169
|
1 |
%!TEX root = ../../blob1.tex
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
2 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
3 |
\section{Comparing $n$-category definitions}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
4 |
\label{sec:comparing-defs}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
5 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
6 |
In this appendix we relate the ``topological" category definitions of Section \ref{sec:ncats}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
7 |
to more traditional definitions, for $n=1$ and 2.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
8 |
|
204
|
9 |
\nn{cases to cover: (a) plain $n$-cats for $n=1,2$; (b) $n$-cat modules for $n=1$, also 2?;
|
|
10 |
(c) $A_\infty$ 1-cat; (b) $A_\infty$ 1-cat module?; (e) tensor products?}
|
|
11 |
|
194
|
12 |
\subsection{$1$-categories over $\Set$ or $\Vect$}
|
|
13 |
\label{ssec:1-cats}
|
|
14 |
Given a topological $1$-category $\cX$ we construct a $1$-category in the conventional sense, $c(\cX)$.
|
345
|
15 |
This construction is quite straightforward, but we include the details for the sake of completeness,
|
|
16 |
because it illustrates the role of structures (e.g. orientations, spin structures, etc)
|
|
17 |
on the underlying manifolds, and
|
194
|
18 |
to shed some light on the $n=2$ case, which we describe in \S \ref{ssec:2-cats}.
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
19 |
|
194
|
20 |
Let $B^k$ denote the \emph{standard} $k$-ball.
|
345
|
21 |
Let the objects of $c(\cX)$ be $c(\cX)^0 = \cX(B^0)$ and the morphisms of $c(\cX)$ be $c(\cX)^1 = \cX(B^1)$.
|
|
22 |
The boundary and restriction maps of $\cX$ give domain and range maps from $c(\cX)^1$ to $c(\cX)^0$.
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
23 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
24 |
Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$.
|
345
|
25 |
Define composition in $c(\cX)$ to be the induced map $c(\cX)^1\times c(\cX)^1 \to c(\cX)^1$
|
|
26 |
(defined only when range and domain agree).
|
194
|
27 |
By isotopy invariance in $\cX$, any other choice of homeomorphism gives the same composition rule.
|
201
|
28 |
Also by isotopy invariance, composition is strictly associative.
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
29 |
|
194
|
30 |
Given $a\in c(\cX)^0$, define $\id_a \deq a\times B^1$.
|
|
31 |
By extended isotopy invariance in $\cX$, this has the expected properties of an identity morphism.
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
32 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
33 |
|
345
|
34 |
If the underlying manifolds for $\cX$ have further geometric structure, then we obtain certain functors.
|
|
35 |
The base case is for oriented manifolds, where we obtain no extra algebraic data.
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
36 |
|
345
|
37 |
For 1-categories based on unoriented manifolds (somewhat confusingly, we're thinking of being
|
|
38 |
unoriented as requiring extra data beyond being oriented, namely the identification between the orientations),
|
|
39 |
there is a map $*:c(\cX)^1\to c(\cX)^1$
|
194
|
40 |
coming from $\cX$ applied to an orientation-reversing homeomorphism (unique up to isotopy)
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
41 |
from $B^1$ to itself.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
42 |
Topological properties of this homeomorphism imply that
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
43 |
$a^{**} = a$ (* is order 2), * reverses domain and range, and $(ab)^* = b^*a^*$
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
44 |
(* is an anti-automorphism).
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
45 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
46 |
For 1-categories based on Spin manifolds,
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
47 |
the the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity
|
194
|
48 |
gives an order 2 automorphism of $c(\cX)^1$.
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
49 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
50 |
For 1-categories based on $\text{Pin}_-$ manifolds,
|
194
|
51 |
we have an order 4 antiautomorphism of $c(\cX)^1$.
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
52 |
For 1-categories based on $\text{Pin}_+$ manifolds,
|
194
|
53 |
we have an order 2 antiautomorphism and also an order 2 automorphism of $c(\cX)^1$,
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
54 |
and these two maps commute with each other.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
55 |
\nn{need to also consider automorphisms of $B^0$ / objects}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
56 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
57 |
\medskip
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
58 |
|
194
|
59 |
In the other direction, given a $1$-category $C$
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
60 |
(with objects $C^0$ and morphisms $C^1$) we will construct a topological
|
194
|
61 |
$1$-category $t(C)$.
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
62 |
|
194
|
63 |
If $X$ is a 0-ball (point), let $t(C)(X) \deq C^0$.
|
|
64 |
If $S$ is a 0-sphere, let $t(C)(S) \deq C^0\times C^0$.
|
|
65 |
If $X$ is a 1-ball, let $t(C)(X) \deq C^1$.
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
66 |
Homeomorphisms isotopic to the identity act trivially.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
67 |
If $C$ has extra structure (e.g.\ it's a *-1-category), we use this structure
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
68 |
to define the action of homeomorphisms not isotopic to the identity
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
69 |
(and get, e.g., an unoriented topological 1-category).
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
70 |
|
194
|
71 |
The domain and range maps of $C$ determine the boundary and restriction maps of $t(C)$.
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
72 |
|
194
|
73 |
Gluing maps for $t(C)$ are determined by composition of morphisms in $C$.
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
74 |
|
194
|
75 |
For $X$ a 0-ball, $D$ a 1-ball and $a\in t(C)(X)$, define the product morphism
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
76 |
$a\times D \deq \id_a$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
77 |
It is not hard to verify that this has the desired properties.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
78 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
79 |
\medskip
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
80 |
|
345
|
81 |
The compositions of the constructions above, $$\cX\to c(\cX)\to t(c(\cX))$$
|
|
82 |
and $$C\to t(C)\to c(t(C)),$$ give back
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
83 |
more or less exactly the same thing we started with.
|
194
|
84 |
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
85 |
As we will see below, for $n>1$ the compositions yield a weaker sort of equivalence.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
86 |
|
115
|
87 |
\medskip
|
|
88 |
|
|
89 |
Similar arguments show that modules for topological 1-categories are essentially
|
|
90 |
the same thing as traditional modules for traditional 1-categories.
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
91 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
92 |
\subsection{Plain 2-categories}
|
194
|
93 |
\label{ssec:2-cats}
|
115
|
94 |
Let $\cC$ be a topological 2-category.
|
|
95 |
We will construct a traditional pivotal 2-category.
|
|
96 |
(The ``pivotal" corresponds to our assumption of strong duality for $\cC$.)
|
|
97 |
|
|
98 |
We will try to describe the construction in such a way the the generalization to $n>2$ is clear,
|
124
|
99 |
though this will make the $n=2$ case a little more complicated than necessary.
|
115
|
100 |
|
125
|
101 |
\nn{Note: We have to decide whether our 2-morphsism are shaped like rectangles or bigons.
|
|
102 |
Each approach has advantages and disadvantages.
|
|
103 |
For better or worse, we choose bigons here.}
|
|
104 |
|
128
|
105 |
\nn{maybe we should do both rectangles and bigons?}
|
|
106 |
|
115
|
107 |
Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard
|
|
108 |
$k$-ball, which we also think of as the standard bihedron.
|
|
109 |
Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$
|
|
110 |
into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$.
|
|
111 |
Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$
|
|
112 |
whose boundary is splittable along $E$.
|
|
113 |
This allows us to define the domain and range of morphisms of $C$ using
|
|
114 |
boundary and restriction maps of $\cC$.
|
|
115 |
|
124
|
116 |
Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $C^1$.
|
|
117 |
This is not associative, but we will see later that it is weakly associative.
|
|
118 |
|
125
|
119 |
Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map
|
|
120 |
on $C^2$ (Figure \ref{fzo1}).
|
124
|
121 |
Isotopy invariance implies that this is associative.
|
|
122 |
We will define a ``horizontal" composition later.
|
127
|
123 |
\nn{maybe no need to postpone?}
|
124
|
124 |
|
126
|
125 |
\begin{figure}[t]
|
|
126 |
\begin{equation*}
|
|
127 |
\mathfig{.73}{tempkw/zo1}
|
|
128 |
\end{equation*}
|
|
129 |
\caption{Vertical composition of 2-morphisms}
|
|
130 |
\label{fzo1}
|
|
131 |
\end{figure}
|
|
132 |
|
125
|
133 |
Given $a\in C^1$, define $\id_a = a\times I \in C^1$ (pinched boundary).
|
|
134 |
Extended isotopy invariance for $\cC$ shows that this morphism is an identity for
|
|
135 |
vertical composition.
|
124
|
136 |
|
125
|
137 |
Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$.
|
|
138 |
We will show that this 1-morphism is a weak identity.
|
|
139 |
This would be easier if our 2-morphisms were shaped like rectangles rather than bigons.
|
201
|
140 |
Let $a: y\to x$ be a 1-morphism.
|
125
|
141 |
Define maps $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$
|
|
142 |
as shown in Figure \ref{fzo2}.
|
126
|
143 |
\begin{figure}[t]
|
|
144 |
\begin{equation*}
|
|
145 |
\mathfig{.73}{tempkw/zo2}
|
|
146 |
\end{equation*}
|
|
147 |
\caption{blah blah}
|
|
148 |
\label{fzo2}
|
|
149 |
\end{figure}
|
125
|
150 |
In that figure, the red cross-hatched areas are the product of $x$ and a smaller bigon,
|
201
|
151 |
while the remainder is a half-pinched version of $a\times I$.
|
126
|
152 |
\nn{the red region is unnecessary; remove it? or does it help?
|
|
153 |
(because it's what you get if you bigonify the natural rectangular picture)}
|
125
|
154 |
We must show that the two compositions of these two maps give the identity 2-morphisms
|
|
155 |
on $a$ and $a\bullet \id_x$, as defined above.
|
|
156 |
Figure \ref{fzo3} shows one case.
|
126
|
157 |
\begin{figure}[t]
|
|
158 |
\begin{equation*}
|
|
159 |
\mathfig{.83}{tempkw/zo3}
|
|
160 |
\end{equation*}
|
|
161 |
\caption{blah blah}
|
|
162 |
\label{fzo3}
|
|
163 |
\end{figure}
|
125
|
164 |
In the first step we have inserted a copy of $id(id(x))$ \nn{need better notation for this}.
|
|
165 |
\nn{also need to talk about (somewhere above)
|
126
|
166 |
how this sort of insertion is allowed by extended isotopy invariance and gluing.
|
|
167 |
Also: maybe half-pinched and unpinched products can be derived from fully pinched
|
|
168 |
products after all (?)}
|
125
|
169 |
Figure \ref{fzo4} shows the other case.
|
126
|
170 |
\begin{figure}[t]
|
|
171 |
\begin{equation*}
|
|
172 |
\mathfig{.83}{tempkw/zo4}
|
|
173 |
\end{equation*}
|
|
174 |
\caption{blah blah}
|
|
175 |
\label{fzo4}
|
|
176 |
\end{figure}
|
|
177 |
We first collapse the red region, then remove a product morphism from the boundary,
|
124
|
178 |
|
127
|
179 |
We define horizontal composition of 2-morphisms as shown in Figure \ref{fzo5}.
|
345
|
180 |
It is not hard to show that this is independent of the arbitrary (left/right)
|
|
181 |
choice made in the definition, and that it is associative.
|
127
|
182 |
\begin{figure}[t]
|
|
183 |
\begin{equation*}
|
|
184 |
\mathfig{.83}{tempkw/zo5}
|
|
185 |
\end{equation*}
|
|
186 |
\caption{Horizontal composition of 2-morphisms}
|
|
187 |
\label{fzo5}
|
|
188 |
\end{figure}
|
125
|
189 |
|
|
190 |
\nn{need to find a list of axioms for pivotal 2-cats to check}
|
124
|
191 |
|
114
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
192 |
\nn{...}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
193 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
194 |
\medskip
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
195 |
\hrule
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
196 |
\medskip
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
197 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
198 |
\nn{to be continued...}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
199 |
\medskip
|
194
|
200 |
|
|
201 |
\subsection{$A_\infty$ $1$-categories}
|
|
202 |
\label{sec:comparing-A-infty}
|
345
|
203 |
In this section, we make contact between the usual definition of an $A_\infty$ algebra
|
|
204 |
and our definition of a topological $A_\infty$ algebra, from Definition \ref{defn:topological-Ainfty-category}.
|
194
|
205 |
|
345
|
206 |
We begin be restricting the data of a topological $A_\infty$ algebra to the standard interval $[0,1]$,
|
|
207 |
which we can alternatively characterise as:
|
194
|
208 |
\begin{defn}
|
345
|
209 |
A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$,
|
|
210 |
and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with
|
194
|
211 |
\begin{itemize}
|
|
212 |
\item an action of the operad of $\Obj(\cC)$-labeled cell decompositions
|
|
213 |
\item and a compatible action of $\CD{[0,1]}$.
|
|
214 |
\end{itemize}
|
|
215 |
\end{defn}
|
345
|
216 |
Here the operad of cell decompositions of $[0,1]$ has operations indexed by a finite set of
|
|
217 |
points $0 < x_1< \cdots < x_k < 1$, cutting $[0,1]$ into subintervals.
|
|
218 |
An $X$-labeled cell decomposition labels $\{0, x_1, \ldots, x_k, 1\}$ by $X$.
|
|
219 |
Given two cell decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$, we can compose
|
|
220 |
them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points
|
|
221 |
of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$.
|
|
222 |
In the $X$-labeled case, we insist that the appropriate labels match up.
|
|
223 |
Saying we have an action of this operad means that for each labeled cell decomposition
|
|
224 |
$0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain
|
|
225 |
map $$\cC_{a_0,a_1} \tensor \cdots \tensor \cC_{a_k,a_{k+1}} \to \cC(a_0,a_{k+1})$$ and these
|
|
226 |
chain maps compose exactly as the cell decompositions.
|
|
227 |
An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad
|
|
228 |
if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which
|
|
229 |
is supported on the subintervals determined by $\pi$, then the two possible operations
|
|
230 |
(glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms
|
|
231 |
separately to the subintervals, then glue) commute (as usual, up to a weakly unique homotopy).
|
194
|
232 |
|
345
|
233 |
Translating between this notion and the usual definition of an $A_\infty$ category is now straightforward.
|
|
234 |
To restrict to the standard interval, define $\cC_{a,b} = \cC([0,1];a,b)$.
|
|
235 |
Given a cell decomposition $0 < x_1< \cdots < x_k < 1$, we use the map (suppressing labels)
|
194
|
236 |
$$\cC([0,1])^{\tensor k+1} \to \cC([0,x_1]) \tensor \cdots \tensor \cC[x_k,1] \to \cC([0,1])$$
|
345
|
237 |
where the factors of the first map are induced by the linear isometries $[0,1] \to [x_i, x_{i+1}]$, and the second map is just gluing.
|
|
238 |
The action of $\CD{[0,1]}$ carries across, and is automatically compatible.
|
|
239 |
Going the other way, we just declare $\cC(J;a,b) = \cC_{a,b}$, pick a diffeomorphism
|
|
240 |
$\phi_J : J \isoto [0,1]$ for every interval $J$, define the gluing map
|
|
241 |
$\cC(J_1) \tensor \cC(J_2) \to \cC(J_1 \cup J_2)$ by the first applying
|
|
242 |
the cell decomposition map for $0 < \frac{1}{2} < 1$, then the self-diffeomorphism of $[0,1]$
|
|
243 |
given by $\frac{1}{2} (\phi_{J_1} \cup (1+ \phi_{J_2})) \circ \phi_{J_1 \cup J_2}^{-1}$.
|
|
244 |
You can readily check that this gluing map is associative on the nose. \todo{really?}
|
194
|
245 |
|
|
246 |
%First recall the \emph{coloured little intervals operad}. Given a set of labels $\cL$, the operations are indexed by \emph{decompositions of the interval}, each of which is a collection of disjoint subintervals $\{(a_i,b_i)\}_{i=1}^k$ of $[0,1]$, along with a labeling of the complementary regions by $\cL$, $\{l_0, \ldots, l_k\}$. Given two decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$ such that $l^{(1)}_{m-1} = l^{(2)}_0$ and $l^{(1)}_{m} = l^{(2)}_{k^{(2)}}$, we can form a new decomposition by inserting the intervals of $\cJ^{(2)}$ linearly inside the $m$-th interval of $\cJ^{(1)}$. We call the resulting decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$.
|
|
247 |
|
|
248 |
%\begin{defn}
|
|
249 |
%A \emph{topological $A_\infty$ category} $\cC$ has a set of objects $\Obj(\cC)$ and for each $a,b \in \Obj(\cC)$ a chain complex $\cC_{a,b}$, along with a compatible `composition map' and an `action of families of diffeomorphisms'.
|
|
250 |
|
|
251 |
%A \emph{composition map} $f$ is a family of chain maps, one for each decomposition of the interval, $f_\cJ : A^{\tensor k} \to A$, making $\cC$ into a category over the coloured little intervals operad, with labels $\cL = \Obj(\cC)$. Thus the chain maps satisfy the identity
|
|
252 |
%\begin{equation*}
|
|
253 |
%f_{\cJ^{(1)} \circ_m \cJ^{(2)}} = f_{\cJ^{(1)}} \circ (\id^{\tensor m-1} \tensor f_{\cJ^{(2)}} \tensor \id^{\tensor k^{(1)} - m}).
|
|
254 |
%\end{equation*}
|
|
255 |
|
|
256 |
%An \emph{action of families of diffeomorphisms} is a chain map $ev: \CD{[0,1]} \tensor A \to A$, such that
|
|
257 |
%\begin{enumerate}
|
|
258 |
%\item The diagram
|
|
259 |
%\begin{equation*}
|
|
260 |
%\xymatrix{
|
|
261 |
%\CD{[0,1]} \tensor \CD{[0,1]} \tensor A \ar[r]^{\id \tensor ev} \ar[d]^{\circ \tensor \id} & \CD{[0,1]} \tensor A \ar[d]^{ev} \\
|
|
262 |
%\CD{[0,1]} \tensor A \ar[r]^{ev} & A
|
|
263 |
%}
|
|
264 |
%\end{equation*}
|
|
265 |
%commutes up to weakly unique homotopy.
|
|
266 |
%\item If $\phi \in \Diff([0,1])$ and $\cJ$ is a decomposition of the interval, we obtain a new decomposition $\phi(\cJ)$ and a collection $\phi_m \in \Diff([0,1])$ of diffeomorphisms obtained by taking the restrictions $\restrict{\phi}{[a_m,b_m]} : [a_m,b_m] \to [\phi(a_m),\phi(b_m)]$ and pre- and post-composing these with the linear diffeomorphisms $[0,1] \to [a_m,b_m]$ and $[\phi(a_m),\phi(b_m)] \to [0,1]$. We require that
|
|
267 |
%\begin{equation*}
|
|
268 |
%\phi(f_\cJ(a_1, \cdots, a_k)) = f_{\phi(\cJ)}(\phi_1(a_1), \cdots, \phi_k(a_k)).
|
|
269 |
%\end{equation*}
|
|
270 |
%\end{enumerate}
|
|
271 |
%\end{defn}
|
|
272 |
|
345
|
273 |
From a topological $A_\infty$ category on $[0,1]$ $\cC$ we can produce a `conventional'
|
|
274 |
$A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}.
|
|
275 |
We'll just describe the algebra case (that is, a category with only one object),
|
|
276 |
as the modifications required to deal with multiple objects are trivial.
|
|
277 |
Define $A = \cC$ as a chain complex (so $m_1 = d$).
|
|
278 |
Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$.
|
|
279 |
To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms
|
|
280 |
of $[0,1]$ that interpolates linearly between the identity and the piecewise linear
|
|
281 |
diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define
|
194
|
282 |
\begin{equation*}
|
|
283 |
m_3(a,b,c) = ev(\phi_3, m_2(m_2(a,b), c)).
|
|
284 |
\end{equation*}
|
|
285 |
|
|
286 |
It's then easy to calculate that
|
|
287 |
\begin{align*}
|
|
288 |
d(m_3(a,b,c)) & = ev(d \phi_3, m_2(m_2(a,b),c)) - ev(\phi_3 d m_2(m_2(a,b), c)) \\
|
|
289 |
& = ev( \phi_3(1), m_2(m_2(a,b),c)) - ev(\phi_3(0), m_2 (m_2(a,b),c)) - \\ & \qquad - ev(\phi_3, m_2(m_2(da, b), c) + (-1)^{\deg a} m_2(m_2(a, db), c) + \\ & \qquad \quad + (-1)^{\deg a+\deg b} m_2(m_2(a, b), dc) \\
|
|
290 |
& = m_2(a , m_2(b,c)) - m_2(m_2(a,b),c) - \\ & \qquad - m_3(da,b,c) + (-1)^{\deg a + 1} m_3(a,db,c) + \\ & \qquad \quad + (-1)^{\deg a + \deg b + 1} m_3(a,b,dc), \\
|
|
291 |
\intertext{and thus that}
|
|
292 |
m_1 \circ m_3 & = m_2 \circ (\id \tensor m_2) - m_2 \circ (m_2 \tensor \id) - \\ & \qquad - m_3 \circ (m_1 \tensor \id \tensor \id) - m_3 \circ (\id \tensor m_1 \tensor \id) - m_3 \circ (\id \tensor \id \tensor m_1)
|
|
293 |
\end{align*}
|
|
294 |
as required (c.f. \cite[p. 6]{MR1854636}).
|
|
295 |
\todo{then the general case.}
|
345
|
296 |
We won't describe a reverse construction (producing a topological $A_\infty$ category
|
|
297 |
from a `conventional' $A_\infty$ category), but we presume that this will be easy for the experts. |