--- a/text/ncat.tex Wed Jul 29 18:23:18 2009 +0000
+++ b/text/ncat.tex Sat Aug 08 22:12:58 2009 +0000
@@ -33,7 +33,7 @@
\xxpar{Morphisms (preliminary version):}
{For any $k$-manifold $X$ homeomorphic
to the standard $k$-ball, we have a set of $k$-morphisms
-$\cC(X)$.}
+$\cC_k(X)$.}
Terminology: By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the
standard $k$-ball.
@@ -41,8 +41,11 @@
preferred homeomorphism to the standard $k$-ball.
The same goes for ``a $k$-sphere" below.
-Given a homeomorphism $f:X\to Y$ between $k$-balls, we want a corresponding
+
+Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on
+the boundary), we want a corresponding
bijection of sets $f:\cC(X)\to \cC(Y)$.
+(This will imply ``strong duality", among other things.)
So we replace the above with
\xxpar{Morphisms:}
@@ -55,6 +58,7 @@
We are being deliberately vague about what flavor of manifolds we are considering.
They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$.
They could be topological or PL or smooth.
+\nn{need to check whether this makes much difference --- see pseudo-isotopy below}
(If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need
to be fussier about corners.)
For each flavor of manifold there is a corresponding flavor of $n$-category.
@@ -64,7 +68,8 @@
of morphisms).
The 0-sphere is unusual among spheres in that it is disconnected.
Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range.
-(Actually, this is only true in the oriented case.)
+(Actually, this is only true in the oriented case, with 1-morphsims parameterized
+by oriented 1-balls.)
For $k>1$ and in the presence of strong duality the domain/range division makes less sense.
\nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah}
We prefer to combine the domain and range into a single entity which we call the
@@ -85,7 +90,7 @@
(Note that the first ``$\bd$" above is part of the data for the category,
while the second is the ordinary boundary of manifolds.)
-Given $c\in\cC(\bd(X))$, let $\cC(X; c) = \bd^{-1}(c)$.
+Given $c\in\cC(\bd(X))$, let $\cC(X; c) \deq \bd^{-1}(c)$.
Most of the examples of $n$-categories we are interested in are enriched in the following sense.
The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and
@@ -97,8 +102,11 @@
$\cC(Y; c)$ is just a plain set if $\dim(Y) < n$.
\medskip
-\nn{At the moment I'm a little confused about orientations, and more specifically
-about the role of orientation-reversing maps of boundaries when gluing oriented manifolds.
+\nn{
+%At the moment I'm a little confused about orientations, and more specifically
+%about the role of orientation-reversing maps of boundaries when gluing oriented manifolds.
+Maybe need a discussion about what the boundary of a manifold with a
+structure (e.g. orientation) means.
Tentatively, I think we need to redefine the oriented boundary of an oriented $n$-manifold.
Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal
first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold
@@ -125,7 +133,10 @@
which is natural with respect to the actions of homeomorphisms.}
Note that we insist on injectivity above.
+
Let $\cC(S)_E$ denote the image of $\gl_E$.
+We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$".
+
We have ``restriction" maps $\cC(S)_E \to \cC(B_i)$, which can be thought of as
domain and range maps, relative to the choice of splitting $S = B_1 \cup_E B_2$.
@@ -158,6 +169,8 @@
\xxpar{Strict associativity:}
{The composition (gluing) maps above are strictly associative.}
+Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$.
+
The above two axioms are equivalent to the following axiom,
which we state in slightly vague form.
@@ -179,9 +192,29 @@
X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\
X \ar[r]^{f} & X'
} \]
-commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.}
+commutes, then we have
+\[
+ \tilde{f}(a\times D) = f(a)\times D' .
+\]
+Product morphisms are compatible with gluing (composition) in both factors:
+\[
+ (a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D
+\]
+and
+\[
+ (a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') .
+\]
+Product morphisms are associative:
+\[
+ (a\times D)\times D' = a\times (D\times D') .
+\]
+(Here we are implicitly using functoriality and the obvious homeomorphism
+$(X\times D)\times D' \to X\times(D\times D')$.)
+}
-\nn{Need to say something about compatibility with gluing (of both $X$ and $D$) above.}
+\nn{need even more subaxioms for product morphisms?
+YES: need compatibility with certain restriction maps
+in order to prove that dimension less than $n$ identities are act like identities.}
All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories.
The last axiom (below), concerning actions of
@@ -254,7 +287,7 @@
\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?}}
We should strengthen the above axiom to apply to families of extended homeomorphisms.
-To do this we need to explain extended homeomorphisms form a space.
+To do this we need to explain how extended homeomorphisms form a topological space.
Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology,
and we can replace the class of all intervals $J$ with intervals contained in $\r$.
\nn{need to also say something about collaring homeomorphisms.}
@@ -281,6 +314,7 @@
The $n$-category can be thought of as the local part of the fields.
Conversely, given an $n$-category we can construct a system of fields via
\nn{gluing, or a universal construction}
+\nn{see subsection below}
\nn{Next, say something about $A_\infty$ $n$-categories and ``homological" systems
of fields.
@@ -418,6 +452,7 @@
Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations,
a.k.a.\ actions).
The definition will be very similar to that of $n$-categories.
+\nn{** need to make sure all revisions of $n$-cat def are also made to module def.}
Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary
in the context of an $m{+}1$-dimensional TQFT.
@@ -628,6 +663,8 @@
and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary
component $\bd_i W$ of $W$.
+\nn{need to generalize to labeling codim 0 submanifolds of the boundary}
+
We will define a set $\cC(W, \cN)$ using a colimit construction similar to above.
\nn{give ref}
(If $k = n$ and our $k$-categories are enriched, then
@@ -681,9 +718,14 @@
Let $\cM$ and $\cM'$ be modules for an $n$-category $\cC$.
(If $k=1$ and manifolds are oriented, then one should be
a left module and the other a right module.)
-We will define an $n{-}1$-category $\cM\ot_\cC\cM'$, which depend (functorially)
+We will define an $n{-}1$-category $\cM\ot_\cC\cM'$, which depends (functorially)
on a choice of 1-ball (interval) $J$.
+
+
+
+
+
Define a {\it doubly marked $k$-ball} to be a triple $(B, N, N')$, where $B$ is a $k$-ball
and $N$ and $N'$ are disjoint $k{-}1$-balls in $\bd B$.