blob: comparison pnas/pnas.tex

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 There are five basic ingredients of an $n$-category definition:
 $k$-morphisms (for $0\le k \le n$), domain and range, composition,
 identity morphisms, and special behavior in dimension $n$ (e.g. enrichment
 in some auxiliary category, or strict associativity instead of weak associativity).
-We will treat each of these it turn.
+We will treat each of these in turn.
 To motivate our morphism axiom, consider the venerable notion of the Moore loop space
-\nn{need citation}.
+\nn{need citation -- \S 2.2 of Adams' ``Infinite Loop Spaces''?}.
 In the standard definition of a loop space, loops are always parameterized by the unit interval $I = [0,1]$,
 so composition of loops requires a reparameterization $I\cup I \cong I$, and this leads to a proliferation
 of higher associativity relations.
 While this proliferation is manageable for 1-categories (and indeed leads to an elegant theory
 of Stasheff polyhedra and $A_\infty$ categories), it becomes undesirably complex for higher categories.
 For each $0 \le k \le n$, we have a functor $\cC_k$ from
 the category of $k$-balls and
 homeomorphisms to the category of sets and bijections.
 \end{axiom}
-Note that the functoriality in the above axiom allows us to operate via
+Note that the functoriality in the above axiom allows us to operate via \nn{fragment?}
 Next we consider domains and ranges of $k$-morphisms.
 Because we assume strong duality, it doesn't make much sense to subdivide the boundary of a morphism
 into domain and range --- the duality operations can convert domain to range and vice-versa.
 Instead, we will use a unified domain/range, which we will call a ``boundary".
 (For $k=n$ in the plain (non-$A_\infty$) case, see below.)
 \end{axiom}
 \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
+Given any decomposition 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.
+any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result.
 \end{axiom}
 \begin{axiom}[Product (identity) morphisms]
 \label{axiom:product}
 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$),
 there is a map $\pi^*:\cC(X)\to \cC(E)$.
 \subsubsection{Decompositions of manifolds}
 \nn{KW: I'm inclined to suppress all discussion of the subtleties of decompositions.
 Maybe just a single remark that we are omitting some details which appear in our
 longer paper.}
+\nn{SM: for now I disagree: the space expense is pretty minor, and it always us to be "in principle" complete. Let's see how we go for length.}
 A \emph{ball decomposition} of $W$ is a
 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls
 $\du_a X_a$ and each $M_i$ is a manifold.
 If $X_a$ is some component of $M_0$, its image in $W$ need not be a ball; $\bd X_a$ may have been glued to itself.
 For $k=2$, we have a two types of generators; they each consists of a field $f$ on $W$, and two balls $B_1$ and $B_2$. In the first case, the balls are disjoint, and $f$ restricted to either of the $B_i$ is a null field. In the second case, the balls are properly nested, say $B_1 \subset B_2$, and $f$ restricted to $B_1$ is null. Note that this implies that $f$ restricted to $B_2$ is also null, by the associativity of the gluing operation. This ensures that the differential is well-defined.
 \section{Properties of the blob complex}
 \subsection{Formal properties}
 \label{sec:properties}
-The blob complex enjoys the following list of formal properties.
+The blob complex enjoys the following list of formal properties. The first three properties are immediate from the definitions.
-The proofs of the first three properties are immediate from the definitions.
 \begin{property}[Functoriality]
 \label{property:functoriality}%
 The blob complex is functorial with respect to homeomorphisms.
 That is,
 obtained by adding an outer $(k{+}1)$-st blob consisting of all $B^n$.
 For $k=0$ we choose a splitting $s: H_0(\bc_*(B^n)) \to \bc_0(B^n)$ and send
 $x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$.
 \end{proof}
-\nn{Properties \ref{property:functoriality} will be immediate from the definition given in
-\S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there.
-Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and
-\ref{property:contractibility} are established in \S \ref{sec:basic-properties}.}
 \subsection{Specializations}
 \label{sec:specializations}
 The blob complex has two important special cases.

changeset 584	7fc1a7ff9667
parent 583	08e80022a881
child 585	e2996d7b4e6c
child 590	6de8871d5786