blob: comparison text/ncat.tex

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 In other words,
 \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.
 We {\it do not} assume that it is equipped with a
 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:}
 {For each $0 \le k \le n$, we have a functor $\cC_k$ from
 the category of $k$-balls and
 (Note: We usually omit the subscript $k$.)
 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.
 We will concentrate of the case of PL unoriented manifolds.
 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries
 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
 boundary of a morphism.
 Morphisms are modeled on balls, so their boundaries are modeled on spheres:
 These maps, for various $X$, comprise a natural transformation of functors.}
 (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
 all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary category
 (e.g.\ vector spaces, or modules over some ring, or chain complexes),
 category structure.
 Note that this auxiliary structure is only in dimension $n$;
 $\cC(Y; c)$ is just a plain set if $\dim(Y) < n$.
 \medskip
-\nn{At the moment I'm a little confused about orientations, and more specifically
+\nn{
-about the role of orientation-reversing maps of boundaries when gluing oriented manifolds.
+%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
 equipped with an orientation of its once-stabilized tangent bundle.
 Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of
 	\gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S)
 \]
 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 $S = B_1 \cup_E B_2$.
 If $B$ is a $k$-ball and $E \sub \bd B$ splits $\bd B$ into two $k{-}1$-balls
 as above, then we define $\cC(B)_E = \bd^{-1}(\cC(\bd B)_E)$.
 (For $k=n$, see below.)}
 \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.
 \xxpar{Multi-composition:}
 {Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball
 If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram
 \[ \xymatrix{
 	X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\
 	X \ar[r]^{f} & X'
 } \]
-commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.}
+commutes, then we have
+\[
-\nn{Need to say something about compatibility with gluing (of both $X$ and $D$) above.}
+	\tilde{f}(a\times D) = f(a)\times D' .
+\]
+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 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
 homeomorphisms in the top dimension $n$, distinguishes the two cases.
 a diagram like the one in Proposition \ref{CDprop} commutes.
 \nn{repeat diagram here?}
 \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.}
 \nn{this paragraph needs work.}
 Thus a system of fields determines an $n$-category simply by restricting our attention to
 balls.
 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.
 The universal (colimit) construction becomes our generalized definition of blob homology.
 Need to explain how it relates to the old definition.}
 \subsection{Modules}
 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.
 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$.
 This will be explained in more detail as we present the axioms.
 Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$),
 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
 $\cC(W, \cN)$ will have additional structure; see below.)
 ** \nn{stuff below needs to be rewritten (shortened), because of new subsections above}
 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 $\bd B$.
 Let $D = (B, N, N')$ be a doubly marked $k$-ball, $1\le k \le n$.

changeset 109	65b291b5e8c8
parent 108	631a082cd21b
child 110	a2444aa1ad31