author | Scott Morrison <scott@tqft.net> |
Thu, 17 Jun 2010 12:35:32 -0700 | |
changeset 371 | 7d15bb9afe01 |
parent 340 | f7da004e1f14 |
child 400 | a02a6158f3bd |
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'' $\cC$. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
6 |
This blob complex provides a simultaneous generalisation 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 $\cC$. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
9 |
(See Property \ref{property:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
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 $\HC_*(\cC)$. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
12 |
(See Property \ref{property:hochschild} and \S \ref{sec:hochschild}.) |
155 | 13 |
\item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have |
14 |
that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
|
280 | 15 |
on the configuration space of unlabeled points in $M$. |
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 $\CH{M}$, |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
26 |
extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Property \ref{property:evaluation}) and a gluing |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
27 |
formula allowing calculations by cutting manifolds into smaller parts (see Property \ref{property: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 |
|
32 |
\subsubsection{Structure of the paper} |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
33 |
The three subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
34 |
summarise the formal properties of the blob complex (see \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
|
35 |
and outline anticipated future directions and applications (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. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
41 |
At first we entirely avoid this problem by introducing the notion of a `system of fields', and define 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
|
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 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
53 |
The basic idea is that each potential definition of an $n$-category makes a choice about the `shape' of morphisms. |
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). |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
64 |
Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an |
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), |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
67 |
in particular the `gluing formula' of Property \ref{property:gluing} below. |
313 | 68 |
|
69 |
The relationship between all these ideas is sketched in Figure \ref{fig:outline}. |
|
150 | 70 |
|
155 | 71 |
\nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.} |
72 |
||
307 | 73 |
\tikzstyle{box} = [rectangle, rounded corners, draw,outer sep = 5pt, inner sep = 5pt, line width=0.5pt] |
74 |
||
313 | 75 |
\begin{figure}[!ht] |
307 | 76 |
{\center |
77 |
||
78 |
\begin{tikzpicture}[align=center,line width = 1.5pt] |
|
79 |
\newcommand{\xa}{2} |
|
80 |
\newcommand{\xb}{10} |
|
81 |
\newcommand{\ya}{14} |
|
82 |
\newcommand{\yb}{10} |
|
83 |
\newcommand{\yc}{6} |
|
84 |
||
85 |
\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
|
86 |
\node[box] at (\xb,\ya) (A) {$\underrightarrow{\cC}(M)$ \\ the (dual) TQFT \\ Hilbert space}; |
307 | 87 |
\node[box] at (\xa,\yb) (FU) {$(\cF, \cU)$ \\ fields and\\ local relations}; |
88 |
\node[box] at (\xb,\yb) (BC) {$\bc_*(M; \cC)$ \\ the blob complex}; |
|
89 |
\node[box] at (\xa,\yc) (Cs) {$\cC_*$ \\ an $A_\infty$ \\$n$-category}; |
|
319
121c580d5ef7
editting all over the place
Scott Morrison <scott@tqft.net>
parents:
314
diff
changeset
|
90 |
\node[box] at (\xb,\yc) (BCs) {$\underrightarrow{\cC_*}(M)$}; |
307 | 91 |
|
92 |
||
93 |
||
314 | 94 |
\draw[->] (C) -- node[above] {$\displaystyle \colim_{\cell(M)} \cC$} node[below] {\S\S \ref{sec:constructing-a-tqft} \& \ref{ss:ncat_fields}} (A); |
307 | 95 |
\draw[->] (FU) -- node[below] {blob complex \\ for $M$} (BC); |
314 | 96 |
\draw[->] (Cs) -- node[above] {$\displaystyle \hocolim_{\cell(M)} \cC_*$} node[below] {\S \ref{ss:ncat_fields}} (BCs); |
307 | 97 |
|
98 |
\draw[->] (FU) -- node[right=10pt] {$\cF(M)/\cU$} (A); |
|
99 |
||
314 | 100 |
\draw[->] (C) -- node[left=10pt] { |
101 |
Example \ref{ex:traditional-n-categories(fields)} \\ and \S \ref{ss:ncat_fields} |
|
307 | 102 |
%$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle \cU(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$ |
103 |
} (FU); |
|
314 | 104 |
\draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Property \ref{property:skein-modules}} (A); |
307 | 105 |
|
106 |
\draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); |
|
107 |
\draw (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); |
|
108 |
\end{tikzpicture} |
|
109 |
||
110 |
} |
|
313 | 111 |
\caption{The main gadgets and constructions of the paper.} |
112 |
\label{fig:outline} |
|
113 |
\end{figure} |
|
307 | 114 |
|
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 |
Finally, later sections address other topics. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
116 |
Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
117 |
thought of as a topological $n$-category, in terms of the topology of $M$. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
118 |
Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
119 |
a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms 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
|
120 |
The appendixes prove technical results about $\CH{M}$ and the `small 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
|
121 |
and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
122 |
as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
150 | 123 |
|
155 | 124 |
|
150 | 125 |
\nn{some more things to cover in the intro} |
131 | 126 |
\begin{itemize} |
127 |
\item related: we are being unsophisticated from a homotopy theory point of |
|
160 | 128 |
view and using chain complexes in many places where we could get by with spaces |
131 | 129 |
\item ? one of the points we make (far) below is that there is not really much |
130 |
difference between (a) systems of fields and local relations and (b) $n$-cats; |
|
131 |
thus we tend to switch between talking in terms of one or the other |
|
132 |
\end{itemize} |
|
133 |
||
134 |
\medskip\hrule\medskip |
|
135 |
||
151 | 136 |
\subsection{Motivations} |
137 |
\label{sec:motivations} |
|
138 |
||
160 | 139 |
We will briefly sketch our original motivation for defining the blob complex. |
140 |
\nn{this is adapted from an old draft of the intro; it needs further modification |
|
141 |
in order to better integrate it into the current intro.} |
|
142 |
||
143 |
As a starting point, consider TQFTs constructed via fields and local relations. |
|
166 | 144 |
(See Section \ref{sec:tqftsviafields} or \cite{kw:tqft}.) |
98 | 145 |
This gives a satisfactory treatment for semisimple TQFTs |
146 |
(i.e.\ TQFTs for which the cylinder 1-category associated to an |
|
147 |
$n{-}1$-manifold $Y$ is semisimple for all $Y$). |
|
160 | 148 |
|
166 | 149 |
For non-semi-simple TQFTs, this approach is less satisfactory. |
98 | 150 |
Our main motivating example (though we will not develop it in this paper) |
160 | 151 |
is the (decapitated) $4{+}1$-dimensional TQFT associated to Khovanov homology. |
98 | 152 |
It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
153 |
with a link $L \subset \bd W$. |
|
154 |
The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
|
160 | 155 |
|
156 |
How would we go about computing $A_{Kh}(W^4, L)$? |
|
98 | 157 |
For $A_{Kh}(B^4, L)$, the main tool is the exact triangle (long exact sequence) |
166 | 158 |
relating resolutions of a crossing. |
98 | 159 |
Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt |
160 |
to compute $A_{Kh}(S^1\times B^3, L)$. |
|
161 |
According to the gluing theorem for TQFTs-via-fields, gluing along $B^3 \subset \bd B^4$ |
|
162 |
corresponds to taking a coend (self tensor product) over the cylinder category |
|
163 |
associated to $B^3$ (with appropriate boundary conditions). |
|
164 |
The coend is not an exact functor, so the exactness of the triangle breaks. |
|
160 | 165 |
|
166 |
||
167 |
The obvious solution to this problem is to replace the coend with its derived counterpart. |
|
98 | 168 |
This presumably works fine for $S^1\times B^3$ (the answer being the Hochschild homology |
169 |
of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired. |
|
170 |
If we build our manifold up via a handle decomposition, the computation |
|
171 |
would be a sequence of derived coends. |
|
172 |
A different handle decomposition of the same manifold would yield a different |
|
173 |
sequence of derived coends. |
|
174 |
To show that our definition in terms of derived coends is well-defined, we |
|
175 |
would need to show that the above two sequences of derived coends yield the same answer. |
|
176 |
This is probably not easy to do. |
|
160 | 177 |
|
178 |
Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$ |
|
98 | 179 |
which is manifestly invariant. |
180 |
(That is, a definition that does not |
|
181 |
involve choosing a decomposition of $W$. |
|
182 |
After all, one of the virtues of our starting point --- TQFTs via field and local relations --- |
|
183 |
is that it has just this sort of manifest invariance.) |
|
160 | 184 |
|
185 |
The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient |
|
98 | 186 |
\[ |
187 |
\text{linear combinations of fields} \;\big/\; \text{local relations} , |
|
188 |
\] |
|
189 |
with an appropriately free resolution (the ``blob complex") |
|
190 |
\[ |
|
191 |
\cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) . |
|
192 |
\] |
|
193 |
Here $\bc_0$ is linear combinations of fields on $W$, |
|
194 |
$\bc_1$ is linear combinations of local relations on $W$, |
|
195 |
$\bc_2$ is linear combinations of relations amongst relations on $W$, |
|
196 |
and so on. |
|
160 | 197 |
|
198 |
None of the above ideas depend on the details of the Khovanov homology example, |
|
166 | 199 |
so we develop the general theory in this paper and postpone specific applications |
98 | 200 |
to later papers. |
160 | 201 |
|
202 |
||
98 | 203 |
|
147 | 204 |
\subsection{Formal properties} |
205 |
\label{sec:properties} |
|
151 | 206 |
We now summarize the results of the paper in the following list of formal properties. |
98 | 207 |
|
208 |
\begin{property}[Functoriality] |
|
209 |
\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
|
210 |
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
|
211 |
That is, |
187 | 212 |
for a fixed $n$-dimensional system of fields $\cC$, the association |
98 | 213 |
\begin{equation*} |
131 | 214 |
X \mapsto \bc_*^{\cC}(X) |
98 | 215 |
\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
|
216 |
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
|
217 |
complexes and isomorphisms between them. |
98 | 218 |
\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
|
219 |
As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*^\cC(X)$; |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
220 |
this action is extended to all of $C_*(\Homeo(X))$ in Property \ref{property:evaluation} below. |
98 | 221 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
222 |
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
|
223 |
although we will not address this in detail here. |
131 | 224 |
|
98 | 225 |
\begin{property}[Disjoint union] |
226 |
\label{property:disjoint-union} |
|
227 |
The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
|
228 |
\begin{equation*} |
|
229 |
\bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
|
230 |
\end{equation*} |
|
231 |
\end{property} |
|
232 |
||
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 |
If an $n$-manifold $X_\text{cut}$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
234 |
write $X_\text{glued} = X_\text{cut} \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
235 |
Note that this includes the case of gluing two disjoint manifolds together. |
131 | 236 |
\begin{property}[Gluing map] |
98 | 237 |
\label{property:gluing-map}% |
136 | 238 |
%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 |
239 |
%\begin{equation*} |
|
240 |
%\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
|
241 |
%\end{equation*} |
|
242 |
Given a gluing $X_\mathrm{cut} \to X_\mathrm{glued}$, there is |
|
131 | 243 |
a natural map |
244 |
\[ |
|
187 | 245 |
\bc_*(X_\mathrm{cut}) \to \bc_*(X_\mathrm{glued}) |
131 | 246 |
\] |
187 | 247 |
(natural with respect to homeomorphisms, and also associative with respect to iterated gluings). |
98 | 248 |
\end{property} |
249 |
||
250 |
\begin{property}[Contractibility] |
|
251 |
\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
|
252 |
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
|
253 |
Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cC$ to balls. |
98 | 254 |
\begin{equation} |
187 | 255 |
\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)} |
98 | 256 |
\end{equation} |
257 |
\end{property} |
|
258 |
||
259 |
\begin{property}[Skein modules] |
|
260 |
\label{property:skein-modules}% |
|
261 |
The $0$-th blob homology of $X$ is the usual |
|
262 |
(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
|
263 |
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
|
264 |
(See \S \ref{sec:local-relations}.) |
98 | 265 |
\begin{equation*} |
131 | 266 |
H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X) |
98 | 267 |
\end{equation*} |
268 |
\end{property} |
|
269 |
||
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
270 |
\todo{Somehow, the Hochschild homology thing isn't a "property". |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
271 |
Let's move it and call it a theorem? -S} |
98 | 272 |
\begin{property}[Hochschild homology when $X=S^1$] |
273 |
\label{property:hochschild}% |
|
274 |
The blob complex for a $1$-category $\cC$ on the circle is |
|
275 |
quasi-isomorphic to the Hochschild complex. |
|
276 |
\begin{equation*} |
|
187 | 277 |
\xymatrix{\bc_*^{\cC}(S^1) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).} |
98 | 278 |
\end{equation*} |
279 |
\end{property} |
|
280 |
||
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
187
diff
changeset
|
281 |
In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
313 | 282 |
\begin{property}[$C_*(\Homeo(-))$ action]\mbox{}\\ |
283 |
\vspace{-0.5cm} |
|
98 | 284 |
\label{property:evaluation}% |
313 | 285 |
\begin{enumerate} |
286 |
\item There is a chain map |
|
98 | 287 |
\begin{equation*} |
166 | 288 |
\ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
98 | 289 |
\end{equation*} |
290 |
||
313 | 291 |
\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
|
292 |
|
313 | 293 |
\item For |
147 | 294 |
any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
313 | 295 |
(using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
147 | 296 |
\begin{equation*} |
297 |
\xymatrix@C+2cm{ |
|
166 | 298 |
\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) \\ |
299 |
\CH{X} \otimes \bc_*(X) |
|
300 |
\ar[r]_{\ev_{X}} \ar[u]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
|
147 | 301 |
\bc_*(X) \ar[u]_{\gl_Y} |
302 |
} |
|
303 |
\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
|
304 |
\item Any such chain map satisfying points 2. and 3. above is unique, up to an iterated homotopy. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
305 |
(That is, any pair of homotopies have a homotopy between them, and so on.) |
313 | 306 |
\item This map is associative, in the sense that the following diagram commutes (up to homotopy). |
307 |
\begin{equation*} |
|
308 |
\xymatrix{ |
|
309 |
\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} \\ |
|
310 |
\CH{X} \tensor \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) |
|
311 |
} |
|
312 |
\end{equation*} |
|
313 |
\end{enumerate} |
|
98 | 314 |
\end{property} |
315 |
||
313 | 316 |
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
|
317 |
$$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
313 | 318 |
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
|
319 |
satisfying corresponding conditions. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
187
diff
changeset
|
320 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
321 |
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
|
322 |
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
|
323 |
Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
151 | 324 |
|
187 | 325 |
\begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] |
151 | 326 |
\label{property: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
|
327 |
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
|
328 |
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
|
329 |
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
|
330 |
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
|
331 |
$$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
332 |
(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
|
333 |
These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
334 |
Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Property \ref{property:evaluation}. |
151 | 335 |
\end{property} |
336 |
\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
|
337 |
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
|
338 |
We think of this $A_\infty$ $n$-category as a free resolution. |
151 | 339 |
\end{rem} |
340 |
||
136 | 341 |
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
|
342 |
instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
343 |
The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. |
136 | 344 |
|
345 |
\begin{property}[Product formula] |
|
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
217
diff
changeset
|
346 |
\label{property:product} |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
347 |
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
|
348 |
Let $\cC$ be an $n$-category. |
338
adc0780aa5e7
updating notation in intro, also deciding that not everything is a 'property'
Scott Morrison <scott@tqft.net>
parents:
332
diff
changeset
|
349 |
Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}). |
136 | 350 |
Then |
351 |
\[ |
|
338
adc0780aa5e7
updating notation in intro, also deciding that not everything is a 'property'
Scott Morrison <scott@tqft.net>
parents:
332
diff
changeset
|
352 |
\bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
136 | 353 |
\] |
354 |
\end{property} |
|
338
adc0780aa5e7
updating notation in intro, also deciding that not everything is a 'property'
Scott Morrison <scott@tqft.net>
parents:
332
diff
changeset
|
355 |
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
|
356 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
357 |
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
|
358 |
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
|
359 |
(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 | 360 |
|
361 |
\begin{property}[Gluing formula] |
|
362 |
\label{property:gluing}% |
|
363 |
\mbox{}% <-- gets the indenting right |
|
364 |
\begin{itemize} |
|
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
217
diff
changeset
|
365 |
\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
|
366 |
$A_\infty$ module for $\bc_*(Y)$. |
98 | 367 |
|
136 | 368 |
\item For any $n$-manifold $X_\text{glued} = X_\text{cut} \bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{glued})$ is the $A_\infty$ self-tensor product of |
338
adc0780aa5e7
updating notation in intro, also deciding that not everything is a 'property'
Scott Morrison <scott@tqft.net>
parents:
332
diff
changeset
|
369 |
$\bc_*(X_\text{cut})$ as an $\bc_*(Y)$-bimodule: |
98 | 370 |
\begin{equation*} |
338
adc0780aa5e7
updating notation in intro, also deciding that not everything is a 'property'
Scott Morrison <scott@tqft.net>
parents:
332
diff
changeset
|
371 |
\bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
98 | 372 |
\end{equation*} |
373 |
\end{itemize} |
|
374 |
\end{property} |
|
375 |
||
338
adc0780aa5e7
updating notation in intro, also deciding that not everything is a 'property'
Scott Morrison <scott@tqft.net>
parents:
332
diff
changeset
|
376 |
Finally, we prove 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
|
377 |
|
338
adc0780aa5e7
updating notation in intro, also deciding that not everything is a 'property'
Scott Morrison <scott@tqft.net>
parents:
332
diff
changeset
|
378 |
\begin{thm}[Mapping spaces] |
187 | 379 |
Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
380 |
$B^n \to T$. |
|
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
217
diff
changeset
|
381 |
(The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
136 | 382 |
Then |
187 | 383 |
$$\bc_*(X, \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
338
adc0780aa5e7
updating notation in intro, also deciding that not everything is a 'property'
Scott Morrison <scott@tqft.net>
parents:
332
diff
changeset
|
384 |
\end{thm} |
117
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
98
diff
changeset
|
385 |
|
187 | 386 |
This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data. |
387 |
||
338
adc0780aa5e7
updating notation in intro, also deciding that not everything is a 'property'
Scott Morrison <scott@tqft.net>
parents:
332
diff
changeset
|
388 |
\begin{thm}[Higher dimensional Deligne conjecture] |
adc0780aa5e7
updating notation in intro, also deciding that not everything is a 'property'
Scott Morrison <scott@tqft.net>
parents:
332
diff
changeset
|
389 |
\label{thm:deligne} |
117
b62214646c4f
preparing for semi-public version soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
98
diff
changeset
|
390 |
The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
338
adc0780aa5e7
updating notation in intro, also deciding that not everything is a 'property'
Scott Morrison <scott@tqft.net>
parents:
332
diff
changeset
|
391 |
\end{thm} |
adc0780aa5e7
updating notation in intro, also deciding that not everything is a 'property'
Scott Morrison <scott@tqft.net>
parents:
332
diff
changeset
|
392 |
See \S \ref{sec:deligne} for a full explanation of the statement, and an outline of the proof. |
98 | 393 |
|
332
160ca7078ae9
fixing some inconsistencies in where the easy basic properties are treated
Scott Morrison <scott@tqft.net>
parents:
319
diff
changeset
|
394 |
Properties \ref{property:functoriality} and \ref{property:skein-modules} will be immediate from the definition given in |
160ca7078ae9
fixing some inconsistencies in where the easy basic properties are treated
Scott Morrison <scott@tqft.net>
parents:
319
diff
changeset
|
395 |
\S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. |
160ca7078ae9
fixing some inconsistencies in where the easy basic properties are treated
Scott Morrison <scott@tqft.net>
parents:
319
diff
changeset
|
396 |
Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
397 |
Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
398 |
in \S \ref{sec:evaluation}, Property \ref{property:blobs-ainfty} as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats}, |
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
217
diff
changeset
|
399 |
and Properties \ref{property:product} and \ref{property:gluing} in \S \ref{sec:ainfblob} as consequences of Theorem \ref{product_thm}. |
148 | 400 |
|
401 |
\subsection{Future directions} |
|
151 | 402 |
\label{sec:future} |
155 | 403 |
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
|
404 |
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
|
405 |
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
|
406 |
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
|
407 |
More could be said about finite characteristic |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
408 |
(there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$, for example). |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
409 |
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
|
410 |
$\operatorname{Spin}$ 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
|
411 |
(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
|
412 |
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
|
413 |
there may be some differences for topological manifolds and smooth manifolds. |
148 | 414 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
415 |
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
|
416 |
interesting to investigate if there is a connection with the material here. |
314 | 417 |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
418 |
Many results in Hochschild homology can be understood `topologically' via 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
|
419 |
For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
420 |
(see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$, |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
338
diff
changeset
|
421 |
but haven't investigated the details. |
148 | 422 |
|
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
217
diff
changeset
|
423 |
Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} |
148 | 424 |
|
425 |
||
426 |
\subsection{Thanks and acknowledgements} |
|
319
121c580d5ef7
editting all over the place
Scott Morrison <scott@tqft.net>
parents:
314
diff
changeset
|
427 |
We'd like to thank David Ben-Zvi, Kevin Costello, Chris Douglas, |
270 | 428 |
Michael Freedman, Vaughan Jones, Justin Roberts, Chris Schommer-Pries, Peter Teichner \nn{and who else?} for many interesting and useful conversations. |
429 |
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 | 430 |