author | Kevin Walker <kevin@canyon23.net> |
Fri, 09 Dec 2011 18:43:11 -0800 | |
changeset 945 | 341c2a09f9a8 |
parent 941 | ee0c940fce59 |
child 947 | ceed680b7f42 |
permissions | -rw-r--r-- |
169 | 1 |
%!TEX root = ../../blob1.tex |
114 | 2 |
|
512
050dba5e7bdd
fixing some (but not all!?) of the hyperref warnings; start on revision of evmap
Kevin Walker <kevin@canyon23.net>
parents:
510
diff
changeset
|
3 |
\section{Comparing \texorpdfstring{$n$}{n}-category definitions} |
114 | 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 |
865
7abe7642265e
relentless adding 'disk-like' everywhere it could possibly go
Scott Morrison <scott@tqft.net>
parents:
854
diff
changeset
|
7 |
a disk-like $n$-category from a traditional $n$-category; the morphisms of the |
7abe7642265e
relentless adding 'disk-like' everywhere it could possibly go
Scott Morrison <scott@tqft.net>
parents:
854
diff
changeset
|
8 |
disk-like $n$-category are string diagrams labeled by the traditional $n$-category. |
451
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. |
685
8efbd2730ef9
"topological n-cat" --> either "disk-like n-cat" or "ordinary n-cat" (when contrasted with A-inf n-cat)
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
10 |
The basic recipe, given a disk-like $n$-category $\cC$, is to define the $k$-morphisms |
451
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. |
685
8efbd2730ef9
"topological n-cat" --> either "disk-like n-cat" or "ordinary n-cat" (when contrasted with A-inf n-cat)
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
14 |
One should also show that composing the two arrows (between traditional and disk-like $n$-categories) |
451
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. |
853
870d6fac5420
several minor corrections, from referee
Scott Morrison <scott@tqft.net>
parents:
792
diff
changeset
|
16 |
Since we haven't given a definition for functors between disk-like $n$-categories, we do not pursue this here. |
114 | 17 |
|
451
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
18 |
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
|
19 |
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
|
20 |
|
680
0591d017e698
plain n-cat -> ordinary n-cat
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
21 |
%\nn{cases to cover: (a) ordinary $n$-cats for $n=1,2$; (b) $n$-cat modules for $n=1$, also 2?; |
451
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
22 |
%(c) $A_\infty$ 1-cat; (b) $A_\infty$ 1-cat module?; (e) tensor products?} |
204 | 23 |
|
512
050dba5e7bdd
fixing some (but not all!?) of the hyperref warnings; start on revision of evmap
Kevin Walker <kevin@canyon23.net>
parents:
510
diff
changeset
|
24 |
\subsection{1-categories over \texorpdfstring{$\Set$ or $\Vect$}{Set or Vect}} |
194 | 25 |
\label{ssec:1-cats} |
685
8efbd2730ef9
"topological n-cat" --> either "disk-like n-cat" or "ordinary n-cat" (when contrasted with A-inf n-cat)
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
26 |
Given a disk-like $1$-category $\cX$ we construct a $1$-category in the conventional sense, $c(\cX)$. |
345 | 27 |
This construction is quite straightforward, but we include the details for the sake of completeness, |
28 |
because it illustrates the role of structures (e.g. orientations, spin structures, etc) |
|
29 |
on the underlying manifolds, and |
|
194 | 30 |
to shed some light on the $n=2$ case, which we describe in \S \ref{ssec:2-cats}. |
114 | 31 |
|
194 | 32 |
Let $B^k$ denote the \emph{standard} $k$-ball. |
345 | 33 |
Let the objects of $c(\cX)$ be $c(\cX)^0 = \cX(B^0)$ and the morphisms of $c(\cX)$ be $c(\cX)^1 = \cX(B^1)$. |
34 |
The boundary and restriction maps of $\cX$ give domain and range maps from $c(\cX)^1$ to $c(\cX)^0$. |
|
114 | 35 |
|
36 |
Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$. |
|
345 | 37 |
Define composition in $c(\cX)$ to be the induced map $c(\cX)^1\times c(\cX)^1 \to c(\cX)^1$ |
38 |
(defined only when range and domain agree). |
|
194 | 39 |
By isotopy invariance in $\cX$, any other choice of homeomorphism gives the same composition rule. |
201 | 40 |
Also by isotopy invariance, composition is strictly associative. |
114 | 41 |
|
194 | 42 |
Given $a\in c(\cX)^0$, define $\id_a \deq a\times B^1$. |
43 |
By extended isotopy invariance in $\cX$, this has the expected properties of an identity morphism. |
|
114 | 44 |
|
790
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
45 |
We have now defined the basic ingredients for the 1-category $c(\cX)$. |
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
46 |
As we explain below, $c(\cX)$ might have additional structure corresponding to the |
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
47 |
unoriented, oriented, Spin, $\text{Pin}_+$ or $\text{Pin}_-$ structure on the 1-balls used to define $\cX$. |
114 | 48 |
|
790
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
49 |
For 1-categories based on unoriented balls, |
737 | 50 |
there is a map $\dagger:c(\cX)^1\to c(\cX)^1$ |
194 | 51 |
coming from $\cX$ applied to an orientation-reversing homeomorphism (unique up to isotopy) |
114 | 52 |
from $B^1$ to itself. |
790
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
53 |
(Of course our $B^1$ is unoriented, i.e.\ not equipped with an orientation. |
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
54 |
We mean the homeomorphism which would reverse the orientation if there were one; |
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
55 |
$B^1$ is not oriented, but it is orientable.) |
114 | 56 |
Topological properties of this homeomorphism imply that |
737 | 57 |
$a^{\dagger\dagger} = a$ ($\dagger$ is order 2), $\dagger$ reverses domain and range, and $(ab)^\dagger = b^\dagger a^\dagger$ |
58 |
($\dagger$ is an anti-automorphism). |
|
790
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
59 |
Recall that in this context 0-balls should be thought of as equipped with a germ of a 1-dimensional neighborhood. |
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
60 |
There is a unique such 0-ball, up to homeomorphism, but it has a non-identity automorphism corresponding to reversing the |
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
61 |
orientation of the germ. |
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
62 |
Consequently, the objects of $c(\cX)$ are equipped with an involution, also denoted $\dagger$. |
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
63 |
If $a:x\to y$ is a morphism of $c(\cX)$ then $a^\dagger: y^\dagger\to x^\dagger$. |
114 | 64 |
|
790
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
65 |
For 1-categories based on oriented balls, there are no non-trivial homeomorphisms of 0- or 1-balls, and thus no |
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
66 |
additional structure on $c(\cX)$. |
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
67 |
|
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
68 |
For 1-categories based on Spin balls, |
791
c5a33223af00
adding remarks on objects to C.1; this is pretty uninspired writing, so feel free to improve
Kevin Walker <kevin@canyon23.net>
parents:
790
diff
changeset
|
69 |
the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity |
194 | 70 |
gives an order 2 automorphism of $c(\cX)^1$. |
791
c5a33223af00
adding remarks on objects to C.1; this is pretty uninspired writing, so feel free to improve
Kevin Walker <kevin@canyon23.net>
parents:
790
diff
changeset
|
71 |
There is a similar involution on the objects $c(\cX)^0$. |
792
966a571daa10
added remark on super algebra
Kevin Walker <kevin@canyon23.net>
parents:
791
diff
changeset
|
72 |
In the case where there is only one object and we are enriching over complex vector spaces, this |
966a571daa10
added remark on super algebra
Kevin Walker <kevin@canyon23.net>
parents:
791
diff
changeset
|
73 |
is just a super algebra. |
966a571daa10
added remark on super algebra
Kevin Walker <kevin@canyon23.net>
parents:
791
diff
changeset
|
74 |
The even elements are the $+1$ eigenspace of the involution on $c(\cX)^1$, |
966a571daa10
added remark on super algebra
Kevin Walker <kevin@canyon23.net>
parents:
791
diff
changeset
|
75 |
and the odd elements are the $-1$ eigenspace of the involution. |
114 | 76 |
|
790
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
77 |
For 1-categories based on $\text{Pin}_-$ balls, |
194 | 78 |
we have an order 4 antiautomorphism of $c(\cX)^1$. |
790
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
79 |
For 1-categories based on $\text{Pin}_+$ balls, |
194 | 80 |
we have an order 2 antiautomorphism and also an order 2 automorphism of $c(\cX)^1$, |
114 | 81 |
and these two maps commute with each other. |
791
c5a33223af00
adding remarks on objects to C.1; this is pretty uninspired writing, so feel free to improve
Kevin Walker <kevin@canyon23.net>
parents:
790
diff
changeset
|
82 |
In both cases there is a similar map on objects. |
790
ec8587c33c0b
more details in C.1; reorganized to-do list
Kevin Walker <kevin@canyon23.net>
parents:
737
diff
changeset
|
83 |
|
114 | 84 |
|
451
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
85 |
\noop{ |
114 | 86 |
\medskip |
87 |
||
194 | 88 |
In the other direction, given a $1$-category $C$ |
685
8efbd2730ef9
"topological n-cat" --> either "disk-like n-cat" or "ordinary n-cat" (when contrasted with A-inf n-cat)
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
89 |
(with objects $C^0$ and morphisms $C^1$) we will construct a disk-like |
194 | 90 |
$1$-category $t(C)$. |
114 | 91 |
|
194 | 92 |
If $X$ is a 0-ball (point), let $t(C)(X) \deq C^0$. |
93 |
If $S$ is a 0-sphere, let $t(C)(S) \deq C^0\times C^0$. |
|
94 |
If $X$ is a 1-ball, let $t(C)(X) \deq C^1$. |
|
114 | 95 |
Homeomorphisms isotopic to the identity act trivially. |
96 |
If $C$ has extra structure (e.g.\ it's a *-1-category), we use this structure |
|
97 |
to define the action of homeomorphisms not isotopic to the identity |
|
685
8efbd2730ef9
"topological n-cat" --> either "disk-like n-cat" or "ordinary n-cat" (when contrasted with A-inf n-cat)
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
98 |
(and get, e.g., an unoriented disk-like 1-category). |
114 | 99 |
|
194 | 100 |
The domain and range maps of $C$ determine the boundary and restriction maps of $t(C)$. |
114 | 101 |
|
194 | 102 |
Gluing maps for $t(C)$ are determined by composition of morphisms in $C$. |
114 | 103 |
|
194 | 104 |
For $X$ a 0-ball, $D$ a 1-ball and $a\in t(C)(X)$, define the product morphism |
114 | 105 |
$a\times D \deq \id_a$. |
106 |
It is not hard to verify that this has the desired properties. |
|
107 |
||
108 |
\medskip |
|
109 |
||
345 | 110 |
The compositions of the constructions above, $$\cX\to c(\cX)\to t(c(\cX))$$ |
111 |
and $$C\to t(C)\to c(t(C)),$$ give back |
|
114 | 112 |
more or less exactly the same thing we started with. |
194 | 113 |
|
114 | 114 |
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
|
115 |
} %end \noop |
114 | 116 |
|
115 | 117 |
\medskip |
118 |
||
685
8efbd2730ef9
"topological n-cat" --> either "disk-like n-cat" or "ordinary n-cat" (when contrasted with A-inf n-cat)
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
119 |
Similar arguments show that modules for disk-like 1-categories are essentially |
115 | 120 |
the same thing as traditional modules for traditional 1-categories. |
114 | 121 |
|
451
bb7e388b9704
starting on comparing_defs.tex
Kevin Walker <kevin@canyon23.net>
parents:
433
diff
changeset
|
122 |
|
481 | 123 |
\subsection{Pivotal 2-categories} |
194 | 124 |
\label{ssec:2-cats} |
685
8efbd2730ef9
"topological n-cat" --> either "disk-like n-cat" or "ordinary n-cat" (when contrasted with A-inf n-cat)
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
125 |
Let $\cC$ be a disk-like 2-category. |
933 | 126 |
We will construct from $\cC$ a traditional pivotal 2-category $D$. |
115 | 127 |
(The ``pivotal" corresponds to our assumption of strong duality for $\cC$.) |
128 |
||
854 | 129 |
We will try to describe the construction in such a way that the generalization to $n>2$ is clear, |
124 | 130 |
though this will make the $n=2$ case a little more complicated than necessary. |
115 | 131 |
|
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
132 |
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
|
133 |
pivotal 2-category are shaped like rectangles or bigons. |
125 | 134 |
Each approach has advantages and disadvantages. |
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
135 |
For better or worse, we choose bigons here. |
128 | 136 |
|
933 | 137 |
Define the $k$-morphisms $D^k$ of $D$ to be $\cC(B^k) \trans E$, where $B^k$ denotes the standard |
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
138 |
$k$-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
|
139 |
(For $k=1$ this is an interval, and for $k=2$ it is a bigon.) |
115 | 140 |
Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$ |
141 |
into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$. |
|
728
83c1ec0aac1f
two more transverse symbols
Scott Morrison <scott@tqft.net>
parents:
685
diff
changeset
|
142 |
Recall that the subscript in $\cC(B^k) \trans E$ means that we consider the subset of $\cC(B^k)$ |
115 | 143 |
whose boundary is splittable along $E$. |
933 | 144 |
This allows us to define the domain and range of morphisms of $D$ using |
115 | 145 |
boundary and restriction maps of $\cC$. |
146 |
||
933 | 147 |
Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $D^1$. |
124 | 148 |
This is not associative, but we will see later that it is weakly associative. |
149 |
||
125 | 150 |
Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map |
933 | 151 |
on $D^2$ (Figure \ref{fzo1}). |
124 | 152 |
Isotopy invariance implies that this is associative. |
153 |
We will define a ``horizontal" composition later. |
|
154 |
||
126 | 155 |
\begin{figure}[t] |
530
b236746e8e4d
futzing with figures (\begin{center|equation} to \centering)
Kevin Walker <kevin@canyon23.net>
parents:
529
diff
changeset
|
156 |
\centering |
510 | 157 |
\begin{tikzpicture} |
158 |
||
159 |
\newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
|
160 |
\newcommand{\nsep}{1.8} |
|
161 |
||
162 |
\node[outer sep=\nsep](A) at (0,0) { |
|
163 |
\begin{tikzpicture} |
|
164 |
\draw (0,0) coordinate (p1); |
|
165 |
\draw (4,0) coordinate (p2); |
|
166 |
\draw (2,1.2) coordinate (pu); |
|
167 |
\draw (2,-1.2) coordinate (pd); |
|
168 |
||
169 |
\draw (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
170 |
\draw (p1)--(p2); |
|
171 |
||
172 |
\draw (p1) \vertex; |
|
173 |
\draw (p2) \vertex; |
|
174 |
||
175 |
\node at (2.1, .44) {$B^2$}; |
|
176 |
\node at (2.1, -.44) {$B^2$}; |
|
177 |
||
178 |
\end{tikzpicture} |
|
179 |
}; |
|
180 |
||
181 |
\node[outer sep=\nsep](B) at (6,0) { |
|
182 |
\begin{tikzpicture} |
|
183 |
\draw (0,0) coordinate (p1); |
|
184 |
\draw (4,0) coordinate (p2); |
|
185 |
\draw (2,.6) coordinate (pu); |
|
186 |
\draw (2,-.6) coordinate (pd); |
|
187 |
||
188 |
\draw (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
189 |
\draw[help lines, dashed] (p1)--(p2); |
|
190 |
||
191 |
\draw (p1) \vertex; |
|
192 |
\draw (p2) \vertex; |
|
193 |
||
194 |
\node at (2.1,0) {$B^2$}; |
|
195 |
||
196 |
\end{tikzpicture} |
|
197 |
}; |
|
198 |
||
199 |
\draw[->, thick, blue!50!green] (A) -- node[black, above] {$\cong$} (B); |
|
200 |
||
201 |
\end{tikzpicture} |
|
126 | 202 |
\caption{Vertical composition of 2-morphisms} |
203 |
\label{fzo1} |
|
204 |
\end{figure} |
|
205 |
||
933 | 206 |
Given $a\in D^1$, define $\id_a = a\times I \in D^2$ (pinched boundary). |
125 | 207 |
Extended isotopy invariance for $\cC$ shows that this morphism is an identity for |
208 |
vertical composition. |
|
124 | 209 |
|
125 | 210 |
Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$. |
211 |
We will show that this 1-morphism is a weak identity. |
|
212 |
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
|
213 |
|
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
214 |
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
|
215 |
rely heavily on the extended isotopy invariance of 2-morphisms in $\cC$. |
934 | 216 |
Extended isotopy invariance implies that adding a product collar to a 2-morphism of $\cC$ has no effect, |
217 |
and by cutting and regluing we can insert (or delete) product regions in the interior of 2-morphisms as well. |
|
218 |
Figure \nn{triangle.pdf 2.a through 2.d} shows some examples. |
|
219 |
||
220 |
||
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
221 |
|
201 | 222 |
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
|
223 |
Define 2-morphsims $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$ |
125 | 224 |
as shown in Figure \ref{fzo2}. |
126 | 225 |
\begin{figure}[t] |
530
b236746e8e4d
futzing with figures (\begin{center|equation} to \centering)
Kevin Walker <kevin@canyon23.net>
parents:
529
diff
changeset
|
226 |
\centering |
498
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
227 |
\begin{tikzpicture} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
228 |
\newcommand{\rr}{6} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
229 |
\newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
503
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
230 |
\newcommand{\namedvertex}[1]{node[circle,fill=black,inner sep=1pt] (#1) {}} |
498
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
231 |
|
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
232 |
\node(A) at (0,0) { |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
233 |
\begin{tikzpicture} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
234 |
\node[red,left] at (0,0) {$y$}; |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
235 |
\draw (0,0) \vertex arc (-120:-105:\rr) node[red,below] {$a$} arc(-105:-90:\rr) \vertex node[red,below](x2) {$x$}; |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
236 |
\draw (0,0) \vertex arc (120:105:\rr) node[red,above] {$a$} arc (105:90:\rr) \vertex node[red,above](x1) {$x$} -- (x2); |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
237 |
\begin{scope} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
238 |
\path[clip] (0,0) arc (-120:-60:\rr) arc (60:120:\rr); |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
239 |
\foreach \x in {0,0.24,...,3} { |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
240 |
\draw[green!50!brown] (\x,1) -- (\x,-1); |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
241 |
} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
242 |
\end{scope} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
243 |
\draw[red, decorate,decoration={brace,amplitude=5pt}] ($(x1)+(0.2,-0.2)$) -- ($(x2)+(0.2,0.2)$) node[midway, xshift=0.7cm] {$x \times I$}; |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
244 |
\end{tikzpicture} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
245 |
}; |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
246 |
|
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
247 |
\node(B) at (-4,-4) { |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
248 |
\begin{tikzpicture} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
249 |
\node[red,left] at (0,0) {$y$}; |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
250 |
\draw (0,0) \vertex |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
251 |
arc (120:105:\rr) node[red,above] {$a$} |
503
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
252 |
arc (105:90:\rr) node[red,above] {$x$} \namedvertex{x1}; |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
253 |
% arc (90:75:\rr) node[red,above] {$x \times I$}; |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
254 |
% arc (75:60:\rr) \vertex node[red,right] {$x$} |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
255 |
% arc (-60:-90:\rr) node[red,below] {$a$} |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
256 |
% arc (-90:-120:\rr); |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
257 |
\draw (0,0) |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
258 |
arc (-120:-90:\rr) node[red,below] {$a$} |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
259 |
arc (-90:-61:\rr) \namedvertex{x2} node[red,right] {$x$}; |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
260 |
\draw (x1) -- node[red, above=3pt] {$x \times I$} (x2); |
498
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
261 |
\begin{scope} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
262 |
\path[clip] (0,0) arc (-120:-60:\rr) arc (60:120:\rr); |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
263 |
\foreach \x in {0,0.48,...,9} { |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
264 |
\draw[green!50!brown] (\x/4,1) -- (\x,-1); |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
265 |
} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
266 |
\end{scope} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
267 |
\end{tikzpicture} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
268 |
}; |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
269 |
|
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
270 |
\node(C) at (4,-4) { |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
271 |
\begin{tikzpicture}[y=-1cm] |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
272 |
\node[red,left] at (0,0) {$y$}; |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
273 |
\draw (0,0) \vertex |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
274 |
arc (120:105:\rr) node[red,below] {$a$} |
503
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
275 |
arc (105:90:\rr) node[red,below] {$x$} \namedvertex{x1}; |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
276 |
% arc (90:75:\rr) node[red,below] {$x \times I$} |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
277 |
% arc (75:60:\rr) \vertex node[red,right] {$x$} |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
278 |
% arc (-60:-90:\rr) node[red,above] {$a$} |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
279 |
% arc (-90:-120:\rr); |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
280 |
\draw (0,0) |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
281 |
arc (-120:-90:\rr) node[red,above] {$a$} |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
282 |
arc (-90:-61:\rr) \namedvertex{x2} node[red,right] {$x$}; |
c221d8331f30
straight lines in figure 42
Scott Morrison <scott@tqft.net>
parents:
498
diff
changeset
|
283 |
\draw (x1) -- node[red, below=3pt] {$x \times I$} (x2); |
498
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
284 |
\begin{scope} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
285 |
\path[clip] (0,0) arc (-120:-60:\rr) arc (60:120:\rr); |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
286 |
\foreach \x in {0,0.48,...,9} { |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
287 |
\draw[green!50!brown] (\x/4,1) -- (\x,-1); |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
288 |
} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
289 |
\end{scope} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
290 |
\end{tikzpicture} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
291 |
}; |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
292 |
|
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
293 |
\draw[->] (A) -- (B); |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
294 |
\draw[->] (A) -- (C); |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
295 |
\end{tikzpicture} |
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
296 |
\caption{Producing weak identities from half pinched products} |
126 | 297 |
\label{fzo2} |
298 |
\end{figure} |
|
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
299 |
As suggested by the figure, these are two different reparameterizations |
934 | 300 |
of a half-pinched version of $a\times I$ |
301 |
(i.e.\ two different homeomorphisms from the half-pinched $I\times I$ to the standard bigon). |
|
125 | 302 |
We must show that the two compositions of these two maps give the identity 2-morphisms |
303 |
on $a$ and $a\bullet \id_x$, as defined above. |
|
304 |
Figure \ref{fzo3} shows one case. |
|
126 | 305 |
\begin{figure}[t] |
530
b236746e8e4d
futzing with figures (\begin{center|equation} to \centering)
Kevin Walker <kevin@canyon23.net>
parents:
529
diff
changeset
|
306 |
\centering |
508 | 307 |
\begin{tikzpicture} |
308 |
||
309 |
\newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
|
310 |
\newcommand{\nsep}{1.8} |
|
311 |
||
312 |
\node(A) at (0,0) { |
|
313 |
\begin{tikzpicture} |
|
314 |
||
315 |
\draw (0,0) coordinate (p1); |
|
316 |
\draw (3.6,0) coordinate (p2); |
|
317 |
\draw (2.3,1) coordinate (p3); |
|
318 |
\draw (2.3,-1) coordinate (p4); |
|
319 |
||
320 |
\begin{scope} |
|
321 |
\clip (p1) .. controls +(.5,-.5) and +(-.8,0) .. (p4) -- |
|
322 |
(p2) -- (p3) .. controls +(-.8,0) and +(.5,.5) .. (p1); |
|
323 |
\foreach \x in {0,0.26,...,4} { |
|
324 |
\draw[green!50!brown] (\x,0) -- (intersection cs: first line={(p2)--(p3)}, second line={(0,0)--(0,1)}); |
|
325 |
\draw[green!50!brown] (\x,0) -- (intersection cs: first line={(p2)--(p4)}, second line={(0,0)--(0,1)}); |
|
326 |
} |
|
327 |
\end{scope} |
|
328 |
||
329 |
\draw (p1) .. controls ($(p1) + (.5,.5)$) and ($(p3) + (-.8,0)$) .. node[red, above=3pt] {$a$} (p3); |
|
330 |
\draw (p1) .. controls ($(p1) + (.5,-.5)$) and ($(p4) + (-.8,0)$) .. node[red, below=3pt] {$a$} (p4); |
|
331 |
\draw (p3) -- node[red, above=7pt, right=1pt] {$x \times I$} (p2); |
|
332 |
\draw (p4) -- node[red, below=7pt, right=1pt] {$x \times I$} (p2); |
|
333 |
\draw (p1) -- (p2); |
|
334 |
||
335 |
\draw (p1) \vertex; |
|
336 |
\draw (p2) \vertex; |
|
337 |
\draw (p3) \vertex; |
|
338 |
\draw (p4) \vertex; |
|
339 |
||
340 |
\end{tikzpicture} |
|
341 |
}; |
|
342 |
||
343 |
\node[outer sep=\nsep](B) at (5.5,0) { |
|
344 |
\begin{tikzpicture} |
|
345 |
||
346 |
\draw (0,0) coordinate (p1); |
|
347 |
\draw (3.6,0) coordinate (p2); |
|
348 |
\draw (2.3,1) coordinate (p3); |
|
349 |
\draw (2.3,-1) coordinate (p4); |
|
350 |
\draw (4.6,0) coordinate (p2b); |
|
351 |
||
352 |
\begin{scope} |
|
353 |
\clip (p1) .. controls +(.5,-.5) and +(-.8,0) .. (p4) -- |
|
354 |
(p2) -- (p3) .. controls +(-.8,0) and +(.5,.5) .. (p1); |
|
355 |
\foreach \x in {0,0.26,...,4} { |
|
356 |
\draw[green!50!brown] (\x,0) -- (intersection cs: first line={(p2)--(p3)}, second line={(0,0)--(0,1)}); |
|
357 |
\draw[green!50!brown] (\x,0) -- (intersection cs: first line={(p2)--(p4)}, second line={(0,0)--(0,1)}); |
|
358 |
} |
|
359 |
\end{scope} |
|
360 |
||
361 |
\begin{scope} |
|
362 |
\clip (p3)--(p2)--(p4)--(p2b)--cycle; |
|
363 |
\draw[blue!50!brown, step=.23] ($(p4)+(0,-1)$) grid +(3,3); |
|
364 |
\end{scope} |
|
365 |
||
366 |
\draw (p1) .. controls ($(p1) + (.5,.5)$) and ($(p3) + (-.8,0)$) .. node[red, above=3pt] {$a$} (p3); |
|
367 |
\draw (p1) .. controls ($(p1) + (.5,-.5)$) and ($(p4) + (-.8,0)$) .. node[red, below=3pt] {$a$} (p4); |
|
368 |
\draw (p3) -- (p2); |
|
369 |
\draw (p4) -- (p2); |
|
370 |
\draw (p3) -- node[red, above=7pt, right=1pt] {$x \times I$} (p2b); |
|
371 |
\draw (p4) -- node[red, below=7pt, right=1pt] {$x \times I$} (p2b); |
|
372 |
||
373 |
\draw (p1) \vertex; |
|
374 |
\draw (p2) \vertex; |
|
375 |
\draw (p3) \vertex; |
|
376 |
\draw (p4) \vertex; |
|
377 |
\draw (p2b) \vertex; |
|
378 |
||
379 |
\end{tikzpicture} |
|
380 |
}; |
|
381 |
||
382 |
\node[outer sep=\nsep](C) at (11,0) { |
|
383 |
\begin{tikzpicture} |
|
384 |
||
385 |
\draw (0,0) coordinate (p1); |
|
386 |
\draw (2.3,0) coordinate (p2); |
|
387 |
\draw (2.3,1) coordinate (p3); |
|
388 |
\draw (2.3,-1) coordinate (p4); |
|
389 |
\draw (3.6,0) coordinate (p2b); |
|
390 |
||
391 |
\begin{scope} |
|
392 |
\clip (p1) .. controls +(.5,-.5) and +(-.8,0) .. (p4) -- |
|
393 |
(p2) -- (p3) .. controls +(-.8,0) and +(.5,.5) .. (p1); |
|
394 |
\foreach \x in {0,0.26,...,4} { |
|
395 |
\draw[green!50!brown] (\x,-1) -- (\x,1); |
|
396 |
} |
|
397 |
\end{scope} |
|
398 |
||
399 |
\begin{scope} |
|
400 |
\clip (p3)--(p2)--(p4)--(p2b)--cycle; |
|
401 |
\draw[blue!50!brown, step=.23] ($(p4)+(0,-1)$) grid +(3,3); |
|
402 |
\end{scope} |
|
403 |
||
404 |
\draw (p1) .. controls ($(p1) + (.5,.5)$) and ($(p3) + (-.8,0)$) .. node[red, above=3pt] {$a$} (p3); |
|
405 |
\draw (p1) .. controls ($(p1) + (.5,-.5)$) and ($(p4) + (-.8,0)$) .. node[red, below=3pt] {$a$} (p4); |
|
406 |
\draw[green!50!brown] (p3) -- (p4); |
|
407 |
\draw (p3) -- node[red, above=7pt, right=1pt] {$x \times I$} (p2b); |
|
408 |
\draw (p4) -- node[red, below=7pt, right=1pt] {$x \times I$} (p2b); |
|
409 |
||
410 |
\draw (p1) \vertex; |
|
411 |
\draw (p3) \vertex; |
|
412 |
\draw (p4) \vertex; |
|
413 |
\draw (p2b) \vertex; |
|
414 |
||
415 |
\end{tikzpicture} |
|
416 |
}; |
|
417 |
||
418 |
\draw[->, thick, blue!50!green] (A) -- (B); |
|
419 |
\draw[->, thick, blue!50!green] (B) -- node[black, above] {$=$} (C); |
|
420 |
||
421 |
\end{tikzpicture} |
|
498
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
422 |
\caption{Composition of weak identities, 1} |
126 | 423 |
\label{fzo3} |
424 |
\end{figure} |
|
457
54328be726e7
comparing_defs.tex 2-cat section
Kevin Walker <kevin@canyon23.net>
parents:
451
diff
changeset
|
425 |
In the first step we have inserted a copy of $(x\times I)\times I$. |
125 | 426 |
Figure \ref{fzo4} shows the other case. |
126 | 427 |
\begin{figure}[t] |
530
b236746e8e4d
futzing with figures (\begin{center|equation} to \centering)
Kevin Walker <kevin@canyon23.net>
parents:
529
diff
changeset
|
428 |
\centering |
509 | 429 |
\begin{tikzpicture} |
430 |
||
431 |
\newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
|
432 |
\newcommand{\nsep}{1.8} |
|
433 |
||
510 | 434 |
\clip (-4,-1.25)--(12,-1.25)--(16,1.25)--(-1,1.25)--cycle; |
435 |
||
436 |
||
509 | 437 |
\node[outer sep=\nsep](A) at (0,0) { |
438 |
\begin{tikzpicture} |
|
439 |
\draw (0,0) coordinate (p1); |
|
440 |
\draw (4,0) coordinate (p2); |
|
441 |
\draw (2.4,0) coordinate (p2a); |
|
442 |
\draw (2,1.2) coordinate (pu); |
|
443 |
\draw (2,-1.2) coordinate (pd); |
|
444 |
||
445 |
\begin{scope} |
|
446 |
\clip (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
447 |
\foreach \t in {0,.065,...,1} { |
|
448 |
\draw[green!50!brown] ($(p1)!\t!(p2a)$) -- +(90 - \t*90 + \t*6 : 4); |
|
449 |
\draw[green!50!brown] ($(p1)!\t!(p2a)$) -- +(-90 + \t*90 - \t*6 : 4); |
|
450 |
} |
|
451 |
\draw[dashed] ($(p2a) + (-.6,3)$)--(p2a)--($(p2a) + (-.6,-3)$); |
|
452 |
\end{scope} |
|
453 |
||
454 |
\draw (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
455 |
\draw (p1)--(p2); |
|
456 |
||
457 |
\draw (p1) \vertex; |
|
458 |
\draw (p2) \vertex; |
|
459 |
\draw (p2a) \vertex; |
|
460 |
\end{tikzpicture} |
|
461 |
}; |
|
462 |
||
463 |
\node[outer sep=\nsep](B) at (5,0) { |
|
464 |
\begin{tikzpicture} |
|
465 |
\draw (0,0) coordinate (p1); |
|
466 |
\draw (4,0) coordinate (p2); |
|
467 |
\draw (2.4,0) coordinate (p2a); |
|
468 |
\draw (2,1.2) coordinate (pu); |
|
469 |
\draw (2,-1.2) coordinate (pd); |
|
470 |
||
471 |
\begin{scope} |
|
472 |
\clip (-.1,3)--($(p2a) + (-.6,3)$)--(p2a)--($(p2a) + (-.6,-3)$)--(-.1,-3)--cycle; |
|
473 |
||
474 |
\begin{scope} |
|
475 |
\clip (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
476 |
\foreach \t in {0,.065,...,1} { |
|
477 |
\draw[green!50!brown] ($(p1)!\t!(p2a)$) -- +(90 - \t*90 + \t*6 : 4); |
|
478 |
\draw[green!50!brown] ($(p1)!\t!(p2a)$) -- +(-90 + \t*90 - \t*6 : 4); |
|
479 |
} |
|
480 |
\draw ($(p2a) + (-.6,3)$)--(p2a)--($(p2a) + (-.6,-3)$); |
|
481 |
\end{scope} |
|
482 |
||
483 |
||
484 |
\draw (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
485 |
%\draw (p1)--(p2); |
|
486 |
\end{scope} |
|
487 |
||
488 |
\begin{scope} |
|
489 |
\clip (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
490 |
\draw ($(p2a) + (-.6,3)$)--(p2a)--($(p2a) + (-.6,-3)$); |
|
491 |
\end{scope} |
|
492 |
||
493 |
\draw (p1) \vertex; |
|
494 |
\draw (p2a) \vertex; |
|
495 |
\end{tikzpicture} |
|
496 |
}; |
|
497 |
||
498 |
\node[outer sep=\nsep](C) at (9,0) { |
|
499 |
\begin{tikzpicture} |
|
500 |
\draw (0,0) coordinate (p1); |
|
501 |
\draw (4,0) coordinate (p2); |
|
502 |
\draw (2,1.2) coordinate (pu); |
|
503 |
\draw (2,-1.2) coordinate (pd); |
|
504 |
||
505 |
\begin{scope} |
|
506 |
\clip (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
507 |
\foreach \t in {0,.045,...,1} { |
|
508 |
\draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); |
|
509 |
} |
|
510 |
\end{scope} |
|
511 |
||
512 |
\draw (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
|
513 |
||
514 |
\draw (p1) \vertex; |
|
515 |
\draw (p2) \vertex; |
|
516 |
\end{tikzpicture} |
|
517 |
}; |
|
518 |
||
519 |
\draw[->, thick, blue!50!green] (A) -- (B); |
|
520 |
\draw[->, thick, blue!50!green] ($(B) + (1,0)$) -- node[black, above] {$=$} (C); |
|
521 |
||
522 |
\end{tikzpicture} |
|
498
b98790f0282e
diagram for producing weak identities
Scott Morrison <scott@tqft.net>
parents:
481
diff
changeset
|
523 |
\caption{Composition of weak identities, 2} |
126 | 524 |
\label{fzo4} |
525 |
\end{figure} |
|
934 | 526 |
We notice that a certain subset of the disk is a product region and remove it. |
124 | 527 |
|
934 | 528 |
Given 2-morphisms $f$ and $g$, we define the horizontal composition $f *_h g$ to be any of the four |
529 |
equal 2-morphisms in Figure \ref{fzo5}. |
|
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
530 |
Figure \ref{fig:horizontal-compositions-equal}illustrates part of the proof that these four 2-morphisms are equal. |
934 | 531 |
Similar arguments show that horizontal composition is associative. |
127 | 532 |
\begin{figure}[t] |
941 | 533 |
\begin{align*} |
510 | 534 |
\raisebox{-.9cm}{ |
535 |
\begin{tikzpicture} |
|
536 |
\draw (0,0) .. controls +(1,.8) and +(-1,.8) .. node[above] {$b$} (2.9,0) |
|
537 |
.. controls +(-1,-.8) and +(1,-.8) .. node[below] {$a$} (0,0); |
|
538 |
\draw[->, thick, orange!50!brown] (1.45,-.4)-- node[left, black] {$f$} +(0,.8); |
|
539 |
\end{tikzpicture}} |
|
540 |
\;\;\;*_h\;\; |
|
541 |
\raisebox{-.9cm}{ |
|
542 |
\begin{tikzpicture} |
|
543 |
\draw (0,0) .. controls +(1,.8) and +(-1,.8) .. node[above] {$d$} (2.9,0) |
|
544 |
.. controls +(-1,-.8) and +(1,-.8) .. node[below] {$c$} (0,0); |
|
545 |
\draw[->, thick, orange!50!brown] (1.45,-.4)-- node[left, black] {$g$} +(0,.8); |
|
546 |
\end{tikzpicture}} |
|
941 | 547 |
\;&=\; |
510 | 548 |
\raisebox{-1.9cm}{ |
549 |
\begin{tikzpicture} |
|
550 |
\draw (0,0) coordinate (p1); |
|
551 |
\draw (5.8,0) coordinate (p2); |
|
552 |
\draw (2.9,.3) coordinate (pu); |
|
553 |
\draw (2.9,-.3) coordinate (pd); |
|
554 |
\begin{scope} |
|
555 |
\clip (p1) .. controls +(.6,.3) and +(-.5,0) .. (pu) |
|
556 |
.. controls +(.5,0) and +(-.6,.3) .. (p2) |
|
557 |
.. controls +(-.6,-.3) and +(.5,0) .. (pd) |
|
558 |
.. controls +(-.5,0) and +(.6,-.3) .. (p1); |
|
559 |
\foreach \t in {0,.03,...,1} { |
|
560 |
\draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); |
|
561 |
} |
|
562 |
\end{scope} |
|
563 |
\draw (p1) .. controls +(.6,.3) and +(-.5,0) .. (pu) |
|
564 |
.. controls +(.5,0) and +(-.6,.3) .. (p2) |
|
565 |
.. controls +(-.6,-.3) and +(.5,0) .. (pd) |
|
566 |
.. controls +(-.5,0) and +(.6,-.3) .. (p1); |
|
567 |
\draw (p1) .. controls +(1,-2) and +(-1,-1) .. (pd); |
|
568 |
\draw (p2) .. controls +(-1,2) and +(1,1) .. (pu); |
|
569 |
\draw[->, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7); |
|
570 |
\draw[->, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7); |
|
571 |
\draw[->, thick, blue!75!yellow] (1.5,.78) node[black, above] {$(b\cdot c)\times I$} -- (2.5,0); |
|
941 | 572 |
\end{tikzpicture}} \\ |
573 |
\;&=\; |
|
574 |
\raisebox{-2.1cm}{ |
|
575 |
\begin{tikzpicture} |
|
576 |
\draw (0,0) coordinate (p1); |
|
577 |
\draw (5.8,0) coordinate (p2); |
|
578 |
\draw (2.9,0) coordinate (pu); |
|
579 |
\draw (2.9,-.9) coordinate (pd); |
|
580 |
\begin{scope} |
|
581 |
\clip (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu) |
|
582 |
.. controls +(.5,-.3) and +(-.6,-.3) .. (p2) |
|
583 |
.. controls +(-.6,-.9) and +(.5,0) .. (pd) |
|
584 |
.. controls +(-.5,0) and +(.6,-.9) .. (p1); |
|
585 |
\foreach \t in {0,.03,...,1} { |
|
586 |
\draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); |
|
587 |
} |
|
588 |
\end{scope} |
|
589 |
\draw (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu) |
|
590 |
.. controls +(.5,-.3) and +(-.6,-.3) .. (p2) |
|
591 |
.. controls +(-.6,-.9) and +(.5,0) .. (pd) |
|
592 |
.. controls +(-.5,0) and +(.6,-.9) .. (p1); |
|
593 |
\draw (p1) .. controls +(1,1) and +(-1,1) .. (pu); |
|
594 |
\draw (p2) .. controls +(-1,1) and +(1,1) .. (pu); |
|
595 |
\draw[->, thick, orange!50!brown] (1.45,-0.1)-- node[left, black] {$f$} +(0,.7); |
|
596 |
\draw[->, thick, orange!50!brown] (4.35,-0.1)-- node[left, black] {$g$} +(0,.7); |
|
597 |
\draw[->, thick, blue!75!yellow] (4.3,-1.5) node[black, below] {$(a\cdot c)\times I$} -- (3.3,-0.5); |
|
598 |
\end{tikzpicture}} \\ |
|
599 |
\;&=\; |
|
600 |
\raisebox{-1.9cm}{ |
|
601 |
\begin{tikzpicture}[y=-1cm] |
|
602 |
\draw (0,0) coordinate (p1); |
|
603 |
\draw (5.8,0) coordinate (p2); |
|
604 |
\draw (2.9,.3) coordinate (pu); |
|
605 |
\draw (2.9,-.3) coordinate (pd); |
|
606 |
\begin{scope} |
|
607 |
\clip (p1) .. controls +(.6,.3) and +(-.5,0) .. (pu) |
|
608 |
.. controls +(.5,0) and +(-.6,.3) .. (p2) |
|
609 |
.. controls +(-.6,-.3) and +(.5,0) .. (pd) |
|
610 |
.. controls +(-.5,0) and +(.6,-.3) .. (p1); |
|
611 |
\foreach \t in {0,.03,...,1} { |
|
612 |
\draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); |
|
613 |
} |
|
614 |
\end{scope} |
|
615 |
\draw (p1) .. controls +(.6,.3) and +(-.5,0) .. (pu) |
|
616 |
.. controls +(.5,0) and +(-.6,.3) .. (p2) |
|
617 |
.. controls +(-.6,-.3) and +(.5,0) .. (pd) |
|
618 |
.. controls +(-.5,0) and +(.6,-.3) .. (p1); |
|
619 |
\draw (p1) .. controls +(1,-2) and +(-1,-1) .. (pd); |
|
620 |
\draw (p2) .. controls +(-1,2) and +(1,1) .. (pu); |
|
621 |
\draw[<-, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7); |
|
622 |
\draw[<-, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7); |
|
623 |
\draw[->, thick, blue!75!yellow] (1.5,.78) node[black, below] {$(a\cdot d)\times I$} -- (2.5,0); |
|
624 |
\end{tikzpicture}} \\ |
|
625 |
\;&=\; |
|
626 |
\raisebox{-1.0cm}{ |
|
627 |
\begin{tikzpicture}[y=-1cm] |
|
628 |
\draw (0,0) coordinate (p1); |
|
629 |
\draw (5.8,0) coordinate (p2); |
|
630 |
\draw (2.9,0) coordinate (pu); |
|
631 |
\draw (2.9,-.9) coordinate (pd); |
|
632 |
\begin{scope} |
|
633 |
\clip (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu) |
|
634 |
.. controls +(.5,-.3) and +(-.6,-.3) .. (p2) |
|
635 |
.. controls +(-.6,-.9) and +(.5,0) .. (pd) |
|
636 |
.. controls +(-.5,0) and +(.6,-.9) .. (p1); |
|
637 |
\foreach \t in {0,.03,...,1} { |
|
638 |
\draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); |
|
639 |
} |
|
640 |
\end{scope} |
|
641 |
\draw (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu) |
|
642 |
.. controls +(.5,-.3) and +(-.6,-.3) .. (p2) |
|
643 |
.. controls +(-.6,-.9) and +(.5,0) .. (pd) |
|
644 |
.. controls +(-.5,0) and +(.6,-.9) .. (p1); |
|
645 |
\draw (p1) .. controls +(1,1) and +(-1,1) .. (pu); |
|
646 |
\draw (p2) .. controls +(-1,1) and +(1,1) .. (pu); |
|
647 |
\draw[<-, thick, orange!50!brown] (1.45,-0.1)-- node[left, black] {$f$} +(0,.7); |
|
648 |
\draw[<-, thick, orange!50!brown] (4.35,-0.1)-- node[left, black] {$g$} +(0,.7); |
|
649 |
\draw[->, thick, blue!75!yellow] (4.3,-1.5) node[black, above] {$(b\cdot d)\times I$} -- (3.3,-0.5); |
|
650 |
\end{tikzpicture}} |
|
651 |
\end{align*} |
|
127 | 652 |
\caption{Horizontal composition of 2-morphisms} |
653 |
\label{fzo5} |
|
654 |
\end{figure} |
|
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
655 |
\begin{figure}[t] |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
656 |
$$ |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
657 |
\mathfig{0.6}{triangle/triangle3c} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
658 |
$$ |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
659 |
\caption{Part of the proof that the four different horizontal compositions of 2-morphisms are equal.} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
660 |
\label{fig:horizontal-compositions-equal} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
661 |
\end{figure} |
125 | 662 |
|
934 | 663 |
Given 1-morphisms $a$, $b$ and $c$ of $D$, we define the associator from $(a\bullet b)\bullet c$ to $a\bullet(b\bullet c)$ |
664 |
as in Figure \nn{like triangle 4.a, but more general; use three colors as in that fig}. |
|
665 |
This is just a reparameterization of the pinched product $(a\bullet b\bullet c)\times I$ of $\cC$. |
|
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
666 |
\begin{figure}[t] |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
667 |
$$ |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
668 |
\mathfig{0.4}{triangle/triangle4a} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
669 |
$$ |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
670 |
\nn{remember to change the labels} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
671 |
\caption{An associator.} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
672 |
\label{fig:associator} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
673 |
\end{figure} |
934 | 674 |
|
935
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
675 |
Let $x,y,z$ be objects of $D$ and let $a:x\to y$ and $b:y\to z$ be 1-morphisms of $D$. |
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
676 |
We have already defined above |
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
677 |
structure maps $u:a\bullet \id_y\to a$ and $v:\id_y\bullet b\to b$, as well as an associator |
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
678 |
$\alpha: (a\bullet \id_y)\bullet b\to a\bullet(\id_y\bullet b)$, as shown in |
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
679 |
Figure \ref{fig:ingredients-triangle-axiom}. |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
680 |
(See also Figures \ref{fzo2} and \ref{fig:associator}.) |
935
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
681 |
We now show that $D$ satisfies the triangle axiom, which states that $u\bullet\id_b$ |
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
682 |
is equal to the composition of $\alpha$ and $\id_a\bullet v$. |
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
683 |
(Both are 2-morphisms from $(a\bullet \id_y)\bullet b$ to $a\bullet b$.) |
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
684 |
\begin{figure}[t] |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
685 |
\begin{align*} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
686 |
\mathfig{0.4}{triangle/triangle4a} \\ |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
687 |
\mathfig{0.4}{triangle/triangle4b} \\ |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
688 |
\mathfig{0.4}{triangle/triangle4c} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
689 |
\end{align*} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
690 |
\nn{remember to change `assoc' to $\alpha$} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
691 |
\caption{Ingredients for the triangle axiom.} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
692 |
\label{fig:ingredients-triangle-axiom} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
693 |
\end{figure} |
935
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
694 |
|
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
695 |
The horizontal compositions $u\bullet\id_b$ and $\id_a\bullet v$ are shown in Figure \ref{fig:horizontal-composition} |
935
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
696 |
(see also Figure \ref{fzo5}). |
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
697 |
The vertical composition of $\alpha$ and $\id_a\bullet v$ is shown in Figure \ref{fig:vertical-composition}. |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
698 |
Figure \ref{fig:adding-a-collar} shows that we can add a collar to $u\bullet\id_b$ so that the result differs from |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
699 |
Figure \ref{fig:vertical-composition} by an isotopy rel boundary. |
935
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
700 |
Note that here we have used in an essential way the associativity of product morphisms (Axiom \ref{axiom:product}.3) |
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
701 |
as well as compatibility of product morphisms with fiber-preserving maps (Axiom \ref{axiom:product}.1). |
936
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
702 |
\begin{figure}[t] |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
703 |
\begin{align*} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
704 |
\mathfig{0.4}{triangle/triangle4d} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
705 |
\mathfig{0.4}{triangle/triangle4e} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
706 |
\end{align*} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
707 |
\caption{Horizontal compositions in the triangle axiom.} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
708 |
\label{fig:horizontal-composition} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
709 |
\end{figure} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
710 |
\begin{figure}[t] |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
711 |
\begin{align*} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
712 |
\mathfig{0.4}{triangle/triangle4f} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
713 |
\end{align*} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
714 |
\caption{Vertical composition in the triangle axiom.} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
715 |
\label{fig:vertical-composition} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
716 |
\end{figure} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
717 |
\begin{figure}[t] |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
718 |
\begin{align*} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
719 |
\mathfig{0.4}{triangle/triangle5} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
720 |
\end{align*} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
721 |
\caption{Adding a collar in the proof of the triangle axiom.} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
722 |
\label{fig:adding-a-collar} |
d73a88d78498
chopping up hand-drawn figures, and getting all the figure environments and references right
Scott Morrison <scott@tqft.net>
parents:
935
diff
changeset
|
723 |
\end{figure} |
935
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
724 |
|
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
725 |
|
d85867a99545
done with appendic revision (except for figures)
Kevin Walker <kevin@canyon23.net>
parents:
934
diff
changeset
|
726 |
|
512
050dba5e7bdd
fixing some (but not all!?) of the hyperref warnings; start on revision of evmap
Kevin Walker <kevin@canyon23.net>
parents:
510
diff
changeset
|
727 |
\subsection{\texorpdfstring{$A_\infty$}{A-infinity} 1-categories} |
194 | 728 |
\label{sec:comparing-A-infty} |
431 | 729 |
In this section, we make contact between the usual definition of an $A_\infty$ category |
888
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
730 |
and our definition of a disk-like $A_\infty$ $1$-category, from \S \ref{ss:n-cat-def}. |
477
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
731 |
|
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
732 |
\medskip |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
733 |
|
888
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
734 |
Given a disk-like $A_\infty$ $1$-category $\cC$, we define an ``$m_k$-style" |
477
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
735 |
$A_\infty$ $1$-category $A$ as follows. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
736 |
The objects of $A$ are $\cC(pt)$. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
737 |
The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$ |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
738 |
($\cC$ applied to the standard interval with boundary labeled by $x$ and $y$). |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
739 |
For simplicity we will now assume there is only one object and suppress it from the notation. |
887
ab0b4827c89c
more referee report stuff, relatively minor
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
740 |
Henceforth $A$ will also denote its unique morphism space. |
477
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
741 |
|
854 | 742 |
A choice of homeomorphism $I\cup I \to I$ induces a chain map $m_2: A\otimes A\to A$. |
477
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
743 |
We now have two different homeomorphisms $I\cup I\cup I \to I$, but they are isotopic. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
744 |
Choose a specific 1-parameter family of homeomorphisms connecting them; this induces |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
745 |
a degree 1 chain homotopy $m_3:A\ot A\ot A\to A$. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
746 |
Proceeding in this way we define the rest of the $m_i$'s. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
747 |
It is straightforward to verify that they satisfy the necessary identities. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
748 |
|
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
749 |
\medskip |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
750 |
|
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
751 |
In the other direction, we start with an alternative conventional definition of an $A_\infty$ algebra: |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
752 |
an algebra $A$ for the $A_\infty$ operad. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
753 |
(For simplicity, we are assuming our $A_\infty$ 1-category has only one object.) |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
754 |
We are free to choose any operad with contractible spaces, so we choose the operad |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
755 |
whose $k$-th space is the space of decompositions of the standard interval $I$ into $k$ |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
756 |
parameterized copies of $I$. |
480 | 757 |
Note in particular that when $k=1$ this implies a $C_*(\Homeo(I))$ action on $A$. |
758 |
(Compare with Example \ref{ex:e-n-alg} and the discussion which precedes it.) |
|
477
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
759 |
Given a non-standard interval $J$, we define $\cC(J)$ to be |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
760 |
$(\Homeo(I\to J) \times A)/\Homeo(I\to I)$, |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
761 |
where $\beta \in \Homeo(I\to I)$ acts via $(f, a) \mapsto (f\circ \beta\inv, \beta_*(a))$. |
480 | 762 |
Note that $\cC(J) \cong A$ (non-canonically) for all intervals $J$. |
477
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
763 |
We define a $\Homeo(J)$ action on $\cC(J)$ via $g_*(f, a) = (g\circ f, a)$. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
764 |
The $C_*(\Homeo(J))$ action is defined similarly. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
765 |
|
887
ab0b4827c89c
more referee report stuff, relatively minor
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
766 |
Let $J_1$ and $J_2$ be intervals, and let $J_1\cup J_2$ denote their union along a single boundary point. |
477
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
767 |
We must define a map $\cC(J_1)\ot\cC(J_2)\to\cC(J_1\cup J_2)$. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
768 |
Choose a homeomorphism $g:I\to J_1\cup J_2$. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
769 |
Let $(f_i, a_i)\in \cC(J_i)$. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
770 |
We have a parameterized decomposition of $I$ into two intervals given by |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
771 |
$g\inv \circ f_i$, $i=1,2$. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
772 |
Corresponding to this decomposition the operad action gives a map $\mu: A\ot A\to A$. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
773 |
Define the gluing map to send $(f_1, a_1)\ot (f_2, a_2)$ to $(g, \mu(a_1\ot a_2))$. |
480 | 774 |
Operad associativity for $A$ implies that this gluing map is independent of the choice of |
775 |
$g$ and the choice of representative $(f_i, a_i)$. |
|
477
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
776 |
|
888
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
777 |
It is straightforward to verify the remaining axioms for a disk-like $A_\infty$ 1-category. |
477
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
778 |
|
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
779 |
|
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
780 |
|
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
781 |
|
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
782 |
|
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
783 |
|
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
784 |
|
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
785 |
\noop { %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
194 | 786 |
|
431 | 787 |
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 | 788 |
\begin{defn} |
345 | 789 |
A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$, |
790 |
and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with |
|
194 | 791 |
\begin{itemize} |
792 |
\item an action of the operad of $\Obj(\cC)$-labeled cell decompositions |
|
793 |
\item and a compatible action of $\CD{[0,1]}$. |
|
794 |
\end{itemize} |
|
795 |
\end{defn} |
|
345 | 796 |
Here the operad of cell decompositions of $[0,1]$ has operations indexed by a finite set of |
797 |
points $0 < x_1< \cdots < x_k < 1$, cutting $[0,1]$ into subintervals. |
|
798 |
An $X$-labeled cell decomposition labels $\{0, x_1, \ldots, x_k, 1\}$ by $X$. |
|
799 |
Given two cell decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$, we can compose |
|
800 |
them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points |
|
801 |
of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$. |
|
802 |
In the $X$-labeled case, we insist that the appropriate labels match up. |
|
803 |
Saying we have an action of this operad means that for each labeled cell decomposition |
|
804 |
$0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain |
|
431 | 805 |
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 | 806 |
chain maps compose exactly as the cell decompositions. |
807 |
An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad |
|
808 |
if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which |
|
809 |
is supported on the subintervals determined by $\pi$, then the two possible operations |
|
810 |
(glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms |
|
811 |
separately to the subintervals, then glue) commute (as usual, up to a weakly unique homotopy). |
|
194 | 812 |
|
345 | 813 |
Translating between this notion and the usual definition of an $A_\infty$ category is now straightforward. |
814 |
To restrict to the standard interval, define $\cC_{a,b} = \cC([0,1];a,b)$. |
|
815 |
Given a cell decomposition $0 < x_1< \cdots < x_k < 1$, we use the map (suppressing labels) |
|
194 | 816 |
$$\cC([0,1])^{\tensor k+1} \to \cC([0,x_1]) \tensor \cdots \tensor \cC[x_k,1] \to \cC([0,1])$$ |
345 | 817 |
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. |
818 |
The action of $\CD{[0,1]}$ carries across, and is automatically compatible. |
|
819 |
Going the other way, we just declare $\cC(J;a,b) = \cC_{a,b}$, pick a diffeomorphism |
|
820 |
$\phi_J : J \isoto [0,1]$ for every interval $J$, define the gluing map |
|
821 |
$\cC(J_1) \tensor \cC(J_2) \to \cC(J_1 \cup J_2)$ by the first applying |
|
822 |
the cell decomposition map for $0 < \frac{1}{2} < 1$, then the self-diffeomorphism of $[0,1]$ |
|
823 |
given by $\frac{1}{2} (\phi_{J_1} \cup (1+ \phi_{J_2})) \circ \phi_{J_1 \cup J_2}^{-1}$. |
|
824 |
You can readily check that this gluing map is associative on the nose. \todo{really?} |
|
194 | 825 |
|
826 |
%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 $\cJ^{(1)} \circ_m \cJ^{(2)}$. |
|
827 |
||
828 |
%\begin{defn} |
|
829 |
%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'. |
|
830 |
||
831 |
%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 |
|
832 |
%\begin{equation*} |
|
833 |
%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}). |
|
834 |
%\end{equation*} |
|
835 |
||
836 |
%An \emph{action of families of diffeomorphisms} is a chain map $ev: \CD{[0,1]} \tensor A \to A$, such that |
|
837 |
%\begin{enumerate} |
|
838 |
%\item The diagram |
|
839 |
%\begin{equation*} |
|
840 |
%\xymatrix{ |
|
841 |
%\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} \\ |
|
842 |
%\CD{[0,1]} \tensor A \ar[r]^{ev} & A |
|
843 |
%} |
|
844 |
%\end{equation*} |
|
845 |
%commutes up to weakly unique homotopy. |
|
846 |
%\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 |
|
847 |
%\begin{equation*} |
|
848 |
%\phi(f_\cJ(a_1, \cdots, a_k)) = f_{\phi(\cJ)}(\phi_1(a_1), \cdots, \phi_k(a_k)). |
|
849 |
%\end{equation*} |
|
850 |
%\end{enumerate} |
|
851 |
%\end{defn} |
|
852 |
||
345 | 853 |
From a topological $A_\infty$ category on $[0,1]$ $\cC$ we can produce a `conventional' |
854 |
$A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}. |
|
855 |
We'll just describe the algebra case (that is, a category with only one object), |
|
856 |
as the modifications required to deal with multiple objects are trivial. |
|
857 |
Define $A = \cC$ as a chain complex (so $m_1 = d$). |
|
858 |
Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$. |
|
859 |
To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms |
|
860 |
of $[0,1]$ that interpolates linearly between the identity and the piecewise linear |
|
861 |
diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define |
|
194 | 862 |
\begin{equation*} |
863 |
m_3(a,b,c) = ev(\phi_3, m_2(m_2(a,b), c)). |
|
864 |
\end{equation*} |
|
865 |
||
866 |
It's then easy to calculate that |
|
867 |
\begin{align*} |
|
868 |
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)) \\ |
|
869 |
& = 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) \\ |
|
870 |
& = 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), \\ |
|
871 |
\intertext{and thus that} |
|
872 |
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) |
|
873 |
\end{align*} |
|
874 |
as required (c.f. \cite[p. 6]{MR1854636}). |
|
875 |
\todo{then the general case.} |
|
345 | 876 |
We won't describe a reverse construction (producing a topological $A_\infty$ category |
477
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
877 |
from a ``conventional" $A_\infty$ category), but we presume that this will be easy for the experts. |
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
878 |
|
86c8e2129355
radically shorter a-inf appendix
Kevin Walker <kevin@canyon23.net>
parents:
457
diff
changeset
|
879 |
} %%%%% end \noop %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |