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 The next axiom says, roughly, that we have strict associativity in dimension $n$,
 even when we reparametrize our $n$-balls.
 \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$]
+\label{axiom:isotopy-preliminary}
 Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which
 acts trivially on the restriction $\bd b$ of $b$ to $\bd X$.
 (Keep in mind the important special case where $f$ restricted to $\bd X$ is the identity.)
 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which act
 trivially on $\bd b$.
 We use this axiom in the proofs of \ref{lem:d-a-acyclic}, \ref{lem:colim-injective} \nn{...}.
 All of the examples of (disk-like) $n$-categories we consider in this paper satisfy the axiom, but
 nevertheless we feel that it is too strong.
 In the future we would like to see this provisional version of the axiom replaced by something less restrictive.
-We give two alternate versions of he axiom, one better suited for smooth examples, and one better suited to PL examples.
+We give two alternate versions of the axiom, one better suited for smooth examples, and one better suited to PL examples.
 \begin{axiom}[Splittings]
 \label{axiom:splittings}
 Let $c\in \cC_k(X)$, with $0\le k < n$.
 Let $X = \cup_i X_i$ be a splitting of $X$.
 Before stating the revised axioms for an $n$-category enriched in a distributive monoidal category,
 we need a preliminary definition.
 Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the
 category $\bbc$ of {\it $n$-balls with boundary conditions}.
 Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \cl\cC(\bd X)$ is the ``boundary condition".
-The morphisms from $(X, c)$ to $(X', c')$, denoted $\Homeo(X,c; X', c')$, are
+The morphisms from $(X, c)$ to $(X', c')$, denoted $\Homeo(X; c \to X'; c')$, are
 homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$.
 %Let $\pi_0(\bbc)$ denote
 \begin{axiom}[Enriched $n$-categories]
 \label{axiom:enriched}
 Instead, we will give a relatively narrow definition which covers the examples we consider in this paper.
 After stating it, we will briefly discuss ways in which it can be made more general.
 }
 Recall the category $\bbc$ of balls with boundary conditions.
-Note that the morphisms $\Homeo(X,c; X', c')$ from $(X, c)$ to $(X', c')$ form a topological space.
+Note that the morphisms $\Homeo(X;c \to X'; c')$ from $(X, c)$ to $(X', c')$ form a topological space.
 Let $\cS$ be an appropriate $\infty$-category (e.g.\ chain complexes)
 and let $\cJ$ be an $\infty$-functor from topological spaces to $\cS$
 (e.g.\ the singular chain functor $C_*$).
 \begin{axiom}[\textup{\textbf{[$A_\infty$ replacement for Axiom \ref{axiom:extended-isotopies}]}} Families of homeomorphisms act in dimension $n$.]
 \label{axiom:families}
 For each pair of $n$-balls $X$ and $X'$ and each pair $c\in \cl{\cC}(\bd X)$ and $c'\in \cl{\cC}(\bd X')$ we have an $\cS$-morphism
 \[
-	\cJ(\Homeo(X,c; X', c')) \ot \cC(X; c) \to \cC(X'; c') .
+	\cJ(\Homeo(X;c \to X'; c')) \ot \cC(X; c) \to \cC(X'; c') .
 \]
 Similarly, we have an $\cS$-morphism
 \[
 	\cJ(\Coll(X,c)) \ot \cC(X; c) \to \cC(X; c),
 \]
 a diagram like the one in Theorem \ref{thm:CH} commutes.
 % say something about compatibility with product morphisms?
 \end{axiom}
 We now describe the topology on $\Coll(X; c)$.
-We retain notation from the above definition of collar map.
+We retain notation from the above definition of collar map (after Axiom \ref{axiom:isotopy-preliminary}).
 Each collaring homeomorphism $X \cup (Y\times J) \to X$ determines a map from points $p$ of $\bd X$ to
 (possibly length zero) embedded intervals in $X$ terminating at $p$.
 If $p \in Y$ this interval is the image of $\{p\}\times J$.
 If $p \notin Y$ then $p$ is assigned the length zero interval $\{p\}$.
 Such collections of intervals have a natural topology, and $\Coll(X; c)$ inherits its topology from this.
 Note in particular that parts of the collar are allowed to shrink continuously to zero length.
 (This is the real content; if nothing shrinks to zero length then the action of families of collar
 maps follows from the action of families of homeomorphisms and compatibility with gluing.)
 The $k=n$ case of Axiom \ref{axiom:morphisms} posits a {\it strictly} associative action of {\it sets}
-$\Homeo(X,c; X', c') \times \cC(X; c) \to \cC(X'; c')$, and at first it might seem that this would force the above
+$\Homeo(X;c\to X'; c') \times \cC(X; c) \to \cC(X'; c')$, and at first it might seem that this would force the above
-action of $\cJ(\Homeo(X,c; X', c'))$ to be strictly associative as well (assuming the two actions are compatible).
+action of $\cJ(\Homeo(X;c\to X'; c'))$ to be strictly associative as well (assuming the two actions are compatible).
 In fact, compatibility implies less than this.
 For simplicity, assume that $\cJ$ is $C_*$, the singular chains functor.
 (This is the example most relevant to this paper.)
-Then compatibility implies that the action of $C_*(\Homeo(X,c; X', c'))$ agrees with the action
+Then compatibility implies that the action of $C_*(\Homeo(X;c\to X'; c'))$ agrees with the action
-of $C_0(\Homeo(X,c; X', c'))$ coming from Axiom \ref{axiom:morphisms}, so we only require associativity in degree zero.
+of $C_0(\Homeo(X;c\to X'; c'))$ coming from Axiom \ref{axiom:morphisms}, so we only require associativity in degree zero.
-And indeed, this is true for our main example of an $A_\infty$ $n$-category based on the blob construction.
+And indeed, this is true for our main example of an $A_\infty$ $n$-category based on the blob construction (see Example \ref{ex:blob-complexes-of-balls} below).
 Stating this sort of compatibility for general $\cS$ and $\cJ$ requires further assumptions,
 such as the forgetful functor from $\cS$ to sets having a left adjoint, and $\cS$ having an internal Hom.
 An alternative (due to Peter Teichner) is to say that Axiom \ref{axiom:families}
 supersedes the $k=n$ case of Axiom \ref{axiom:morphisms}; in dimension $n$ we just have a
 (and their boundaries), while for fields we consider all manifolds.
 Second,  in the category definition we directly impose isotopy
 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 $(\cF,U)$ determines an $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to
+Thus a \nn{lemma-ize} 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,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.
 Let $\cF$ be a TQFT in the sense of \S\ref{sec:fields}: an $n$-dimensional
 system of fields (also denoted $\cF$) and local relations.
 Let $W$ be an $n{-}j$-manifold.
 Define the $j$-category $\cF(W)$ as follows.
 If $X$ is a $k$-ball with $k<j$, let $\cF(W)(X) \deq \cF(W\times X)$.
 If $X$ is a $j$-ball and $c\in \cl{\cF(W)}(\bd X)$,
 let $\cF(W)(X; c) \deq A_\cF(W\times X; c)$.
 \end{example}
 The next example is only intended to be illustrative, as we don't specify
 which definition of a ``traditional $n$-category" we intend.
 The above operad-like structure is analogous to the swiss cheese operad
 \cite{MR1718089}.
 \medskip
-We can define marked pinched products $\pi:E\to M$ of marked balls analogously to the
+We can define marked pinched products $\pi:E\to M$ of marked balls similarly to the
-plain ball case.
+plain ball case. A marked pinched product $\pi: E \to M$ is a pinched product (that is, locally modeled on degeneracy maps) which restricts to a map between the markings which is also a pinched product, and in a neighborhood of the markings is the product of the map between the markings with an interval.
+\nn{figure, 2 examples}
 Note that a marked pinched product can be decomposed into either
 two marked pinched products or a plain pinched product and a marked pinched product.
-%\nn{should maybe give figure}
+\nn{should  give figure}
 \begin{module-axiom}[Product (identity) morphisms]
 For each pinched product $\pi:E\to M$, with $M$ a marked $k$-ball and $E$ a marked
 $k{+}m$-ball ($m\ge 1$),
 there is a map $\pi^*:\cM(M)\to \cM(E)$.

changeset 897	9ba67422f1b9
parent 896	deeff619087e
child 899	b04070fc937b
child 900	2efd26072c91