55
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parents:
diff
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|
1 |
\section{Gluing - needs to be rewritten/replaced}
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scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
2 |
\label{sec:gluing}%
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scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
3 |
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scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
4 |
We now turn to establishing the gluing formula for blob homology, restated from Property \ref{property:gluing} in the Introduction
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
5 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
6 |
%\mbox{}% <-- gets the indenting right
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
7 |
\item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
8 |
naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
9 |
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scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
10 |
\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
11 |
$A_\infty$ module for $\bc_*(Y \times I)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
12 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
13 |
\item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
14 |
$0$-submanifold of its boundary, the blob homology of $X'$, obtained from
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
15 |
$X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
16 |
$\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
17 |
\begin{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
18 |
\bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
19 |
\end{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
20 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
21 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
22 |
Although this gluing formula is stated in terms of $A_\infty$ categories and their (bi-)modules, it will be more natural for us to give alternative
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
23 |
definitions of `topological' $A_\infty$-categories and their bimodules, explain how to translate between the `algebraic' and `topological' definitions,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
24 |
and then prove the gluing formula in the topological langauge. Section \ref{sec:topological-A-infty} below explains these definitions, and establishes
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
25 |
the desired equivalence. This is quite involved, and in particular requires us to generalise the definition of blob homology to allow $A_\infty$ algebras
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
26 |
as inputs, and to re-establish many of the properties of blob homology in this generality. Many readers may prefer to read the
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
27 |
Definitions \ref{defn:topological-algebra} and \ref{defn:topological-module} of `topological' $A_\infty$-categories, and Definition \ref{???} of the
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
28 |
self-tensor product of a `topological' $A_\infty$-bimodule, then skip to \S \ref{sec:boundary-action} and \S \ref{sec:gluing-formula} for the proofs
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
29 |
of the gluing formula in the topological context.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
30 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
31 |
\subsection{`Topological' $A_\infty$ $n$-categories}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
32 |
\label{sec:topological-A-infty}%
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
33 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
34 |
This section prepares the ground for establishing Property \ref{property:gluing} by defining the notion of a \emph{topological $A_\infty$-$n$-category}.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
35 |
The main result of this section is
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
36 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
37 |
\begin{thm}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
38 |
Topological $A_\infty$-$1$-categories are equivalent to the usual notion of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
39 |
$A_\infty$-$1$-categories.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
40 |
\end{thm}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
41 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
42 |
Before proving this theorem, we embark upon a long string of definitions.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
43 |
For expository purposes, we begin with the $n=1$ special cases,\scott{Why are we treating the $n>1$ cases at all?} and define
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
44 |
first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
45 |
to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
46 |
\nn{Something about duals?}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
47 |
\todo{Explain that we're not making contact with any previous notions for the general $n$ case?}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
48 |
\kevin{probably we should say something about the relation
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
49 |
to [framed] $E_\infty$ algebras
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
50 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
51 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
52 |
\todo{}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
53 |
Various citations we might want to make:
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
54 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
55 |
\item \cite{MR2061854} McClure and Smith's review article
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
56 |
\item \cite{MR0420610} May, (inter alia, definition of $E_\infty$ operad)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
57 |
\item \cite{MR0236922,MR0420609} Boardman and Vogt
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
58 |
\item \cite{MR1256989} definition of framed little-discs operad
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
59 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
60 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
61 |
\begin{defn}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
62 |
\label{defn:topological-algebra}%
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
63 |
A ``topological $A_\infty$-algebra'' $A$ consists of the following data.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
64 |
\begin{enumerate}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
65 |
\item For each $1$-manifold $J$ diffeomorphic to the standard interval
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
66 |
$I=\left[0,1\right]$, a complex of vector spaces $A(J)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
67 |
% either roll functoriality into the evaluation map
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
68 |
\item For each pair of intervals $J,J'$ an `evaluation' chain map
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
69 |
$\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
70 |
\item For each decomposition of intervals $J = J'\cup J''$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
71 |
a gluing map $\gl_{J',J''} : A(J') \tensor A(J'') \to A(J)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
72 |
% or do it as two separate pieces of data
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
73 |
%\item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
74 |
%\item for each diffeomorphism $\phi : J \to J'$, an isomorphism $A(\phi) : A(J) \isoto A(J')$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
75 |
%\item and for each pair of intervals $J,J'$ a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
76 |
\end{enumerate}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
77 |
This data is required to satisfy the following conditions.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
78 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
79 |
\item The evaluation chain map is associative, in that the diagram
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
80 |
\begin{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
81 |
\xymatrix{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
82 |
& \quad \mathclap{\CD{J' \to J''} \tensor \CD{J \to J'} \tensor A(J)} \quad \ar[dr]^{\id \tensor \ev_{J \to J'}} \ar[dl]_{\compose \tensor \id} & \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
83 |
\CD{J' \to J''} \tensor A(J') \ar[dr]^{\ev_{J' \to J''}} & & \CD{J \to J''} \tensor A(J) \ar[dl]_{\ev_{J \to J''}} \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
84 |
& A(J'') &
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
85 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
86 |
\end{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
87 |
commutes up to homotopy.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
88 |
Here the map $$\compose : \CD{J' \to J''} \tensor \CD{J \to J'} \to \CD{J \to J''}$$ is a composition: take products of singular chains first, then compose diffeomorphisms.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
89 |
%% or the version for separate pieces of data:
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
90 |
%\item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
91 |
%\item The evaluation chain map is associative, in that the diagram
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
92 |
%\begin{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
93 |
%\xymatrix{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
94 |
%\CD{J} \tensor \CD{J} \tensor A(J) \ar[r]^{\id \tensor \ev_J} \ar[d]_{\compose \tensor \id} &
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
95 |
%\CD{J} \tensor A(J) \ar[d]^{\ev_J} \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
96 |
%\CD{J} \tensor A(J) \ar[r]_{\ev_J} &
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
97 |
%A(J)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
98 |
%}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
99 |
%\end{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
100 |
%commutes. (Here the map $\compose : \CD{J} \tensor \CD{J} \to \CD{J}$ is a composition: take products of singular chains first, then use the group multiplication in $\Diff(J)$.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
101 |
\item The gluing maps are \emph{strictly} associative. That is, given $J$, $J'$ and $J''$, the diagram
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
102 |
\begin{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
103 |
\xymatrix{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
104 |
A(J) \tensor A(J') \tensor A(J'') \ar[rr]^{\gl_{J,J'} \tensor \id} \ar[d]_{\id \tensor \gl_{J',J''}} &&
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
105 |
A(J \cup J') \tensor A(J'') \ar[d]^{\gl_{J \cup J', J''}} \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
106 |
A(J) \tensor A(J' \cup J'') \ar[rr]_{\gl_{J, J' \cup J''}} &&
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
107 |
A(J \cup J' \cup J'')
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
108 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
109 |
\end{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
110 |
commutes.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
111 |
\item The gluing and evaluation maps are compatible.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
112 |
\nn{give diagram, or just say ``in the obvious way", or refer to diagram in blob eval map section?}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
113 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
114 |
\end{defn}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
115 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
116 |
\begin{rem}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
117 |
We can restrict the evaluation map to $0$-chains, and see that $J \mapsto A(J)$ and $(\phi:J \to J') \mapsto \ev_{J \to J'}(\phi, \bullet)$ together
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
118 |
constitute a functor from the category of intervals and diffeomorphisms between them to the category of complexes of vector spaces.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
119 |
Further, once this functor has been specified, we only need to know how the evaluation map acts when $J = J'$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
120 |
\end{rem}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
121 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
122 |
%% if we do things separately, we should say this:
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
123 |
%\begin{rem}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
124 |
%Of course, the first and third pieces of data (the complexes, and the isomorphisms) together just constitute a functor from the category of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
125 |
%intervals and diffeomorphisms between them to the category of complexes of vector spaces.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
126 |
%Further, one can combine the second and third pieces of data, asking instead for a map
|
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|
127 |
%\begin{equation*}
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|
128 |
%\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J').
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parents:
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|
129 |
%\end{equation*}
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|
130 |
%(Any $k$-parameter family of diffeomorphisms in $C_k(\Diff(J \to J'))$ factors into a single diffeomorphism $J \to J'$ and a $k$-parameter family of
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|
131 |
%diffeomorphisms in $\CD{J'}$.)
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|
132 |
%\end{rem}
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|
133 |
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|
134 |
To generalise the definition to that of a category, we simply introduce a set of objects which we call $A(pt)$. Now we associate complexes to each
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|
135 |
interval with boundary conditions $(J, c_-, c_+)$, with $c_-, c_+ \in A(pt)$, and only ask for gluing maps when the boundary conditions match up:
|
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parents:
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|
136 |
\begin{equation*}
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parents:
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|
137 |
\gl : A(J, c_-, c_0) \tensor A(J', c_0, c_+) \to A(J \cup J', c_-, c_+).
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parents:
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|
138 |
\end{equation*}
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|
139 |
The action of diffeomorphisms (and of $k$-parameter families of diffeomorphisms) ignores the boundary conditions.
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|
140 |
\todo{we presumably need to say something about $\id_c \in A(J, c, c)$.}
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|
141 |
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142 |
At this point we can give two motivating examples. The first is `chains of maps to $M$' for some fixed target space $M$.
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parents:
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|
143 |
\begin{defn}
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|
144 |
Define the topological $A_\infty$ category $C_*(\Maps(\bullet \to M))$ by
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|
145 |
\begin{enumerate}
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|
146 |
\item $A(J) = C_*(\Maps(J \to M))$, singular chains on the space of smooth maps from $J$ to $M$,
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|
147 |
\item $\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J')$ is the composition
|
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parents:
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|
148 |
\begin{align*}
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|
149 |
\CD{J \to J'} \tensor C_*(\Maps(J \to M)) & \to C_*(\Diff(J \to J') \times \Maps(J \to M)) \\ & \to C_*(\Maps(J' \to M)),
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|
150 |
\end{align*}
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parents:
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|
151 |
where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism,
|
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|
152 |
\item $\gl_{J,J'} : A(J) \tensor A(J')$ takes the product of singular chains, then glues maps to $M$ together.
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parents:
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|
153 |
\end{enumerate}
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parents:
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|
154 |
The associativity conditions are trivially satisfied.
|
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parents:
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|
155 |
\end{defn}
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|
156 |
|
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|
157 |
The second example is simply the blob complex of $Y \times J$, for any $n-1$ manifold $Y$. We define $A(J) = \bc_*(Y \times J)$.
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parents:
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|
158 |
Observe $\Diff(J \to J')$ embeds into $\Diff(Y \times J \to Y \times J')$. The evaluation and gluing maps then come directly from Properties
|
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parents:
diff
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|
159 |
\ref{property:evaluation} and \ref{property:gluing-map} respectively. We'll often write $bc_*(Y)$ for this algebra.
|
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parents:
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|
160 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
161 |
The definition of a module follows closely the definition of an algebra or category.
|
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parents:
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|
162 |
\begin{defn}
|
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|
163 |
\label{defn:topological-module}%
|
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|
164 |
A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$
|
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|
165 |
consists of the following data.
|
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|
166 |
\begin{enumerate}
|
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parents:
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|
167 |
\item A functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with the upper boundary point `marked', to complexes of vector spaces.
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parents:
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|
168 |
\item For each pair of such marked intervals,
|
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parents:
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|
169 |
an `evaluation' chain map $\ev_{K\to K'} : \CD{K \to K'} \tensor M(K) \to M(K')$.
|
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|
170 |
\item For each decomposition $K = J\cup K'$ of the marked interval
|
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parents:
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|
171 |
$K$ into an unmarked interval $J$ and a marked interval $K'$, a gluing map
|
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parents:
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|
172 |
$\gl_{J,K'} : A(J) \tensor M(K') \to M(K)$.
|
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parents:
diff
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|
173 |
\end{enumerate}
|
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parents:
diff
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|
174 |
The above data is required to satisfy
|
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|
175 |
conditions analogous to those in Definition \ref{defn:topological-algebra}.
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parents:
diff
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|
176 |
\end{defn}
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parents:
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|
177 |
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|
178 |
For any manifold $X$ with $\bdy X = Y$ (or indeed just with $Y$ a codimension $0$-submanifold of $\bdy X$) we can think of $\bc_*(X)$ as
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parents:
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|
179 |
a topological $A_\infty$ module over $\bc_*(Y)$, the topological $A_\infty$ category described above.
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parents:
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|
180 |
For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$.
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parents:
diff
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|
181 |
(Here we glue $Y \times pt$ to $Y \subset \bdy X$, where $pt$ is the marked point of $K$.) Again, the evaluation and gluing maps come directly from Properties
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
182 |
\ref{property:evaluation} and \ref{property:gluing-map} respectively.
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parents:
diff
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|
183 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
184 |
The definition of a bimodule is like the definition of a module,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
185 |
except that we have two disjoint marked intervals $K$ and $L$, one with a marked point
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parents:
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|
186 |
on the upper boundary and the other with a marked point on the lower boundary.
|
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parents:
diff
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|
187 |
There are evaluation maps corresponding to gluing unmarked intervals
|
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parents:
diff
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|
188 |
to the unmarked ends of $K$ and $L$.
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parents:
diff
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|
189 |
|
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parents:
diff
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|
190 |
Let $X$ be an $n$-manifold with a copy of $Y \du -Y$ embedded as a
|
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parents:
diff
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|
191 |
codimension-0 submanifold of $\bdy X$.
|
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parents:
diff
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|
192 |
Then the the assignment $K,L \mapsto \bc_*(X \cup_Y (Y\times K) \cup_{-Y} (-Y\times L))$ has the
|
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parents:
diff
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|
193 |
structure of a topological $A_\infty$ bimodule over $\bc_*(Y)$.
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parents:
diff
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|
194 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
195 |
Next we define the coend
|
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parents:
diff
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|
196 |
(or gluing or tensor product or self tensor product, depending on the context)
|
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parents:
diff
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|
197 |
$\gl(M)$ of a topological $A_\infty$ bimodule $M$. This will be an `initial' or `universal' object satisfying various properties.
|
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parents:
diff
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|
198 |
\begin{defn}
|
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parents:
diff
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|
199 |
We define a category $\cG(M)$. Objects consist of the following data.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
200 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
201 |
\item For each interval $N$ with both endpoints marked, a complex of vector spaces C(N).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
202 |
\item For each pair of intervals $N,N'$ an evaluation chain map
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
203 |
$\ev_{N \to N'} : \CD{N \to N'} \tensor C(N) \to C(N')$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
204 |
\item For each decomposition of intervals $N = K\cup L$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
205 |
a gluing map $\gl_{K,L} : M(K,L) \to C(N)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
206 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
207 |
This data must satisfy the following conditions.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
208 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
209 |
\item The evaluation maps are associative.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
210 |
\nn{up to homotopy?}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
211 |
\item Gluing is strictly associative.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
212 |
That is, given a decomposition $N = K\cup J\cup L$, the chain maps associated to
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
213 |
$K\du J\du L \to (K\cup J)\du L \to N$ and $K\du J\du L \to K\du (J\cup L) \to N$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
214 |
agree.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
215 |
\item the gluing and evaluation maps are compatible.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
216 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
217 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
218 |
A morphism $f$ between such objects $C$ and $C'$ is a chain map $f_N : C(N) \to C'(N)$ for each interval $N$ with both endpoints marked,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
219 |
satisfying the following conditions.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
220 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
221 |
\item For each pair of intervals $N,N'$, the diagram
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
222 |
\begin{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
223 |
\xymatrix{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
224 |
\CD{N \to N'} \tensor C(N) \ar[d]_{\ev} \ar[r]^{\id \tensor f_N} & \CD{N \to N'} \tensor C'(N) \ar[d]^{\ev} \\
|
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parents:
diff
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|
225 |
C(N) \ar[r]_{f_N} & C'(N)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
226 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
227 |
\end{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
228 |
commutes.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
229 |
\item For each decomposition of intervals $N = K \cup L$, the gluing map for $C'$, $\gl'_{K,L} : M(K,L) \to C'(N)$ is the composition
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
230 |
$$M(K,L) \xto{\gl_{K,L}} C(N) \xto{f_N} C'(N).$$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
231 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
232 |
\end{defn}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
233 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
234 |
We now define $\gl(M)$ to be an initial object in the category $\cG{M}$. This just says that for any other object $C'$ in $\cG{M}$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
235 |
there are chain maps $f_N: \gl(M)(N) \to C'(N)$, compatible with the action of families of diffeomorphisms, so that the gluing maps $M(K,L) \to C'(N)$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
236 |
factor through the gluing maps for $\gl(M)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
237 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
238 |
We return to our two favourite examples. First, the coend of the topological $A_\infty$ category $C_*(\Maps(\bullet \to M))$ as a bimodule over itself
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
239 |
is essentially $C_*(\Maps(S^1 \to M))$. \todo{}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
240 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
241 |
For the second example, given $X$ and $Y\du -Y \sub \bdy X$, the assignment
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
242 |
$$N \mapsto \bc_*(X \cup_{Y\du -Y} (N\times Y))$$ clearly gives an object in $\cG{M}$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
243 |
Showing that it is an initial object is the content of the gluing theorem proved below.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
244 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
245 |
The definitions for a topological $A_\infty$-$n$-category are very similar to the above
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
246 |
$n=1$ case.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
247 |
One replaces intervals with manifolds diffeomorphic to the ball $B^n$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
248 |
Marked points are replaced by copies of $B^{n-1}$ in $\bdy B^n$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
249 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
250 |
\nn{give examples: $A(J^n) = \bc_*(Z\times J)$ and $A(J^n) = C_*(\Maps(J \to M))$.}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
251 |
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parents:
diff
changeset
|
252 |
\todo{the motivating example $C_*(\maps(X, M))$}
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scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
253 |
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scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
254 |
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scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
255 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
256 |
\newcommand{\skel}[1]{\operatorname{skeleton}(#1)}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
257 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
258 |
Given a topological $A_\infty$-category $\cC$, we can construct an `algebraic' $A_\infty$ category $\skel{\cC}$. First, pick your
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
259 |
favorite diffeomorphism $\phi: I \cup I \to I$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
260 |
\begin{defn}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
261 |
We'll write $\skel{\cC} = (A, m_k)$. Define $A = \cC(I)$, and $m_2 : A \tensor A \to A$ by
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
262 |
\begin{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
263 |
m_2 \cC(I) \tensor \cC(I) \xrightarrow{\gl_{I,I}} \cC(I \cup I) \xrightarrow{\cC(\phi)} \cC(I).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
264 |
\end{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
265 |
Next, we define all the `higher associators' $m_k$ by
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
266 |
\todo{}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
267 |
\end{defn}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
268 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
269 |
Give an `algebraic' $A_\infty$ category $(A, m_k)$, we can construct a topological $A_\infty$-category, which we call $\bc_*^A$. You should
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
270 |
think of this as a generalisation of the blob complex, although the construction we give will \emph{not} specialise to exactly the usual definition
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
271 |
in the case the $A$ is actually an associative category.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
272 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
273 |
We'll first define $\cT_{k,n}$ to be the set of planar forests consisting of $n-k$ trees, with a total of $n$ leaves. Thus
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
274 |
\todo{$\cT_{0,n}$ has 1 element, with $n$ vertical lines, $\cT_{1,n}$ has $n-1$ elements, each with a single trivalent vertex, $\cT_{2,n}$ etc...}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
275 |
\begin{align*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
276 |
\end{align*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
277 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
278 |
\begin{defn}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
279 |
The topological $A_\infty$ category $\bc_*^A$ is doubly graded, by `blob degree' and `internal degree'. We'll write $\bc_k^A$ for the blob degree $k$ piece.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
280 |
The homological degree of an element $a \in \bc_*^A(J)$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
281 |
is the sum of the blob degree and the internal degree.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
282 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
283 |
We first define $\bc_0^A(J)$ as a vector space by
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
284 |
\begin{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
285 |
\bc_0^A(J) = \DirectSum_{\substack{\{J_i\}_{i=1}^n \\ \mathclap{\bigcup_i J_i = J}}} \Tensor_{i=1}^n (\CD{J_i \to I} \tensor A).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
286 |
\end{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
287 |
(That is, for each division of $J$ into finitely many subintervals,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
288 |
we have the tensor product of chains of diffeomorphisms from each subinterval to the standard interval,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
289 |
and a copy of $A$ for each subinterval.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
290 |
The internal degree of an element $(f_1 \tensor a_1, \ldots, f_n \tensor a_n)$ is the sum of the dimensions of the singular chains
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
291 |
plus the sum of the homological degrees of the elements of $A$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
292 |
The differential is defined just by the graded Leibniz rule and the differentials on $\CD{J_i \to I}$ and on $A$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
293 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
294 |
Next,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
295 |
\begin{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
296 |
\bc_1^A(J) = \DirectSum_{\substack{\{J_i\}_{i=1}^n \\ \mathclap{\bigcup_i J_i = J}}} \DirectSum_{T \in \cT_{1,n}} \Tensor_{i=1}^n (\CD{J_i \to I} \tensor A).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
297 |
\end{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
298 |
\end{defn}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
299 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
300 |
\begin{figure}[!ht]
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
301 |
\begin{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
302 |
\mathfig{0.7}{associahedron/A4-vertices}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
303 |
\end{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
304 |
\caption{The vertices of the $k$-dimensional associahedron are indexed by binary trees on $k+2$ leaves.}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
305 |
\label{fig:A4-vertices}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
306 |
\end{figure}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
307 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
308 |
\begin{figure}[!ht]
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
309 |
\begin{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
310 |
\mathfig{0.7}{associahedron/A4-faces}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
311 |
\end{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
312 |
\caption{The faces of the $k$-dimensional associahedron are indexed by trees with $2$ vertices on $k+2$ leaves.}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
313 |
\label{fig:A4-vertices}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
314 |
\end{figure}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
315 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
316 |
\newcommand{\tm}{\widetilde{m}}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
317 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
318 |
Let $\tm_1(a) = a$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
319 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
320 |
We now define $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$, first giving an opaque formula, then explaining the combinatorics behind it.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
321 |
\begin{align}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
322 |
\notag \bdy(\tm_k(a_1 & \tensor \cdots \tensor a_k)) = \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
323 |
\label{eq:bdy-tm-k-1} & \phantom{+} \sum_{\ell'=0}^{k-1} (-1)^{\abs{\tm_k}+\sum_{j=1}^{\ell'} \abs{a_j}} \tm_k(a_1 \tensor \cdots \tensor \bdy a_{\ell'+1} \tensor \cdots \tensor a_k) + \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
324 |
\label{eq:bdy-tm-k-2} & + \sum_{\ell=1}^{k-1} \tm_{\ell}(a_1 \tensor \cdots \tensor a_{\ell}) \tensor \tm_{k-\ell}(a_{\ell+1} \tensor \cdots \tensor a_k) + \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
325 |
\label{eq:bdy-tm-k-3} & + \sum_{\ell=1}^{k-1} \sum_{\ell'=0}^{l-1} (-1)^{\abs{\tm_k}+\sum_{j=1}^{\ell'} \abs{a_j}} \tm_{\ell}(a_1 \tensor \cdots \tensor m_{k-\ell + 1}(a_{\ell' + 1} \tensor \cdots \tensor a_{\ell' + k - \ell + 1}) \tensor \cdots \tensor a_k)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
326 |
\end{align}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
327 |
The first set of terms in $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ just have $\bdy$ acting on each argument $a_i$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
328 |
The terms appearing in \eqref{eq:bdy-tm-k-2} and \eqref{eq:bdy-tm-k-3} are indexed by trees with $2$ vertices on $k+1$ leaves.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
329 |
Note here that we have one more leaf than there arguments of $\tm_k$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
330 |
(See Figure \ref{fig:A4-vertices}, in which the rightmost branches are helpfully drawn in red.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
331 |
We will treat the vertices which involve a rightmost (red) branch differently from the vertices which only involve the first $k$ leaves.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
332 |
The terms in \eqref{eq:bdy-tm-k-2} arise in the cases in which both
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
333 |
vertices are rightmost, and the corresponding term in $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ is a tensor product of the form
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
334 |
$$\tm_{\ell}(a_1 \tensor \cdots \tensor a_{\ell}) \tensor \tm_{k-\ell}(a_{\ell+1} \tensor \cdots \tensor a_k)$$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
335 |
where $\ell + 1$ and $k - \ell + 1$ are the number of branches entering the vertices.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
336 |
If only one vertex is rightmost, we get the term $$\tm_{\ell}(a_1 \tensor \cdots \tensor m_{k-\ell+1}(a_{\ell' + 1} \tensor \cdots \tensor a_{\ell' + k - \ell}) \tensor \cdots \tensor a_k)$$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
337 |
in \eqref{eq:bdy-tm-k-3},
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
338 |
where again $\ell + 1$ is the number of branches entering the rightmost vertex, $k-\ell+1$ is the number of branches entering the other vertex, and $\ell'$ is the number of edges meeting the rightmost vertex which start to the left of the other vertex.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
339 |
For example, we have
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
340 |
\begin{align*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
341 |
\bdy(\tm_2(a \tensor b)) & = \left(\tm_2(\bdy a \tensor b) + (-1)^{\abs{a}} \tm_2(a \tensor \bdy b)\right) + \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
342 |
& \qquad - a \tensor b + m_2(a \tensor b) \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
343 |
\bdy(\tm_3(a \tensor b \tensor c)) & = \left(- \tm_3(\bdy a \tensor b \tensor c) + (-1)^{\abs{a} + 1} \tm_3(a \tensor \bdy b \tensor c) + (-1)^{\abs{a} + \abs{b} + 1} \tm_3(a \tensor b \tensor \bdy c)\right) + \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
344 |
& \qquad + \left(- \tm_2(a \tensor b) \tensor c + a \tensor \tm_2(b \tensor c)\right) + \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
345 |
& \qquad + \left(- \tm_2(m_2(a \tensor b) \tensor c) + \tm_2(a, m_2(b \tensor c)) + m_3(a \tensor b \tensor c)\right)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
346 |
\end{align*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
347 |
\begin{align*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
348 |
\bdy(& \tm_4(a \tensor b \tensor c \tensor d)) = \left(\tm_4(\bdy a \tensor b \tensor c \tensor d) + \cdots + \tm_4(a \tensor b \tensor c \tensor \bdy d)\right) + \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
349 |
& + \left(\tm_3(a \tensor b \tensor c) \tensor d + \tm_2(a \tensor b) \tensor \tm_2(c \tensor d) + a \tensor \tm_3(b \tensor c \tensor d)\right) + \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
350 |
& + \left(\tm_3(m_2(a \tensor b) \tensor c \tensor d) + \tm_3(a \tensor m_2(b \tensor c) \tensor d) + \tm_3(a \tensor b \tensor m_2(c \tensor d))\right. + \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
351 |
& + \left.\tm_2(m_3(a \tensor b \tensor c) \tensor d) + \tm_2(a \tensor m_3(b \tensor c \tensor d)) + m_4(a \tensor b \tensor c \tensor d)\right) \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
352 |
\end{align*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
353 |
See Figure \ref{fig:A4-terms}, comparing it against Figure \ref{fig:A4-faces}, to see this illustrated in the case $k=4$. There the $3$ faces closest
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
354 |
to the top of the diagram have two rightmost vertices, while the other $6$ faces have only one.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
355 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
356 |
\begin{figure}[!ht]
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
357 |
\begin{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
358 |
\mathfig{1.0}{associahedron/A4-terms}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
359 |
\end{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
360 |
\caption{The terms of $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ correspond to the faces of the $k-1$ dimensional associahedron.}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
361 |
\label{fig:A4-terms}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
362 |
\end{figure}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
363 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
364 |
\begin{lem}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
365 |
This definition actually results in a chain complex, that is $\bdy^2 = 0$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
366 |
\end{lem}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
367 |
\begin{proof}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
368 |
\newcommand{\T}{\text{---}}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
369 |
\newcommand{\ssum}[1]{{\sum}^{(#1)}}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
370 |
For the duration of this proof, inside a summation over variables $l_1, \ldots, l_m$, an expression with $m$ dashes will be interpreted
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
371 |
by replacing each dash with contiguous factors from $a_1 \tensor \cdots \tensor a_k$, so the first dash takes the first $l_1$ factors, the second
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
372 |
takes the next $l_2$ factors, and so on. Further, we'll write $\ssum{m}$ for $\sum_{\sum_{i=1}^m l_i = k}$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
373 |
In this notation, the formula for the differential becomes
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
374 |
\begin{align}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
375 |
\notag
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
376 |
\bdy \tm(\T) & = \ssum{2} \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2} + \ssum{3} \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3} \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
377 |
\intertext{and we calculate}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
378 |
\notag
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
379 |
\bdy^2 \tm(\T) & = \ssum{2} \bdy \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2} \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
380 |
\notag & \qquad + \ssum{2} \tm(\T) \tensor \bdy \tm(\T) \times \sigma_{0;l_1,l_2} \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
381 |
\notag & \qquad + \ssum{3} \bdy \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3} \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
382 |
\label{eq:d21} & = \ssum{3} \tm(\T) \tensor \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1+l_2,l_3} \sigma_{0;l_1,l_2} \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
383 |
\label{eq:d22} & \qquad + \ssum{4} \tm(\T \tensor m(\T) \tensor \T) \tensor \tm(\T) \times \sigma_{0;l_1+l_2+l_3,l_4} \tau_{0;l_1,l_2,l_3} \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
384 |
\label{eq:d23} & \qquad + \ssum{3} \tm(\T) \tensor \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2+l_3} \sigma_{l_1;l_2,l_3} \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
385 |
\label{eq:d24} & \qquad + \ssum{4} \tm(\T) \tensor \tm(\T \tensor m(\T) \tensor \T) \times \sigma_{0;l_1,l_2+l_3+l_4} \tau_{l_1;l_2,l_3,l_4} \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
386 |
\label{eq:d25} & \qquad + \ssum{4} \tm(\T \tensor m(\T) \tensor \T) \tensor \tm(\T) \times \tau_{0;l_1,l_2,l_3+l_4} ??? \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
387 |
\label{eq:d26} & \qquad + \ssum{4} \tm(\T) \tensor \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1+l_2,l_3,l_4} \sigma_{0;l_1,l_2} \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
388 |
\label{eq:d27} & \qquad + \ssum{5} \tm(\T \tensor m(\T) \tensor \T \tensor m(\T) \tensor \T) \times \tau_{0;l_1+l_2+l_3,l_4,l_5} \tau_{0;l_1,l_2,l_3} \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
389 |
\label{eq:d28} & \qquad + \ssum{5} \tm(\T \tensor m(\T \tensor m(\T) \tensor \T) \tensor \T) \times \tau_{0;l_1,l_2+l_3+l_4,l_5} ??? \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
390 |
\label{eq:d29} & \qquad + \ssum{5} \tm(\T \tensor m(\T) \tensor \T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3+l_4+l_5} ???
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
391 |
\end{align}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
392 |
Now, we see the the expressions on the right hand side of line \eqref{eq:d21} and those on \eqref{eq:d23} cancel. Similarly, line \eqref{eq:d22} cancels
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
393 |
with \eqref{eq:d25}, \eqref{eq:d24} with \eqref{eq:d26}, and \eqref{eq:d27} with \eqref{eq:d29}. Finally, we need to see that \eqref{eq:d28} gives $0$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
394 |
by the usual relations between the $m_k$ in an $A_\infty$ algebra.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
395 |
\end{proof}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
396 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
397 |
\nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
398 |
$n$-category or $A_\infty$ $n$-category). DG $n$-category case is pretty
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
399 |
easy, I think, so maybe it should be done earlier??}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
400 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
401 |
\bigskip
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
402 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
403 |
Outline:
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
404 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
405 |
\item recall defs of $A_\infty$ category (1-category only), modules, (self-) tensor product.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
406 |
use graphical/tree point of view, rather than following Keller exactly
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
407 |
\item define blob complex in $A_\infty$ case; fat mapping cones? tree decoration?
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
408 |
\item topological $A_\infty$ cat def (maybe this should go first); also modules gluing
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
409 |
\item motivating example: $C_*(\maps(X, M))$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
410 |
\item maybe incorporate dual point of view (for $n=1$), where points get
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
411 |
object labels and intervals get 1-morphism labels
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
412 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
413 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
414 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
415 |
\subsection{$A_\infty$ action on the boundary}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
416 |
\label{sec:boundary-action}%
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
417 |
Let $Y$ be an $n{-}1$-manifold.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
418 |
The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
419 |
conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
420 |
of an $A_\infty$ category.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
421 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
422 |
Composition of morphisms (multiplication) depends of a choice of homeomorphism
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
423 |
$I\cup I \cong I$. Given this choice, gluing gives a map
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
424 |
\eq{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
425 |
\bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
426 |
\cong \bc_*(Y\times I; a, c)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
427 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
428 |
Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
429 |
higher associators of the $A_\infty$ structure, more or less canonically.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
430 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
431 |
\nn{is this obvious? does more need to be said?}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
432 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
433 |
Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
434 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
435 |
Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
436 |
$(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
437 |
(variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
438 |
$A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
439 |
Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
440 |
of $Y$ in $X$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
441 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
442 |
In the next section we use the above $A_\infty$ actions to state and prove
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
443 |
a gluing theorem for the blob complexes of $n$-manifolds.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
444 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
445 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
446 |
\subsection{The gluing formula}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
447 |
\label{sec:gluing-formula}%
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
448 |
Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
449 |
of $Y \du -Y$ contained in its boundary.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
450 |
Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
451 |
We wish to describe the blob complex of $X\sgl$ in terms of the blob complex
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
452 |
of $X$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
453 |
More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
454 |
where $c\sgl \in \cC(\bd X\sgl)$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
455 |
in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
456 |
of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
457 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
458 |
\begin{thm}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
459 |
$\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
460 |
of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
461 |
\end{thm}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
462 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
463 |
The proof will occupy the remainder of this section.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
464 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
465 |
\nn{...}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
466 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
467 |
\bigskip
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
468 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
469 |
\nn{need to define/recall def of (self) tensor product over an $A_\infty$ category}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
470 |
|