--- a/text/ncat.tex Wed Jul 28 12:15:58 2010 -0700
+++ b/text/ncat.tex Wed Jul 28 12:16:27 2010 -0700
@@ -10,7 +10,7 @@
\label{ss:n-cat-def}
Before proceeding, we need more appropriate definitions of $n$-categories,
-$A_\infty$ $n$-categories, modules for these, and tensor products of these modules.
+$A_\infty$ $n$-categories, as well as modules for these, and tensor products of these modules.
(As is the case throughout this paper, by ``$n$-category" we mean some notion of
a ``weak" $n$-category with ``strong duality".)
@@ -24,6 +24,8 @@
For examples of a more purely algebraic origin, one would typically need the combinatorial
results that we have avoided here.
+\nn{Say something explicit about Lurie's work here? It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen}
+
\medskip
There are many existing definitions of $n$-categories, with various intended uses.
@@ -58,7 +60,7 @@
\end{axiom}
-(Note: We usually omit the subscript $k$.)
+(Note: We often omit the subscript $k$.)
We are being deliberately vague about what flavor of $k$-balls
we are considering.
@@ -70,14 +72,14 @@
For each flavor of manifold there is a corresponding flavor of $n$-category.
For simplicity, we will concentrate on the case of PL unoriented manifolds.
-(The ambitious reader may want to keep in mind two other classes of balls.
+An ambitious reader may want to keep in mind two other classes of balls.
The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}).
-This will be used below to describe the blob complex of a fiber bundle with
+This will be used below (see the end of \S \ref{ss:product-formula}) to describe the blob complex of a fiber bundle with
base space $Y$.
The second is balls equipped with a section of the tangent bundle, or the frame
-bundle (i.e.\ framed balls), or more generally some flag bundle associated to the tangent bundle.
+bundle (i.e.\ framed balls), or more generally some partial flag bundle associated to the tangent bundle.
These can be used to define categories with less than the ``strong" duality we assume here,
-though we will not develop that idea fully in this paper.)
+though we will not develop that idea fully in this paper.
Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries
of morphisms).
@@ -88,7 +90,7 @@
For $k>1$ and in the presence of strong duality the division into domain and range makes less sense.
For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc.
(sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary.
-We prefer to not make the distinction in the first place.
+We prefer not to make the distinction in the first place.
Instead, we will combine the domain and range into a single entity which we call the
boundary of a morphism.
@@ -118,9 +120,8 @@
These maps, for various $X$, comprise a natural transformation of functors.
\end{axiom}
-(Note that the first ``$\bd$" above is part of the data for the category,
-while the second is the ordinary boundary of manifolds.)
-
+Note that the first ``$\bd$" above is part of the data for the category,
+while the second is the ordinary boundary of manifolds.
Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$.
Most of the examples of $n$-categories we are interested in are enriched in the following sense.
@@ -130,14 +131,14 @@
\nn{actually, need both disj-union/sub and product/tensor-product; what's the name for this sort of cat?}
and all the structure maps of the $n$-category should be compatible with the auxiliary
category structure.
-Note that this auxiliary structure is only in dimension $n$;
-$\cC(Y; c)$ is just a plain set if $\dim(Y) < n$.
+Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then
+$\cC(Y; c)$ is just a plain set.
\medskip
-(In order to simplify the exposition we have concentrated on the case of
+In order to simplify the exposition we have concentrated on the case of
unoriented PL manifolds and avoided the question of what exactly we mean by
-the boundary a manifold with extra structure, such as an oriented manifold.
+the boundary of a manifold with extra structure, such as an oriented manifold.
In general, all manifolds of dimension less than $n$ should be equipped with the germ
of a thickening to dimension $n$, and this germ should carry whatever structure we have
on $n$-manifolds.
@@ -147,7 +148,7 @@
For example, the boundary of an oriented $n$-ball
should be an $n{-}1$-sphere equipped with an orientation of its once stabilized tangent
bundle and a choice of direction in this bundle indicating
-which side the $n$-ball lies on.)
+which side the $n$-ball lies on.
\medskip
@@ -188,7 +189,7 @@
\caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure}
Note that we insist on injectivity above.
-The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}.
+The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. \nn{we might want a more official looking proof...}
Let $\cl{\cC}(S)_E$ denote the image of $\gl_E$.
We will refer to elements of $\cl{\cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$".
@@ -199,7 +200,7 @@
We will call the projection $\cl{\cC}(S)_E \to \cC(B_i)$
a {\it restriction} map and write $\res_{B_i}(a)$
(or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)_E$.
-More generally, we also include under the rubric ``restriction map" the
+More generally, we also include under the rubric ``restriction map"
the boundary maps of Axiom \ref{nca-boundary} above,
another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition
of restriction maps.
@@ -251,15 +252,16 @@
\begin{axiom}[Strict associativity] \label{nca-assoc}
The composition (gluing) maps above are strictly associative.
-Given any splitting of a ball $B$ into smaller balls $B_1,\ldots,B_m$,
-any sequence of gluings of the smaller balls yields the same result.
+Given any splitting of a ball $B$ into smaller balls
+$$\bigsqcup B_i \to B,$$
+any sequence of gluings (in the sense of Definition \ref{defn:gluing-decomposition}, where all the intermediate steps are also disjoint unions of balls) yields the same result.
\end{axiom}
\begin{figure}[!ht]
$$\mathfig{.65}{ncat/strict-associativity}$$
\caption{An example of strict associativity.}\label{blah6}\end{figure}
-We'll use the notations $a\bullet b$ as well as $a \cup b$ for the glued together field $\gl_Y(a, b)$.
+We'll use the notation $a\bullet b$ for the glued together field $\gl_Y(a, b)$.
In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$
a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$.
%Compositions of boundary and restriction maps will also be called restriction maps.
@@ -271,22 +273,22 @@
``splittable along $Y$'' or ``transverse to $Y$''.
We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$.
-More generally, let $\alpha$ be a subdivision of a ball $X$ into smaller balls.
+More generally, let $\alpha$ be a splitting of $X$ into smaller balls.
Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from
the smaller balls to $X$.
We say that elements of $\cC(X)_\alpha$ are morphisms which are ``splittable along $\alpha$".
-In situations where the subdivision is notationally anonymous, we will write
+In situations where the splitting is notationally anonymous, we will write
$\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to)
-the unnamed subdivision.
-If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$;
+the unnamed splitting.
+If $\beta$ is a ball decomposition of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$;
this can also be denoted $\cC(X)\spl$ if the context contains an anonymous
-subdivision of $\bd X$ and no competing subdivision of $X$.
+decomposition of $\bd X$ and no competing splitting of $X$.
The above two composition axioms are equivalent to the following one,
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
+{Given any splitting $B_1 \sqcup \cdots \sqcup B_m \to B$ of a $k$-ball
into small $k$-balls, there is a
map from an appropriate subset (like a fibered product)
of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$,
@@ -384,11 +386,9 @@
(We thank Kevin Costello for suggesting this approach.)
Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball,
-and for for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension
+and for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension
$l \le m$, with $l$ depending on $x$.
-
It is easy to see that a composition of pinched products is again a pinched product.
-
A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction
$\pi:E'\to \pi(E')$ is again a pinched product.
A {union} of pinched products is a decomposition $E = \cup_i E_i$
@@ -523,15 +523,14 @@
\begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$]
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
to the identity on $\bd X$ and is isotopic (rel boundary) to the identity.
-Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$.
+Then $f$ acts trivially on $\cC(X)$; that is $f(a) = a$ for all $a\in \cC(X)$.
\end{axiom}
This axiom needs to be strengthened to force product morphisms to act as the identity.
Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball.
Let $J$ be a 1-ball (interval).
We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$.
-(Here we use the ``pinched" version of $Y\times J$.
-\nn{do we need notation for this?})
+(Here we use $Y\times J$ with boundary entirely pinched.)
We define a map
\begin{eqnarray*}
\psi_{Y,J}: \cC(X) &\to& \cC(X) \\
@@ -625,7 +624,7 @@
%\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?}
\end{axiom}
-We should strengthen the above axiom to apply to families of collar maps.
+We should strengthen the above $A_\infty$ axiom to apply to families of collar maps.
To do this we need to explain how collar maps form a topological space.
Roughly, the set of collared $n{-}1$-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 $\r$.
@@ -652,10 +651,10 @@
invariance in dimension $n$, while in the fields definition we
instead remember a subspace of local relations which contain differences of isotopic fields.
(Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.)
-Thus a system of fields and local relations $(\cF,\cU)$ determines an $n$-category $\cC_ {\cF,\cU}$ simply by restricting our attention to
+Thus a system of fields and local relations $(\cF,U)$ determines an $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to
balls and, at level $n$, quotienting out by the local relations:
\begin{align*}
-\cC_{\cF,\cU}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / \cU(B) & \text{when $k=n$.}\end{cases}
+\cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases}
\end{align*}
This $n$-category can be thought of as the local part of the fields.
Conversely, given a topological $n$-category we can construct a system of fields via
@@ -807,7 +806,7 @@
\end{example}
See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to
-homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$.
+homotopy as the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$.
\begin{example}[Blob complexes of balls (with a fiber)]
\rm
@@ -973,7 +972,7 @@
\begin{defn}
Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows.
-For a decomposition $x = \bigcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset
+For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset
\begin{equation}
\label{eq:psi-C}
\psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl
@@ -996,7 +995,7 @@
$\cS$ and the coproduct and product in Equation \ref{eq:psi-CC} should be replaced by the approriate
operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect).
-Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$.
+Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$:
\begin{defn}[System of fields functor]
\label{def:colim-fields}
@@ -1020,14 +1019,14 @@
In the non-enriched case (e.g.\ $k<n$), where each $\cC(X_a; \beta)$ is just a set,
the colimit is
\[
- \cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) / \sim ,
+ \cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) \Bigg/ \sim ,
\]
where $x$ runs through decomposition of $W$, and $\sim$ is the obvious equivalence relation
induced by refinement and gluing.
If $\cC$ is enriched over vector spaces and $W$ is an $n$-manifold,
we can take
\begin{equation*}
- \cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) / K
+ \cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K,
\end{equation*}
where $K$ is the vector space spanned by elements $a - g(a)$, with
$a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x)
@@ -1041,7 +1040,7 @@
\[
\cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] ,
\]
-where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$.
+where the sum is over all $m$ and all $m$-sequences $(x_i)$, and each summand is degree shifted by $m$.
Elements of a summand indexed by an $m$-sequence will be call $m$-simplices.
We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$
summands plus another term using the differential of the simplicial set of $m$-sequences.
@@ -1095,7 +1094,7 @@
Throughout, we fix an $n$-category $\cC$.
For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category.
-We state the final axiom, on actions of homeomorphisms, differently in the two cases.
+We state the final axiom, regarding actions of homeomorphisms, differently in the two cases.
Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair
$$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$
@@ -1122,7 +1121,7 @@
%the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.)
\begin{figure}[!ht]
-$$\mathfig{.8}{ncat/boundary-collar}$$
+$$\mathfig{.55}{ncat/boundary-collar}$$
\caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure}
Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$.
@@ -1258,12 +1257,11 @@
The above three axioms are equivalent to the following axiom,
which we state in slightly vague form.
-\nn{need figure for this}
\xxpar{Module multi-composition:}
-{Given any decomposition
+{Given any splitting
\[
- M = X_1 \cup\cdots\cup X_p \cup M_1\cup\cdots\cup M_q
+ X_1 \sqcup\cdots\sqcup X_p \sqcup M_1\sqcup\cdots\sqcup M_q \to M
\]
of a marked $k$-ball $M$
into small (marked and plain) $k$-balls $M_i$ and $X_j$, there is a
@@ -1422,7 +1420,7 @@
\begin{example}[Examples from the blob complex] \label{bc-module-example}
\rm
In the previous example, we can instead define
-$\cF(Y)(M)\deq \bc_*^\cF((B\times W) \cup (N\times Y); c)$ (when $\dim(M) = n$)
+$\cF(Y)(M)\deq \bc_*((B\times W) \cup (N\times Y), c; \cF)$ (when $\dim(M) = n$)
and get a module for the $A_\infty$ $n$-category associated to $\cF$ as in
Example \ref{ex:blob-complexes-of-balls}.
\end{example}
@@ -1449,18 +1447,18 @@
let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$,
and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$.
-We will define a set $\cC(W, \cN)$ using a colimit construction similar to
+We will define a set $\cC(W, \cN)$ using a colimit construction very similar to
the one appearing in \S \ref{ss:ncat_fields} above.
(If $k = n$ and our $n$-categories are enriched, then
$\cC(W, \cN)$ will have additional structure; see below.)
-Define a permissible decomposition of $W$ to be a decomposition
+Define a permissible decomposition of $W$ to be a map
\[
- W = \left(\bigcup_a X_a\right) \cup \left(\bigcup_{i,b} M_{ib}\right) ,
+ \left(\bigsqcup_a X_a\right) \sqcup \left(\bigsqcup_{i,b} M_{ib}\right) \to W,
\]
-where each $X_a$ is a plain $k$-ball (disjoint from $\cup Y_i$) and
-each $M_{ib}$ is a marked $k$-ball intersecting $Y_i$,
-with $M_{ib}\cap Y_i$ being the marking.
+where each $X_a$ is a plain $k$-ball disjoint, in $W$, from $\cup Y_i$, and
+each $M_{ib}$ is a marked $k$-ball intersecting $Y_i$ (once mapped into $W$),
+with $M_{ib}\cap Y_i$ being the marking, which extends to a ball decomposition in the sense of Definition \ref{defn:gluing-decomposition}.
(See Figure \ref{mblabel}.)
\begin{figure}[t]
\begin{equation*}