author | Kevin Walker <kevin@canyon23.net> |
Thu, 22 Jul 2010 15:35:26 -0600 | |
changeset 475 | 07c18e2abd8f |
parent 457 | 54328be726e7 |
child 477 | 86c8e2129355 |
permissions | -rw-r--r-- |
169 | 1 |
%!TEX root = ../../blob1.tex |
114 | 2 |
|
3 |
\section{Comparing $n$-category definitions} |
|
4 |
\label{sec:comparing-defs} |
|
5 |
||
451
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
6 |
In \S\ref{sec:example:traditional-n-categories(fields)} we showed how to construct |
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
7 |
a topological $n$-category from a traditional $n$-category; the morphisms of the |
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
8 |
topological $n$-category are string diagrams labeled by the traditional $n$-category. |
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
9 |
In this appendix we sketch how to go the other direction, for $n=1$ and 2. |
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
10 |
The basic recipe, given a topological $n$-category $\cC$, is to define the $k$-morphisms |
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
11 |
of the corresponding traditional $n$-category to be $\cC(B^k)$, where |
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
12 |
$B^k$ is the {\it standard} $k$-ball. |
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
13 |
One must then show that the axioms of \S\ref{ss:n-cat-def} imply the traditional $n$-category axioms. |
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
14 |
One should also show that composing the two arrows (between traditional and topological $n$-categories) |
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
15 |
yields the appropriate sort of equivalence on each side. |
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
16 |
Since we haven't given a definition for functors between topological $n$-categories |
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
17 |
(the paper is already too long!), we do not pursue this here. |
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
18 |
\nn{say something about modules and tensor products?} |
114 | 19 |
|
451
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
20 |
We emphasize that we are just sketching some of the main ideas in this appendix --- |
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
21 |
it falls well short of proving the definitions are equivalent. |
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
22 |
|
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
23 |
%\nn{cases to cover: (a) plain $n$-cats for $n=1,2$; (b) $n$-cat modules for $n=1$, also 2?; |
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
24 |
%(c) $A_\infty$ 1-cat; (b) $A_\infty$ 1-cat module?; (e) tensor products?} |
204 | 25 |
|
194 | 26 |
\subsection{$1$-categories over $\Set$ or $\Vect$} |
27 |
\label{ssec:1-cats} |
|
28 |
Given a topological $1$-category $\cX$ we construct a $1$-category in the conventional sense, $c(\cX)$. |
|
345 | 29 |
This construction is quite straightforward, but we include the details for the sake of completeness, |
30 |
because it illustrates the role of structures (e.g. orientations, spin structures, etc) |
|
31 |
on the underlying manifolds, and |
|
194 | 32 |
to shed some light on the $n=2$ case, which we describe in \S \ref{ssec:2-cats}. |
114 | 33 |
|
194 | 34 |
Let $B^k$ denote the \emph{standard} $k$-ball. |
345 | 35 |
Let the objects of $c(\cX)$ be $c(\cX)^0 = \cX(B^0)$ and the morphisms of $c(\cX)$ be $c(\cX)^1 = \cX(B^1)$. |
36 |
The boundary and restriction maps of $\cX$ give domain and range maps from $c(\cX)^1$ to $c(\cX)^0$. |
|
114 | 37 |
|
38 |
Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$. |
|
345 | 39 |
Define composition in $c(\cX)$ to be the induced map $c(\cX)^1\times c(\cX)^1 \to c(\cX)^1$ |
40 |
(defined only when range and domain agree). |
|
194 | 41 |
By isotopy invariance in $\cX$, any other choice of homeomorphism gives the same composition rule. |
201 | 42 |
Also by isotopy invariance, composition is strictly associative. |
114 | 43 |
|
194 | 44 |
Given $a\in c(\cX)^0$, define $\id_a \deq a\times B^1$. |
45 |
By extended isotopy invariance in $\cX$, this has the expected properties of an identity morphism. |
|
114 | 46 |
|
47 |
||
345 | 48 |
If the underlying manifolds for $\cX$ have further geometric structure, then we obtain certain functors. |
49 |
The base case is for oriented manifolds, where we obtain no extra algebraic data. |
|
114 | 50 |
|
451
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
51 |
For 1-categories based on unoriented manifolds, |
345 | 52 |
there is a map $*:c(\cX)^1\to c(\cX)^1$ |
194 | 53 |
coming from $\cX$ applied to an orientation-reversing homeomorphism (unique up to isotopy) |
114 | 54 |
from $B^1$ to itself. |
55 |
Topological properties of this homeomorphism imply that |
|
56 |
$a^{**} = a$ (* is order 2), * reverses domain and range, and $(ab)^* = b^*a^*$ |
|
57 |
(* is an anti-automorphism). |
|
58 |
||
59 |
For 1-categories based on Spin manifolds, |
|
60 |
the the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity |
|
194 | 61 |
gives an order 2 automorphism of $c(\cX)^1$. |
114 | 62 |
|
63 |
For 1-categories based on $\text{Pin}_-$ manifolds, |
|
194 | 64 |
we have an order 4 antiautomorphism of $c(\cX)^1$. |
114 | 65 |
For 1-categories based on $\text{Pin}_+$ manifolds, |
194 | 66 |
we have an order 2 antiautomorphism and also an order 2 automorphism of $c(\cX)^1$, |
114 | 67 |
and these two maps commute with each other. |
451
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
68 |
%\nn{need to also consider automorphisms of $B^0$ / objects} |
114 | 69 |
|
451
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
70 |
\noop{ |
114 | 71 |
\medskip |
72 |
||
194 | 73 |
In the other direction, given a $1$-category $C$ |
114 | 74 |
(with objects $C^0$ and morphisms $C^1$) we will construct a topological |
194 | 75 |
$1$-category $t(C)$. |
114 | 76 |
|
194 | 77 |
If $X$ is a 0-ball (point), let $t(C)(X) \deq C^0$. |
78 |
If $S$ is a 0-sphere, let $t(C)(S) \deq C^0\times C^0$. |
|
79 |
If $X$ is a 1-ball, let $t(C)(X) \deq C^1$. |
|
114 | 80 |
Homeomorphisms isotopic to the identity act trivially. |
81 |
If $C$ has extra structure (e.g.\ it's a *-1-category), we use this structure |
|
82 |
to define the action of homeomorphisms not isotopic to the identity |
|
83 |
(and get, e.g., an unoriented topological 1-category). |
|
84 |
||
194 | 85 |
The domain and range maps of $C$ determine the boundary and restriction maps of $t(C)$. |
114 | 86 |
|
194 | 87 |
Gluing maps for $t(C)$ are determined by composition of morphisms in $C$. |
114 | 88 |
|
194 | 89 |
For $X$ a 0-ball, $D$ a 1-ball and $a\in t(C)(X)$, define the product morphism |
114 | 90 |
$a\times D \deq \id_a$. |
91 |
It is not hard to verify that this has the desired properties. |
|
92 |
||
93 |
\medskip |
|
94 |
||
345 | 95 |
The compositions of the constructions above, $$\cX\to c(\cX)\to t(c(\cX))$$ |
96 |
and $$C\to t(C)\to c(t(C)),$$ give back |
|
114 | 97 |
more or less exactly the same thing we started with. |
194 | 98 |
|
114 | 99 |
As we will see below, for $n>1$ the compositions yield a weaker sort of equivalence. |
451
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
100 |
} %end \noop |
114 | 101 |
|
115 | 102 |
\medskip |
103 |
||
104 |
Similar arguments show that modules for topological 1-categories are essentially |
|
105 |
the same thing as traditional modules for traditional 1-categories. |
|
114 | 106 |
|
451
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
107 |
|
114 | 108 |
\subsection{Plain 2-categories} |
194 | 109 |
\label{ssec:2-cats} |
115 | 110 |
Let $\cC$ be a topological 2-category. |
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
111 |
We will construct from $\cC$ a traditional pivotal 2-category. |
115 | 112 |
(The ``pivotal" corresponds to our assumption of strong duality for $\cC$.) |
113 |
||
114 |
We will try to describe the construction in such a way the the generalization to $n>2$ is clear, |
|
124 | 115 |
though this will make the $n=2$ case a little more complicated than necessary. |
115 | 116 |
|
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
117 |
Before proceeding, we must decide whether the 2-morphisms of our |
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
118 |
pivotal 2-category are shaped like rectangles or bigons. |
125 | 119 |
Each approach has advantages and disadvantages. |
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
120 |
For better or worse, we choose bigons here. |
128 | 121 |
|
115 | 122 |
Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard |
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
123 |
$k$-ball, which we also think of as the standard bihedron (a.k.a.\ globe). |
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
124 |
(For $k=1$ this is an interval, and for $k=2$ it is a bigon.) |
115 | 125 |
Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$ |
126 |
into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$. |
|
127 |
Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$ |
|
128 |
whose boundary is splittable along $E$. |
|
129 |
This allows us to define the domain and range of morphisms of $C$ using |
|
130 |
boundary and restriction maps of $\cC$. |
|
131 |
||
124 | 132 |
Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $C^1$. |
133 |
This is not associative, but we will see later that it is weakly associative. |
|
134 |
||
125 | 135 |
Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map |
136 |
on $C^2$ (Figure \ref{fzo1}). |
|
124 | 137 |
Isotopy invariance implies that this is associative. |
138 |
We will define a ``horizontal" composition later. |
|
139 |
||
126 | 140 |
\begin{figure}[t] |
141 |
\begin{equation*} |
|
142 |
\mathfig{.73}{tempkw/zo1} |
|
143 |
\end{equation*} |
|
144 |
\caption{Vertical composition of 2-morphisms} |
|
145 |
\label{fzo1} |
|
146 |
\end{figure} |
|
147 |
||
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
148 |
Given $a\in C^1$, define $\id_a = a\times I \in C^2$ (pinched boundary). |
125 | 149 |
Extended isotopy invariance for $\cC$ shows that this morphism is an identity for |
150 |
vertical composition. |
|
124 | 151 |
|
125 | 152 |
Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$. |
153 |
We will show that this 1-morphism is a weak identity. |
|
154 |
This would be easier if our 2-morphisms were shaped like rectangles rather than bigons. |
|
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
155 |
|
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
156 |
In showing that identity 1-morphisms have the desired properties, we will |
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
157 |
rely heavily on the extended isotopy invariance of 2-morphisms in $\cC$. |
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
158 |
This means we are free to add or delete product regions from 2-morphisms. |
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
159 |
|
201 | 160 |
Let $a: y\to x$ be a 1-morphism. |
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
161 |
Define 2-morphsims $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ |
125 | 162 |
as shown in Figure \ref{fzo2}. |
126 | 163 |
\begin{figure}[t] |
164 |
\begin{equation*} |
|
165 |
\mathfig{.73}{tempkw/zo2} |
|
166 |
\end{equation*} |
|
167 |
\caption{blah blah} |
|
168 |
\label{fzo2} |
|
169 |
\end{figure} |
|
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
170 |
As suggested by the figure, these are two different reparameterizations |
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
171 |
of a half-pinched version of $a\times I$. |
125 | 172 |
We must show that the two compositions of these two maps give the identity 2-morphisms |
173 |
on $a$ and $a\bullet \id_x$, as defined above. |
|
174 |
Figure \ref{fzo3} shows one case. |
|
126 | 175 |
\begin{figure}[t] |
176 |
\begin{equation*} |
|
177 |
\mathfig{.83}{tempkw/zo3} |
|
178 |
\end{equation*} |
|
179 |
\caption{blah blah} |
|
180 |
\label{fzo3} |
|
181 |
\end{figure} |
|
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
182 |
In the first step we have inserted a copy of $(x\times I)\times I$. |
125 | 183 |
Figure \ref{fzo4} shows the other case. |
126 | 184 |
\begin{figure}[t] |
185 |
\begin{equation*} |
|
186 |
\mathfig{.83}{tempkw/zo4} |
|
187 |
\end{equation*} |
|
188 |
\caption{blah blah} |
|
189 |
\label{fzo4} |
|
190 |
\end{figure} |
|
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
191 |
We identify a product region and remove it. |
124 | 192 |
|
127 | 193 |
We define horizontal composition of 2-morphisms as shown in Figure \ref{fzo5}. |
345 | 194 |
It is not hard to show that this is independent of the arbitrary (left/right) |
195 |
choice made in the definition, and that it is associative. |
|
127 | 196 |
\begin{figure}[t] |
197 |
\begin{equation*} |
|
198 |
\mathfig{.83}{tempkw/zo5} |
|
199 |
\end{equation*} |
|
200 |
\caption{Horizontal composition of 2-morphisms} |
|
201 |
\label{fzo5} |
|
202 |
\end{figure} |
|
125 | 203 |
|
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
204 |
%\nn{need to find a list of axioms for pivotal 2-cats to check} |
114 | 205 |
|
194 | 206 |
|
207 |
\subsection{$A_\infty$ $1$-categories} |
|
208 |
\label{sec:comparing-A-infty} |
|
431 | 209 |
In this section, we make contact between the usual definition of an $A_\infty$ category |
210 |
and our definition of a topological $A_\infty$ $1$-category, from \S \ref{???}. |
|
194 | 211 |
|
431 | 212 |
That definition associates a chain complex to every interval, and we begin by giving an alternative definition that is entirely in terms of the chain complex associated to the standard interval $[0,1]$. |
194 | 213 |
\begin{defn} |
345 | 214 |
A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$, |
215 |
and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with |
|
194 | 216 |
\begin{itemize} |
217 |
\item an action of the operad of $\Obj(\cC)$-labeled cell decompositions |
|
218 |
\item and a compatible action of $\CD{[0,1]}$. |
|
219 |
\end{itemize} |
|
220 |
\end{defn} |
|
345 | 221 |
Here the operad of cell decompositions of $[0,1]$ has operations indexed by a finite set of |
222 |
points $0 < x_1< \cdots < x_k < 1$, cutting $[0,1]$ into subintervals. |
|
223 |
An $X$-labeled cell decomposition labels $\{0, x_1, \ldots, x_k, 1\}$ by $X$. |
|
224 |
Given two cell decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$, we can compose |
|
225 |
them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points |
|
226 |
of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$. |
|
227 |
In the $X$-labeled case, we insist that the appropriate labels match up. |
|
228 |
Saying we have an action of this operad means that for each labeled cell decomposition |
|
229 |
$0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain |
|
431 | 230 |
map $$\cC_{a_0,a_1} \tensor \cdots \tensor \cC_{a_k,a_{k+1}} \to \cC_{a_0,a_{k+1}}$$ and these |
345 | 231 |
chain maps compose exactly as the cell decompositions. |
232 |
An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad |
|
233 |
if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which |
|
234 |
is supported on the subintervals determined by $\pi$, then the two possible operations |
|
235 |
(glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms |
|
236 |
separately to the subintervals, then glue) commute (as usual, up to a weakly unique homotopy). |
|
194 | 237 |
|
345 | 238 |
Translating between this notion and the usual definition of an $A_\infty$ category is now straightforward. |
239 |
To restrict to the standard interval, define $\cC_{a,b} = \cC([0,1];a,b)$. |
|
240 |
Given a cell decomposition $0 < x_1< \cdots < x_k < 1$, we use the map (suppressing labels) |
|
194 | 241 |
$$\cC([0,1])^{\tensor k+1} \to \cC([0,x_1]) \tensor \cdots \tensor \cC[x_k,1] \to \cC([0,1])$$ |
345 | 242 |
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. |
243 |
The action of $\CD{[0,1]}$ carries across, and is automatically compatible. |
|
244 |
Going the other way, we just declare $\cC(J;a,b) = \cC_{a,b}$, pick a diffeomorphism |
|
245 |
$\phi_J : J \isoto [0,1]$ for every interval $J$, define the gluing map |
|
246 |
$\cC(J_1) \tensor \cC(J_2) \to \cC(J_1 \cup J_2)$ by the first applying |
|
247 |
the cell decomposition map for $0 < \frac{1}{2} < 1$, then the self-diffeomorphism of $[0,1]$ |
|
248 |
given by $\frac{1}{2} (\phi_{J_1} \cup (1+ \phi_{J_2})) \circ \phi_{J_1 \cup J_2}^{-1}$. |
|
249 |
You can readily check that this gluing map is associative on the nose. \todo{really?} |
|
194 | 250 |
|
251 |
%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 $\cJ^{(1)} \circ_m \cJ^{(2)}$. |
|
252 |
||
253 |
%\begin{defn} |
|
254 |
%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'. |
|
255 |
||
256 |
%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 |
|
257 |
%\begin{equation*} |
|
258 |
%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}). |
|
259 |
%\end{equation*} |
|
260 |
||
261 |
%An \emph{action of families of diffeomorphisms} is a chain map $ev: \CD{[0,1]} \tensor A \to A$, such that |
|
262 |
%\begin{enumerate} |
|
263 |
%\item The diagram |
|
264 |
%\begin{equation*} |
|
265 |
%\xymatrix{ |
|
266 |
%\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} \\ |
|
267 |
%\CD{[0,1]} \tensor A \ar[r]^{ev} & A |
|
268 |
%} |
|
269 |
%\end{equation*} |
|
270 |
%commutes up to weakly unique homotopy. |
|
271 |
%\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 |
|
272 |
%\begin{equation*} |
|
273 |
%\phi(f_\cJ(a_1, \cdots, a_k)) = f_{\phi(\cJ)}(\phi_1(a_1), \cdots, \phi_k(a_k)). |
|
274 |
%\end{equation*} |
|
275 |
%\end{enumerate} |
|
276 |
%\end{defn} |
|
277 |
||
345 | 278 |
From a topological $A_\infty$ category on $[0,1]$ $\cC$ we can produce a `conventional' |
279 |
$A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}. |
|
280 |
We'll just describe the algebra case (that is, a category with only one object), |
|
281 |
as the modifications required to deal with multiple objects are trivial. |
|
282 |
Define $A = \cC$ as a chain complex (so $m_1 = d$). |
|
283 |
Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$. |
|
284 |
To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms |
|
285 |
of $[0,1]$ that interpolates linearly between the identity and the piecewise linear |
|
286 |
diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define |
|
194 | 287 |
\begin{equation*} |
288 |
m_3(a,b,c) = ev(\phi_3, m_2(m_2(a,b), c)). |
|
289 |
\end{equation*} |
|
290 |
||
291 |
It's then easy to calculate that |
|
292 |
\begin{align*} |
|
293 |
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)) \\ |
|
294 |
& = 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) \\ |
|
295 |
& = 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), \\ |
|
296 |
\intertext{and thus that} |
|
297 |
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) |
|
298 |
\end{align*} |
|
299 |
as required (c.f. \cite[p. 6]{MR1854636}). |
|
300 |
\todo{then the general case.} |
|
345 | 301 |
We won't describe a reverse construction (producing a topological $A_\infty$ category |
417
d3b05641e7ca
making quotation marks consistently "American style"
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
302 |
from a ``conventional" $A_\infty$ category), but we presume that this will be easy for the experts. |