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 $$
 \caption{From two balls to one ball.}\label{blah5}\end{figure}
 \begin{axiom}[Strict associativity] \label{nca-assoc}
 The composition (gluing) maps above are strictly associative.
+Given any splitting of a ball $B$ into smaller balls $B_1,\ldots,B_m$,
+any sequence of gluings of the smaller balls yields the same result.
 \end{axiom}
-\nn{should say this means $N$ at a time, not just 3 at a time}
 \begin{figure}[!ht]
 $$\mathfig{.65}{ncat/strict-associativity}$$
 \caption{An example of strict associativity.}\label{blah6}\end{figure}
 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.]
 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes
 \[
 	C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) .
 \]
-These action maps are required to be associative up to homotopy
+These action maps are required to be associative up to homotopy,
-\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that
+%\nn{iterated homotopy?}
+and also compatible with composition (gluing) in the sense that
 a diagram like the one in Theorem \ref{thm:CH} commutes.
-\nn{repeat diagram here?}
+%\nn{repeat diagram here?}
-\nn{restate this with $\Homeo(X\to X')$?  what about boundary fixing property?}
+%\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.
 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,
 This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product.
 Notice that with $F$ a point, the above example is a construction turning a topological
 $n$-category $\cC$ into an $A_\infty$ $n$-category.
 We think of this as providing a ``free resolution"
 of the topological $n$-category.
-\nn{say something about cofibrant replacements?}
+%\nn{say something about cofibrant replacements?}
 In fact, there is also a trivial, but mostly uninteresting, way to do this:
 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$,
 and take $\CD{B}$ to act trivially.
 Be careful that the ``free resolution" of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$.
 submanifolds $W$ of $X\times \Real^\infty$ such that
 $W$ coincides with $c$ at $\bd X \times \Real^\infty$.
 (The topology on this space is induced by ambient isotopy rel boundary.
 This is homotopy equivalent to a disjoint union of copies $\mathrm{B}\!\Homeo(W')$, where
 $W'$ runs though representatives of homeomorphism types of such manifolds.)
-\nn{check this}
 \end{example}
 Let $\cE\cB_n$ be the operad of smooth embeddings of $k$ (little)
 copies of the standard $n$-ball $B^n$ into another (big) copy of $B^n$.
 (We require that the interiors of the little balls be disjoint, but their
 boundaries are allowed to meet.
 Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely
 the embeddings of a ``little" ball with image all of the big ball $B^n$.
-\nn{should we warn that the inclusion of this copy of $\Diff(B^n)$ is not a homotopy equivalence?})
+(But note also that this inclusion is not
+necessarily a homotopy equivalence.)
 The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad:
 by shrinking the little balls (precomposing them with dilations),
 we see that both operads are homotopic to the space of $k$ framed points
 in $B^n$.
 It is easy to see that $n$-fold loop spaces $\Omega^n(T)$  have
 an action of $\cE\cB_n$.
-\nn{add citation for this operad if we can find one}
+%\nn{add citation for this operad if we can find one}
 \begin{example}[$E_n$ algebras]
 \rm
 \label{ex:e-n-alg}
 --- composition and $\Diff(X\to X')$ action ---
 also comes from the $\cE\cB_n$ action on $A$.
 \nn{should we spell this out?}
 \nn{Should remark that the associated hocolim for manifolds
-is agrees with Lurie's topological chiral homology construction; maybe wait
+agrees with Lurie's topological chiral homology construction; maybe wait
 until next subsection to say that?}
 Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms
 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to
 an $\cE\cB_n$-algebra.
 In the case of plain $n$-categories, this construction factors into a construction of a
 system of fields and local relations, followed by the usual TQFT definition of a
 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}.
 For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead.
 Recall that we can take a plain $n$-category $\cC$ and pass to the ``free resolution",
-an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls (recall Example \ref{ex:blob-complexes-of-balls} above).
+an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls
+(recall Example \ref{ex:blob-complexes-of-balls} above).
 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant
-for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the same as the original blob complex  for $M$ with coefficients in $\cC$.
+for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the
+same as the original blob complex  for $M$ with coefficients in $\cC$.
-We will first define the ``cell-decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$.
+We will first define the ``decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$.
 An $n$-category $\cC$ provides a functor from this poset to the category of sets,
 and we  will define $\cl{\cC}(W)$ as a suitable colimit
 (or homotopy colimit in the $A_\infty$ case) of this functor.
 We'll later give a more explicit description of this colimit.
 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-balls with boundary data),
 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex).
-Define a {\it permissible decomposition} of $W$ to be a cell decomposition
+Recall (Definition \ref{defn:gluing-decomposition}) that a {\it ball decomposition} of $W$ is a
-\[
+sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls
-	W = \bigcup_a X_a ,
+$\du_a X_a$.
-\]
+Abusing notation, we let $X_a$ denote both the ball (component of $M_0$) and
-where each closed top-dimensional cell $X_a$ is an embedded $k$-ball.
+its image in $W$ (which is not necessarily a ball --- parts of $\bd X_a$ may have been glued together).
-\nn{need to define this more carefully}
+Define a {\it permissible decomposition} of $W$ to be a map
-Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement
+\[
-of $y$, or write $x \le y$, if each $k$-ball of $y$ is a union of $k$-balls of $x$.
+	\coprod_a X_a \to W,
+\]
+which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$.
+Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls
+are glued up to yield $W$, so long as there is some (non-pathological) way to glue them.
+Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ or $W$, we say that $x$ is a refinement
+of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$
+with $\du_b Y_b = M_i$ for some $i$.
 \begin{defn}
 The category (poset) $\cell(W)$ has objects the permissible decompositions of $W$,
 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.
 See Figure \ref{partofJfig} for an example.
 (For $k=n$, see below.)}
 \end{module-axiom}
 \begin{module-axiom}[Strict associativity]
 The composition and action maps above are strictly associative.
+Given any decomposition of a large marked ball into smaller marked and unmarked balls
+any sequence of pairwise gluings yields (via composition and action maps) the same result.
 \end{module-axiom}
-\nn{should say that this is multifold, not just 3-fold}
 Note that the above associativity axiom applies to mixtures of module composition,
 action maps and $n$-category composition.
 See Figure \ref{zzz1b}.

changeset 475	07c18e2abd8f
parent 463	982eca6e2663
child 476	4d0ca2fc4f2b