94
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1 |
%!TEX root = ../blob1.tex
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
2 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
3 |
\def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
4 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
5 |
\section{$n$-categories (maybe)}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
6 |
\label{sec:ncats}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
7 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
8 |
\nn{experimental section. maybe this should be rolled into other sections.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
9 |
maybe it should be split off into a separate paper.}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
10 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
11 |
Before proceeding, we need more appropriate definitions of $n$-categories,
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
12 |
$A_\infty$ $n$-categories, modules for these, and tensor products of these modules.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
13 |
(As is the case throughout this paper, by ``$n$-category" we mean
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
14 |
a weak $n$-category with strong duality.)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
15 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
16 |
Consider first ordinary $n$-categories.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
17 |
We need a set (or sets) of $k$-morphisms for each $0\le k \le n$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
18 |
We must decide on the ``shape" of the $k$-morphisms.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
19 |
Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, ...).
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
20 |
Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$,
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
21 |
a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$,
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
22 |
and so on.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
23 |
(This allows for strict associativity.)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
24 |
Still other definitions \nn{need refs for all these; maybe the Leinster book}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
25 |
model the $k$-morphisms on more complicated combinatorial polyhedra.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
26 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
27 |
We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to a $k$-ball.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
28 |
In other words,
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
29 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
30 |
\xxpar{Morphisms (preliminary version):}{For any $k$-manifold $X$ homeomorphic
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
31 |
to a $k$-ball, we have a set of $k$-morphisms
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
32 |
$\cC(X)$.}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
33 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
34 |
Given a homeomorphism $f:X\to Y$ between such $k$-manifolds, we want a corresponding
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
35 |
bijection of sets $f:\cC(X)\to \cC(Y)$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
36 |
So we replace the above with
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
37 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
38 |
\xxpar{Morphisms:}{For each $0 \le k \le n$, we have a functor $\cC_k$ from
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
39 |
the category of manifolds homeomorphic to the $k$-ball and
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
40 |
homeomorphisms to the category of sets and bijections.}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
41 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
42 |
(Note: We usually omit the subscript $k$.)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
43 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
44 |
We are being deliberately vague about what flavor of manifolds we are considering.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
45 |
They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
46 |
They could be topological or PL or smooth.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
47 |
(If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
48 |
to be fussier about corners.)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
49 |
For each flavor of manifold there is a corresponding flavor of $n$-category.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
50 |
We will concentrate of the case of PL unoriented manifolds.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
51 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
52 |
Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
53 |
of morphisms).
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
54 |
The 0-sphere is unusual among spheres in that it is disconnected.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
55 |
Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
56 |
For $k>1$ and in the presence of strong duality the domain/range division makes less sense.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
57 |
\nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
58 |
We prefer to combine the domain and range into a single entity which we call the
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
59 |
boundary of a morphism.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
60 |
Morphisms are modeled on balls, so their boundaries are modeled on spheres:
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
61 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
62 |
\xxpar{Boundaries (domain and range), part 1:}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
63 |
{For each $0 \le k \le n-1$, we have a functor $\cC_k$ from
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
64 |
the category of manifolds homeomorphic to the $k$-sphere and
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
65 |
homeomorphisms to the category of sets and bijections.}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
66 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
67 |
(In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
68 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
69 |
\xxpar{Boundaries, part 2:}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
70 |
{For each $X$ homeomorphic to a $k$-ball, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
71 |
These maps, for various $X$, comprise a natural transformation of functors.}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
72 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
73 |
(Note that the first ``$\bd$" above is part of the data for the category,
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
74 |
while the second is the ordinary boundary of manifolds.)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
75 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
76 |
Given $c\in\cC(\bd(X))$, let $\cC(X; c) = \bd^{-1}(c)$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
77 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
78 |
Most of the examples of $n$-categories we are interested in are enriched in the following sense.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
79 |
The various sets of $n$-morphisms $\cC(X; c)$, for all $X$ homeomorphic to an $n$-ball and
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
80 |
all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary category
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
81 |
(e.g.\ vector spaces, or modules over some ring, or chain complexes),
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
82 |
and all the structure maps of the $n$-category should be compatible with the auxiliary
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
83 |
category structure.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
84 |
Note that this auxiliary structure is only in dimension $n$;
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
85 |
$\cC(Y; c)$ is just a plain set if $\dim(Y) < n$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
86 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
87 |
\medskip
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
88 |
\nn{At the moment I'm a little confused about orientations, and more specifically
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
89 |
about the role of orientation-reversing maps of boundaries when gluing oriented manifolds.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
90 |
Tentatively, I think we need to redefine the oriented boundary of an oriented $n$-manifold.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
91 |
Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
92 |
first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
93 |
equipped with an orientation of its once-stabilized tangent bundle.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
94 |
Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
95 |
their $k$ times stabilized tangent bundles.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
96 |
For the moment just stick with unoriented manifolds.}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
97 |
\medskip
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
98 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
99 |
We have just argued that the boundary of a morphism has no preferred splitting into
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
100 |
domain and range, but the converse meets with our approval.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
101 |
That is, given compatible domain and range, we should be able to combine them into
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
102 |
the full boundary of a morphism:
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
103 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
104 |
\xxpar{Domain $+$ range $\to$ boundary:}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
105 |
{Let $S = B_1 \cup_E B_2$, where $S$ is homeomorphic to a $k$-sphere ($0\le k\le n-1$),
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
106 |
$B_i$ is homeomorphic to a $k$-ball, and $E = B_1\cap B_2$ is homeomorphic to a $k{-}1$-sphere.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
107 |
Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
108 |
two maps $\bd: \cC(B_i)\to \cC(E)$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
109 |
Then (axiom) we have an injective map
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
110 |
\[
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
111 |
\gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
112 |
\]
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
113 |
which is natural with respect to the actions of homeomorphisms.}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
114 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
115 |
Note that we insist on injectivity above.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
116 |
Let $\cC(S)_E$ denote the image of $\gl_E$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
117 |
We have ``restriction" maps $\cC(S)_E \to \cC(B_i)$, which can be thought of as
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
118 |
domain and range maps, relative to the choice of splitting $S = B_1 \cup_E B_2$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
119 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
120 |
If $B$ is homeomorphic to a $k$-ball and $E \sub \bd B$ splits $\bd B$ into two $k{-}1$-balls
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
121 |
as above, then we define $\cC(B)_E = \bd^{-1}(\cC(\bd B)_E)$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
122 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
123 |
Next we consider composition of morphisms.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
124 |
For $n$-categories which lack strong duality, one usually considers
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
125 |
$k$ different types of composition of $k$-morphisms, each associated to a different direction.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
126 |
(For example, vertical and horizontal composition of 2-morphisms.)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
127 |
In the presence of strong duality, these $k$ distinct compositions are subsumed into
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
128 |
one general type of composition which can be in any ``direction".
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
129 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
130 |
\xxpar{Composition:}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
131 |
{Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are homeomorphic to $k$-balls ($0\le k\le n$)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
132 |
and $Y = B_1\cap B_2$ is homeomorphic to a $k{-}1$-ball.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
133 |
Let $E = \bd Y$, which is homeomorphic to a $k{-}2$-sphere.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
134 |
Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
135 |
We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
136 |
Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
137 |
Then (axiom) we have a map
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
138 |
\[
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
139 |
\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
140 |
\]
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
141 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
142 |
to the intersection of the boundaries of $B$ and $B_i$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
143 |
If $k < n$ we require that $\gl_Y$ is injective.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
144 |
(For $k=n$, see below.)}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
145 |
|
95
|
146 |
\xxpar{Strict associativity:}
|
|
147 |
{The composition (gluing) maps above are strictly associative.
|
|
148 |
It follows that given a decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball
|
|
149 |
into small $k$-balls, there is a well-defined
|
|
150 |
map from an appropriate subset of $\cC(B_1)\times\cdots\times\cC(B_m)$ to $\cC(B)$,
|
|
151 |
and these various $m$-fold composition maps satisfy an
|
|
152 |
operad-type associativity condition.}
|
|
153 |
|
|
154 |
\nn{above maybe needs some work}
|
|
155 |
|
|
156 |
The next axiom is related to identity morphisms, though that might not be immediately obvious.
|
|
157 |
|
|
158 |
\xxpar{Product (identity) morphisms:}
|
|
159 |
{Let $X$ be homeomorphic to a $k$-ball and $D$ be homeomorphic to an $m$-ball, with $k+m \le n$.
|
|
160 |
Then we have a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$.
|
|
161 |
If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram
|
|
162 |
\[ \xymatrix{
|
96
|
163 |
X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\
|
95
|
164 |
X \ar[r]^{f} & X'
|
|
165 |
} \]
|
|
166 |
commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.}
|
|
167 |
|
|
168 |
\nn{Need to say something about compatibility with gluing (of both $X$ and $D$) above.}
|
|
169 |
|
|
170 |
All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories.
|
|
171 |
The last axiom (below), concerning actions of
|
|
172 |
homeomorphisms in the top dimension $n$, distinguishes the two cases.
|
|
173 |
|
|
174 |
We start with the plain $n$-category case.
|
|
175 |
|
|
176 |
\xxpar{Isotopy invariance in dimension $n$ (preliminary version):}
|
|
177 |
{Let $X$ be homeomorphic to the $n$-ball and $f: X\to X$ be a homeomorphism which restricts
|
|
178 |
to the identity on $\bd X$ and is isotopic (rel boundary) to the identity.
|
96
|
179 |
Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$.}
|
|
180 |
|
|
181 |
We will strengthen the above axiom in two ways.
|
|
182 |
(Amusingly, these two ways are related to each of the two senses of the term
|
|
183 |
``pseudo-isotopy".)
|
|
184 |
|
|
185 |
First, we require that $f$ act trivially on $\cC(X)$ if it is pseudo-isotopic to the identity
|
|
186 |
in the sense of homeomorphisms of mapping cylinders.
|
|
187 |
This is motivated by TQFT considerations:
|
|
188 |
If the mapping cylinder of $f$ is homeomorphic to the mapping cylinder of the identity,
|
|
189 |
then these two $n{+}1$-manifolds should induce the same map from $\cC(X)$ to itself.
|
|
190 |
\nn{is there a non-TQFT reason to require this?}
|
94
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
191 |
|
96
|
192 |
Second, we require that product (a.k.a.\ identity) $n$-morphisms act as the identity.
|
|
193 |
Let $X$ be an $n$-ball and $Y\sub\bd X$ be at $n{-}1$-ball.
|
|
194 |
Let $J$ be a 1-ball (interval).
|
|
195 |
We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$.
|
|
196 |
We define a map
|
|
197 |
\begin{eqnarray*}
|
|
198 |
\psi_{Y,J}: \cC(X) &\to& \cC(X) \\
|
|
199 |
a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) .
|
|
200 |
\end{eqnarray*}
|
|
201 |
\nn{need to say something somewhere about pinched boundary convention for products}
|
|
202 |
We will call $\psi_{Y,J}$ an extended isotopy.
|
97
|
203 |
\nn{or extended homeomorphism? see below.}
|
|
204 |
\nn{maybe remark that in some examples (e.g.\ ones based on sub cell complexes)
|
|
205 |
extended isotopies are also plain isotopies, so
|
|
206 |
no extension necessary}
|
96
|
207 |
It can be thought of as the action of the inverse of
|
|
208 |
a map which projects a collar neighborhood of $Y$ onto $Y$.
|
|
209 |
(This sort of collapse map is the other sense of ``pseudo-isotopy".)
|
|
210 |
\nn{need to check this}
|
|
211 |
|
|
212 |
The revised axiom is
|
|
213 |
|
|
214 |
\xxpar{Pseudo and extended isotopy invariance in dimension $n$:}
|
|
215 |
{Let $X$ be homeomorphic to the $n$-ball and $f: X\to X$ be a homeomorphism which restricts
|
|
216 |
to the identity on $\bd X$ and is pseudo-isotopic or extended isotopic (rel boundary) to the identity.
|
|
217 |
Then $f$ acts trivially on $\cC(X)$.}
|
|
218 |
|
|
219 |
\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.}
|
94
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
220 |
|
97
|
221 |
\smallskip
|
|
222 |
|
|
223 |
For $A_\infty$ $n$-categories, we replace
|
|
224 |
isotopy invariance with the requirement that families of homeomorphisms act.
|
|
225 |
For the moment, assume that our $n$-morphisms are enriched over chain complexes.
|
|
226 |
|
|
227 |
\xxpar{Families of homeomorphisms act.}
|
|
228 |
{For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes
|
|
229 |
\[
|
|
230 |
C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) .
|
|
231 |
\]
|
|
232 |
Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$
|
|
233 |
which fix $\bd X$.
|
|
234 |
These action maps are required to be associative up to homotopy
|
|
235 |
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that
|
|
236 |
a diagram like the one in Proposition \ref{CDprop} commutes.
|
|
237 |
\nn{repeat diagram here?}
|
|
238 |
\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?}}
|
|
239 |
|
|
240 |
We should strengthen the above axiom to apply to families of extended homeomorphisms.
|
|
241 |
To do this we need to explain extended homeomorphisms form a space.
|
|
242 |
Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology,
|
|
243 |
and we can replace the class of all intervals $J$ with intervals contained in $\r$.
|
|
244 |
\nn{need to also say something about collaring homeomorphisms.}
|
|
245 |
\nn{this paragraph needs work.}
|
|
246 |
|
|
247 |
Note that if take homology of chain complexes, we turn an $A_\infty$ $n$-category
|
|
248 |
into a plain $n$-category.
|
|
249 |
\nn{say more here?}
|
|
250 |
In the other direction, if we enrich over topological spaces instead of chain complexes,
|
|
251 |
we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting
|
|
252 |
instead of $C_*(\Homeo_\bd(X))$.
|
|
253 |
Taking singular chains converts a space-type $A_\infty$ $n$-category into a chain complex
|
|
254 |
type $A_\infty$ $n$-category.
|
|
255 |
|
|
256 |
|
|
257 |
|
|
258 |
|
|
259 |
|
|
260 |
|
|
261 |
|
95
|
262 |
|
|
263 |
|
|
264 |
|
|
265 |
\medskip
|
|
266 |
|
|
267 |
\hrule
|
|
268 |
|
|
269 |
\medskip
|
|
270 |
|
|
271 |
\nn{to be continued...}
|
|
272 |
|