimensional system of fields $\cC$, the association |
98 | 211 |
\begin{equation*} |
131 | 212 |
X \mapsto \bc_*^{\cC}(X) |
98 | 213 |
\end{equation*} |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
214 |
is a functor from $n$-manifolds and homeomorphisms between them to chain |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
215 |
complexes and isomorphisms between them. |
98 | 216 |
\end{property} |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
217 |
As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*^\cC(X)$; |
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
218 |
this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:evaluation} below. |
98 | 219 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
220 |
The blob complex is also functorial (indeed, exact) with respect to $\cC$, |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
221 |
although we will not address this in detail here. |
131 | 222 |
|
98 | 223 |
\begin{property}[Disjoint union] |
224 |
\label{property:disjoint-union} |
|
225 |
The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
|
226 |
\begin{equation*} |
|
227 |
\bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
|
228 |
\end{equation*} |
|
229 |
\end{property} |
|
230 |
||
437 | 231 |
If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, |
232 |
write $X_\text{gl} = X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
233 |
Note that this includes the case of gluing two disjoint manifolds together. |
131 | 234 |
\begin{property}[Gluing map] |
98 | 235 |
\label{property:gluing-map}% |
136 | 236 |
%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 |
237 |
%\begin{equation*} |
|
238 |
%\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
|
239 |
%\end{equation*} |
|
437 | 240 |
Given a gluing $X \to X_\mathrm{gl}$, there is |
131 | 241 |
a natural map |
242 |
\[ |
|
437 | 243 |
\bc_*(X) \to \bc_*(X_\mathrm{gl}) |
131 | 244 |
\] |
187 | 245 |
(natural with respect to homeomorphisms, and also associative with respect to iterated gluings). |
98 | 246 |
\end{property} |
247 |
||
248 |
\begin{property}[Contractibility] |
|
249 |
\label{property:contractibility}% |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
250 |
With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
251 |
Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cC$ to balls. |
437 | 252 |
\begin{equation*} |
187 | 253 |
\xymatrix{\bc_*^{\cC}(B^n) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*^{\cC}(B^n)) \ar[r]^(0.6)\iso & \cC(B^n)} |
437 | 254 |
\end{equation*} |
98 | 255 |
\end{property} |
256 |
||
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
257 |
Properties \ref{property:functoriality} will be immediate from the definition given in |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
258 |
\S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there. |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
259 |
Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
260 |
|
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
261 |
\subsection{Specializations} |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
262 |
\label{sec:specializations} |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
263 |
|
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
264 |
The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology. |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
265 |
|
437 | 266 |
\newtheorem*{thm:skein-modules}{Theorem \ref{thm:skein-modules}} |
267 |
||
268 |
\begin{thm:skein-modules}[Skein modules] |
|
98 | 269 |
The $0$-th blob homology of $X$ is the usual |
270 |
(dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
271 |
by $\cC$. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
272 |
(See \S \ref{sec:local-relations}.) |
98 | 273 |
\begin{equation*} |
131 | 274 |
H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X) |
98 | 275 |
\end{equation*} |
437 | 276 |
\end{thm:skein-modules} |
98 | 277 |
|
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
278 |
\newtheorem*{thm:hochschild}{Theorem \ref{thm:hochschild}} |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
279 |
|
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
280 |
\begin{thm:hochschild}[Hochschild homology when $X=S^1$] |
98 | 281 |
The blob complex for a $1$-category $\cC$ on the circle is |
282 |
quasi-isomorphic to the Hochschild complex. |
|
283 |
\begin{equation*} |
|
187 | 284 |
\xymatrix{\bc_*^{\cC}(S^1) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).} |
98 | 285 |
\end{equation*} |
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
286 |
\end{thm:hochschild} |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
287 |
|
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
288 |
Theorem \ref{thm:skein-modules} is immediate from the definition, and |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
289 |
Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}. |
437 | 290 |
We also note \S \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of certain commutative algebras thought of as an $n$-category. |
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
291 |
|
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
292 |
|
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
293 |
\subsection{Structure of the blob complex} |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
294 |
\label{sec:structure} |
98 | 295 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
187
diff
changeset
|
296 |
In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
437 | 297 |
|
298 |
\newtheorem*{thm:CH}{Theorem \ref{thm:CH}} |
|
299 |
||
300 |
\begin{thm:CH}[$C_*(\Homeo(-))$ action]\mbox{}\\ |
|
313 | 301 |
\vspace{-0.5cm} |
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
302 |
\label{thm:evaluation}% |
437 | 303 |
There is a chain map |
98 | 304 |
\begin{equation*} |
166 | 305 |
\ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
98 | 306 |
\end{equation*} |
437 | 307 |
such that |
308 |
\begin{enumerate} |
|
313 | 309 |
\item Restricted to $C_0(\Homeo(X))$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
187
diff
changeset
|
310 |
|
313 | 311 |
\item For |
147 | 312 |
any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
313 | 313 |
(using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
147 | 314 |
\begin{equation*} |
315 |
\xymatrix@C+2cm{ |
|
166 | 316 |
\CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{\ev_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) \\ |
317 |
\CH{X} \otimes \bc_*(X) |
|
318 |
\ar[r]_{\ev_{X}} \ar[u]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
|
147 | 319 |
\bc_*(X) \ar[u]_{\gl_Y} |
320 |
} |
|
321 |
\end{equation*} |
|
437 | 322 |
\end{enumerate} |
323 |
Moreover any such chain map is unique, up to an iterated homotopy. |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
324 |
(That is, any pair of homotopies have a homotopy between them, and so on.) |
437 | 325 |
\end{thm:CH} |
326 |
||
327 |
\newtheorem*{thm:CH-associativity}{Theorem \ref{thm:CH-associativity}} |
|
328 |
||
329 |
||
330 |
Further, |
|
331 |
\begin{thm:CH-associativity} |
|
332 |
\item The chain map of Theorem \ref{thm:CH} is associative, in the sense that the following diagram commutes (up to homotopy). |
|
313 | 333 |
\begin{equation*} |
334 |
\xymatrix{ |
|
335 |
\CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor \ev_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{\ev_X} \\ |
|
336 |
\CH{X} \tensor \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) |
|
337 |
} |
|
338 |
\end{equation*} |
|
437 | 339 |
\end{thm:CH-associativity} |
98 | 340 |
|
313 | 341 |
Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
187
diff
changeset
|
342 |
$$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
313 | 343 |
for any homeomorphic pair $X$ and $Y$, |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
187
diff
changeset
|
344 |
satisfying corresponding conditions. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
187
diff
changeset
|
345 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
346 |
In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
347 |
Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
348 |
Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
151 | 349 |
|
437 | 350 |
\todo{Give this a number inside the text} |
351 |
\begin{thm}[Blob complexes of products with balls form an $A_\infty$ $n$-category] |
|
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
352 |
\label{thm:blobs-ainfty} |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
353 |
Let $\cC$ be a topological $n$-category. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
354 |
Let $Y$ be an $n{-}k$-manifold. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
355 |
There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
356 |
to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
357 |
$$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
358 |
(When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
359 |
These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in |
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
360 |
Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
361 |
\end{thm} |
151 | 362 |
\begin{rem} |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
363 |
Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from a topological $n$-category. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
364 |
We think of this $A_\infty$ $n$-category as a free resolution. |
151 | 365 |
\end{rem} |
437 | 366 |
Theorem \ref{thm:blobs-ainfty} appears as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats} |
151 | 367 |
|
136 | 368 |
There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
369 |
instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. |
437 | 370 |
The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
136 | 371 |
|
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
372 |
\newtheorem*{thm:product}{Theorem \ref{thm:product}} |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
373 |
|
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
374 |
\begin{thm:product}[Product formula] |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
375 |
Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
376 |
Let $\cC$ be an $n$-category. |
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
377 |
Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Theorem \ref{thm:blobs-ainfty}). |
136 | 378 |
Then |
379 |
\[ |
|
338
adc0780aa5e7
updating notation in intro, also deciding that not everything is a 'property'
Scott Morrison <scott@tqft.net>
parents:
332
diff
changeset
|
380 |
\bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
136 | 381 |
\] |
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
382 |
\end{thm:product} |
338
adc0780aa5e7
updating notation in intro, also deciding that not everything is a 'property'
Scott Morrison <scott@tqft.net>
parents:
332
diff
changeset
|
383 |
We also give a generalization of this statement for arbitrary fibre bundles, in \S \ref{moddecss}, and a sketch of a statement for arbitrary maps. |
adc0780aa5e7
updating notation in intro, also deciding that not everything is a 'property'
Scott Morrison <scott@tqft.net>
parents:
332
diff
changeset
|
384 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
385 |
Fix a topological $n$-category $\cC$, which we'll omit from the notation. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
386 |
Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
387 |
(See Appendix \ref{sec:comparing-A-infty} for the translation between topological $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.) |
98 | 388 |
|
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
389 |
\newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}} |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
390 |
|
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
391 |
\begin{thm:gluing}[Gluing formula] |
98 | 392 |
\mbox{}% <-- gets the indenting right |
393 |
\begin{itemize} |
|
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
217
diff
changeset
|
394 |
\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an |
338
adc0780aa5e7
updating notation in intro, also deciding that not everything is a 'property'
Scott Morrison <scott@tqft.net>
parents:
332
diff
changeset
|
395 |
$A_\infty$ module for $\bc_*(Y)$. |
98 | 396 |
|
437 | 397 |
\item For any $n$-manifold $X_\text{gl} = X\bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{gl})$ is the $A_\infty$ self-tensor product of |
398 |
$\bc_*(X)$ as an $\bc_*(Y)$-bimodule: |
|
98 | 399 |
\begin{equation*} |
437 | 400 |
\bc_*(X_\text{gl}) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
98 | 401 |
\end{equation*} |
402 |
\end{itemize} |
|
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
403 |
\end{thm:gluing} |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
404 |
|
437 | 405 |
Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, with Theorem \ref{thm:gluing} then a relatively straightforward consequence of the proof, explained in \S \ref{sec:gluing}. |
98 | 406 |
|
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
407 |
\subsection{Applications} |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
408 |
\label{sec:applications} |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
409 |
Finally, we give two theorems which we consider as applications. |
117
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
98
diff
changeset
|
410 |
|
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
411 |
\newtheorem*{thm:map-recon}{Theorem \ref{thm:map-recon}} |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
412 |
|
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
413 |
\begin{thm:map-recon}[Mapping spaces] |
187 | 414 |
Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
415 |
$B^n \to T$. |
|
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
217
diff
changeset
|
416 |
(The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
136 | 417 |
Then |
187 | 418 |
$$\bc_*(X, \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
419 |
\end{thm:map-recon} |
117
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
98
diff
changeset
|
420 |
|
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
421 |
This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data. The proof appears in \S \ref{sec:map-recon}. |
187 | 422 |
|
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
423 |
\newtheorem*{thm:deligne}{Theorem \ref{thm:deligne}} |
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
424 |
|
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
425 |
\begin{thm:deligne}[Higher dimensional Deligne conjecture] |
117
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
98
diff
changeset
|
426 |
The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
427 |
\end{thm:deligne} |
338
adc0780aa5e7
updating notation in intro, also deciding that not everything is a 'property'
Scott Morrison <scott@tqft.net>
parents:
332
diff
changeset
|
428 |
See \S \ref{sec:deligne} for a full explanation of the statement, and an outline of the proof. |
98 | 429 |
|
400
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
430 |
|
a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
parents:
340
diff
changeset
|
431 |
|
148 | 432 |
|
433 |
\subsection{Future directions} |
|
151 | 434 |
\label{sec:future} |
155 | 435 |
Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard). |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
436 |
In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
437 |
We could presumably also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories), |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
438 |
and likely it will prove useful to think about the connections between what we do here and $(\infty,k)$-categories. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
439 |
More could be said about finite characteristic |
437 | 440 |
(there appears in be $2$-torsion in $\bc_1(S^2; \cC)$ for any spherical $2$-category $\cC$, for example). |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
441 |
Much more could be said about other types of manifolds, in particular oriented, |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
442 |
$\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
443 |
(We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
444 |
We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
445 |
there may be some differences for topological manifolds and smooth manifolds. |
148 | 446 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
447 |
The paper ``Skein homology'' \cite{MR1624157} has similar motivations, and it may be |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
448 |
interesting to investigate if there is a connection with the material here. |
314 | 449 |
|
417
d3b05641e7ca
making quotation marks consistently "American style"
Kevin Walker <kevin@canyon23.net>
parents:
415
diff
changeset
|
450 |
Many results in Hochschild homology can be understood ``topologically" via the blob complex. |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
451 |
For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ |
437 | 452 |
(see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1]; A)$, |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
453 |
but haven't investigated the details. |
148 | 454 |
|
437 | 455 |
Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} \nn{stabilization} \nn{stable categories, generalized cohomology theories} |
148 | 456 |
|
457 |
||
458 |
\subsection{Thanks and acknowledgements} |
|
411
98b8559b0b7a
starting to work on tqdftreview.tex
Kevin Walker <kevin@canyon23.net>
parents:
407
diff
changeset
|
459 |
% attempting to make this chronological rather than alphabetical |
98b8559b0b7a
starting to work on tqdftreview.tex
Kevin Walker <kevin@canyon23.net>
parents:
407
diff
changeset
|
460 |
We'd like to thank |
98b8559b0b7a
starting to work on tqdftreview.tex
Kevin Walker <kevin@canyon23.net>
parents:
407
diff
changeset
|
461 |
Justin Roberts, |
98b8559b0b7a
starting to work on tqdftreview.tex
Kevin Walker <kevin@canyon23.net>
parents:
407
diff
changeset
|
462 |
Michael Freedman, |
98b8559b0b7a
starting to work on tqdftreview.tex
Kevin Walker <kevin@canyon23.net>
parents:
407
diff
changeset
|
463 |
Peter Teichner, |
98b8559b0b7a
starting to work on tqdftreview.tex
Kevin Walker <kevin@canyon23.net>
parents:
407
diff
changeset
|
464 |
David Ben-Zvi, |
98b8559b0b7a
starting to work on tqdftreview.tex
Kevin Walker <kevin@canyon23.net>
parents:
407
diff
changeset
|
465 |
Vaughan Jones, |
98b8559b0b7a
starting to work on tqdftreview.tex
Kevin Walker <kevin@canyon23.net>
parents:
407
diff
changeset
|
466 |
Chris Schommer-Pries, |
98b8559b0b7a
starting to work on tqdftreview.tex
Kevin Walker <kevin@canyon23.net>
parents:
407
diff
changeset
|
467 |
Thomas Tradler, |
98b8559b0b7a
starting to work on tqdftreview.tex
Kevin Walker <kevin@canyon23.net>
parents:
407
diff
changeset
|
468 |
Kevin Costello, |
98b8559b0b7a
starting to work on tqdftreview.tex
Kevin Walker <kevin@canyon23.net>
parents:
407
diff
changeset
|
469 |
Chris Douglas, |
98b8559b0b7a
starting to work on tqdftreview.tex
Kevin Walker <kevin@canyon23.net>
parents:
407
diff
changeset
|
470 |
and |
98b8559b0b7a
starting to work on tqdftreview.tex
Kevin Walker <kevin@canyon23.net>
parents:
407
diff
changeset
|
471 |
Alexander Kirillov |
98b8559b0b7a
starting to work on tqdftreview.tex
Kevin Walker <kevin@canyon23.net>
parents:
407
diff
changeset
|
472 |
for many interesting and useful conversations. |
270 | 473 |
During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. |
155 | 474 |