--- a/text/a_inf_blob.tex Tue Aug 09 23:22:07 2011 -0700
+++ b/text/a_inf_blob.tex Tue Aug 09 23:55:13 2011 -0700
@@ -1,8 +1,8 @@
%!TEX root = ../blob1.tex
-\section{The blob complex for \texorpdfstring{$A_\infty$}{A-infinity} \texorpdfstring{$n$}{n}-categories}
+\section{The blob complex for \texorpdfstring{$A_\infty$}{A-infinity} disk-like \texorpdfstring{$n$}{n}-categories}
\label{sec:ainfblob}
-Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the
+Given an $A_\infty$ disk-like $n$-category $\cC$ and an $n$-manifold $M$, we make the
anticlimactically tautological definition of the blob
complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}.
@@ -32,7 +32,7 @@
Given an $n$-dimensional system of fields $\cE$ and a $n{-}k$-manifold $F$, recall from
-Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $\cC_F$
+Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ disk-like $k$-category $\cC_F$
defined by $\cC_F(X) = \cE(X\times F)$ if $\dim(X) < k$ and
$\cC_F(X) = \bc_*(X\times F;\cE)$ if $\dim(X) = k$.
@@ -219,11 +219,11 @@
%\nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category}
-If $Y$ has dimension $k-m$, then we have an $m$-category $\cC_{Y\times F}$ whose value at
+If $Y$ has dimension $k-m$, then we have a disk-like $m$-category $\cC_{Y\times F}$ whose value at
a $j$-ball $X$ is either $\cE(X\times Y\times F)$ (if $j<m$) or $\bc_*(X\times Y\times F)$
(if $j=m$).
(See Example \ref{ex:blob-complexes-of-balls}.)
-Similarly we have an $m$-category whose value at $X$ is $\cl{\cC_F}(X\times Y)$.
+Similarly we have a disk-like $m$-category whose value at $X$ is $\cl{\cC_F}(X\times Y)$.
These two categories are equivalent, but since we do not define functors between
disk-like $n$-categories in this paper we are unable to say precisely
what ``equivalent" means in this context.
@@ -235,7 +235,7 @@
\begin{cor}
\label{cor:new-old}
-Let $\cE$ be a system of fields (with local relations) and let $\cC_\cE$ be the $A_\infty$
+Let $\cE$ be a system of fields (with local relations) and let $\cC_\cE$ be the $A_\infty$ disk-like
$n$-category obtained from $\cE$ by taking the blob complex of balls.
Then for all $n$-manifolds $Y$ the old-fashioned and new-fangled blob complexes are
homotopy equivalent:
@@ -261,18 +261,18 @@
We can generalize the definition of a $k$-category by replacing the categories
of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$
(c.f. \cite{MR2079378}).
-Call this a $k$-category over $Y$.
-A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$:
+Call this a disk-like $k$-category over $Y$.
+A fiber bundle $F\to E\to Y$ gives an example of a disk-like $k$-category over $Y$:
assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$, if $\dim(D) = k$,
or the fields $\cE(p^*(E))$, if $\dim(D) < k$.
(Here $p^*(E)$ denotes the pull-back bundle over $D$.)
-Let $\cF_E$ denote this $k$-category over $Y$.
+Let $\cF_E$ denote this disk-like $k$-category over $Y$.
We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to
get a chain complex $\cl{\cF_E}(Y)$.
The proof of Theorem \ref{thm:product} goes through essentially unchanged
to show the following result.
\begin{thm}
-Let $F \to E \to Y$ be a fiber bundle and let $\cF_E$ be the $k$-category over $Y$ defined above.
+Let $F \to E \to Y$ be a fiber bundle and let $\cF_E$ be the disk-like $k$-category over $Y$ defined above.
Then
\[
\bc_*(E) \simeq \cl{\cF_E}(Y) .
@@ -287,13 +287,13 @@
$D\widetilde{\times} M$ is a manifold of dimension $n-k+j$ with a collar structure along the boundary of $D$.
(If $D\to Y$ is an embedding then $D\widetilde{\times} M$ is just the part of $M$
lying above $D$.)
-We can define a $k$-category $\cF_M$ based on maps of balls into $Y$ which are good with respect to $M$.
+We can define a disk-like $k$-category $\cF_M$ based on maps of balls into $Y$ which are good with respect to $M$.
We can again adapt the homotopy colimit construction to
get a chain complex $\cl{\cF_M}(Y)$.
The proof of Theorem \ref{thm:product} again goes through essentially unchanged
to show that
\begin{thm}
-Let $M \to Y$ be a map of manifolds and let $\cF_M$ be the $k$-category over $Y$ defined above.
+Let $M \to Y$ be a map of manifolds and let $\cF_M$ be the disk-like $k$-category over $Y$ defined above.
Then
\[
\bc_*(M) \simeq \cl{\cF_M}(Y) .
@@ -315,7 +315,7 @@
such that the restriction of $E$ to $X_i$ is homeomorphic to a product $F\times X_i$,
and choose trivializations of these products as well.
-Let $\cF$ be the $k$-category associated to $F$.
+Let $\cF$ be the disk-like $k$-category associated to $F$.
To each codimension-1 face $X_i\cap X_j$ we have a bimodule ($S^0$-module) for $\cF$.
More generally, to each codimension-$m$ face we have an $S^{m-1}$-module for a $(k{-}m{+}1)$-category
associated to the (decorated) link of that face.
@@ -341,22 +341,22 @@
$X = X_1\cup (Y\times J) \cup X_2$.
Given this data we have:
\begin{itemize}
-\item An $A_\infty$ $n{-}k$-category $\bc(X)$, which assigns to an $m$-ball
+\item An $A_\infty$ disk-like $n{-}k$-category $\bc(X)$, which assigns to an $m$-ball
$D$ fields on $D\times X$ (for $m+k < n$) or the blob complex $\bc_*(D\times X; c)$
(for $m+k = n$).
(See Example \ref{ex:blob-complexes-of-balls}.)
%\nn{need to explain $c$}.
-\item An $A_\infty$ $n{-}k{+}1$-category $\bc(Y)$, defined similarly.
+\item An $A_\infty$ disk-like $n{-}k{+}1$-category $\bc(Y)$, defined similarly.
\item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked
$m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$)
or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$).
(See Example \ref{bc-module-example}.)
\item The tensor product $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$, which is
-an $A_\infty$ $n{-}k$-category.
+an $A_\infty$ disk-like $n{-}k$-category.
(See \S \ref{moddecss}.)
\end{itemize}
-It is the case that the $n{-}k$-categories $\bc(X)$ and $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$
+It is the case that the disk-like $n{-}k$-categories $\bc(X)$ and $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$
are equivalent for all $k$, but since we do not develop a definition of functor between $n$-categories
in this paper, we cannot state this precisely.
(It will appear in a future paper.)
@@ -403,7 +403,7 @@
The next theorem shows how to reconstruct a mapping space from local data.
Let $T$ be a topological space, let $M$ be an $n$-manifold,
-and recall the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$
+and recall the $A_\infty$ disk-like $n$-category $\pi^\infty_{\leq n}(T)$
of Example \ref{ex:chains-of-maps-to-a-space}.
Think of $\pi^\infty_{\leq n}(T)$ as encoding everything you would ever
want to know about spaces of maps of $k$-balls into $T$ ($k\le n$).
--- a/text/intro.tex Tue Aug 09 23:22:07 2011 -0700
+++ b/text/intro.tex Tue Aug 09 23:55:13 2011 -0700
@@ -64,34 +64,34 @@
definition of an $n$-category, or rather a definition of an $n$-category with strong duality.
(Removing the duality conditions from our definition would make it more complicated rather than less.)
We call these ``disk-like $n$-categories'', to differentiate them from previous versions.
-Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms.
+Moreover, we find that we need analogous $A_\infty$ disk-like $n$-categories, and we define these as well following very similar axioms.
(See \S \ref{n-cat-names} below for a discussion of $n$-category terminology.)
The basic idea is that each potential definition of an $n$-category makes a choice about the ``shape" of morphisms.
We try to be as lax as possible: a disk-like $n$-category associates a
vector space to every $B$ homeomorphic to the $n$-ball.
These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid.
-For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to
+For an $A_\infty$ disk-like $n$-category, we associate a chain complex instead of a vector space to
each such $B$ and ask that the action of
homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms.
-The axioms for an $A_\infty$ $n$-category are designed to capture two main examples:
+The axioms for an $A_\infty$ disk-like $n$-category are designed to capture two main examples:
the blob complexes of $n$-balls labelled by a
disk-like $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$.
-In \S \ref{ssec:spherecat} we explain how $n$-categories can be viewed as objects in an $n{+}1$-category
+In \S \ref{ssec:spherecat} we explain how disk-like $n$-categories can be viewed as objects in a disk-like $n{+}1$-category
of sphere modules.
When $n=1$ this just the familiar 2-category of 1-categories, bimodules and intertwiners.
In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a disk-like $n$-category
(using a colimit along certain decompositions of a manifold into balls).
With this in hand, we write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$
-with the system of fields constructed from the $n$-category $\cC$.
+with the system of fields constructed from the disk-like $n$-category $\cC$.
%\nn{KW: I don't think we use this notational convention any more, right?}
In \S \ref{sec:ainfblob} we give an alternative definition
-of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit).
-Using these definitions, we show how to use the blob complex to ``resolve" any ordinary $n$-category as an
-$A_\infty$ $n$-category, and relate the first and second definitions of the blob complex.
-We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants),
+of the blob complex for an $A_\infty$ disk-like $n$-category on an $n$-manifold (analogously, using a homotopy colimit).
+Using these definitions, we show how to use the blob complex to ``resolve" any ordinary disk-like $n$-category as an
+$A_\infty$ disk-like $n$-category, and relate the first and second definitions of the blob complex.
+We use the blob complex for $A_\infty$ disk-like $n$-categories to establish important properties of the blob complex (in both variants),
in particular the ``gluing formula" of Theorem \ref{thm:gluing} below.
The relationship between all these ideas is sketched in Figure \ref{fig:outline}.
@@ -155,8 +155,8 @@
a higher dimensional generalization of the Deligne conjecture
(that the little discs operad acts on Hochschild cochains) in terms of the blob complex.
The appendices prove technical results about $\CH{M}$ and
-make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$,
-as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras.
+make connections between our definitions of disk-like $n$-categories and familiar definitions for $n=1$ and $n=2$,
+as well as relating the $n=1$ case of our $A_\infty$ disk-like $n$-categories with usual $A_\infty$ algebras.
%Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra,
%thought of as a disk-like $n$-category, in terms of the topology of $M$.
@@ -373,42 +373,42 @@
from which we can construct systems of fields.
Below, when we talk about the blob complex for a disk-like $n$-category,
we are implicitly passing first to this associated system of fields.
-Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category.
+Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ disk-like $n$-category.
In that section we describe how to use the blob complex to
-construct $A_\infty$ $n$-categories from ordinary $n$-categories:
+construct $A_\infty$ disk-like $n$-categories from ordinary disk-like $n$-categories:
\newtheorem*{ex:blob-complexes-of-balls}{Example \ref{ex:blob-complexes-of-balls}}
-\begin{ex:blob-complexes-of-balls}[Blob complexes of products with balls form an $A_\infty$ $n$-category]
+\begin{ex:blob-complexes-of-balls}[Blob complexes of products with balls form an $A_\infty$ disk-like $n$-category]
%\label{thm:blobs-ainfty}
-Let $\cC$ be an ordinary $n$-category.
+Let $\cC$ be an ordinary disk-like $n$-category.
Let $Y$ be an $n{-}k$-manifold.
-There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$,
+There is an $A_\infty$ disk-like $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$,
to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set
$$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$
(When $m=k$ the subsets with fixed boundary conditions form a chain complex.)
-These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in
+These sets have the structure of an $A_\infty$ disk-like $k$-category, with compositions coming from the gluing map in
Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}.
\end{ex:blob-complexes-of-balls}
\begin{rem}
Perhaps the most interesting case is when $Y$ is just a point;
-then we have a way of building an $A_\infty$ $n$-category from an ordinary $n$-category.
-We think of this $A_\infty$ $n$-category as a free resolution.
+then we have a way of building an $A_\infty$ disk-like $n$-category from an ordinary disk-like $n$-category.
+We think of this $A_\infty$ disk-like $n$-category as a free resolution.
\end{rem}
-There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category
-instead of an ordinary $n$-category; this is described in \S \ref{sec:ainfblob}.
+There is a version of the blob complex for $\cC$ an $A_\infty$ disk-like $n$-category
+instead of an ordinary disk-like $n$-category; this is described in \S \ref{sec:ainfblob}.
The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$.
The next theorem describes the blob complex for product manifolds,
-in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as in the previous example.
+in terms of the $A_\infty$ blob complex of the $A_\infty$ disk-like $n$-categories constructed as in the previous example.
%The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit.
\newtheorem*{thm:product}{Theorem \ref{thm:product}}
\begin{thm:product}[Product formula]
Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold.
-Let $\cC$ be an $n$-category.
-Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology
+Let $\cC$ be a disk-like $n$-category.
+Let $\bc_*(Y;\cC)$ be the $A_\infty$ disk-like $k$-category associated to $Y$ via blob homology
(see Example \ref{ex:blob-complexes-of-balls}).
Then
\[
@@ -420,7 +420,7 @@
Fix a disk-like $n$-category $\cC$, which we'll omit from the notation.
Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category.
-(See Appendix \ref{sec:comparing-A-infty} for the translation between disk-like $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.)
+(See Appendix \ref{sec:comparing-A-infty} for the translation between $A_\infty$ disk-like $1$-categories and the usual algebraic notion of an $A_\infty$ category.)
\newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}}
@@ -447,7 +447,7 @@
\newtheorem*{thm:map-recon}{Theorem \ref{thm:map-recon}}
\begin{thm:map-recon}[Mapping spaces]
-Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps
+Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ disk-like $n$-category based on maps
$B^n \to T$.
(The case $n=1$ is the usual $A_\infty$-category of paths in $T$.)
Then
@@ -512,11 +512,11 @@
since we think of the higher homotopies not as morphisms of the $n$-category but
rather as belonging to some auxiliary category (like chain complexes)
that we are enriching in.
-We have decided to call them ``$A_\infty$ $n$-categories", since they are a natural generalization
+We have decided to call them ``$A_\infty$ disk-like $n$-categories", since they are a natural generalization
of the familiar $A_\infty$ 1-categories.
We also considered the names ``homotopy $n$-categories" and ``infinity $n$-categories".
When we need to emphasize that we are talking about an $n$-category which is not $A_\infty$ in this sense
-we will say ``ordinary $n$-category".
+we will say ``ordinary disk-like $n$-category".
% small problem: our n-cats are of course strictly associative, since we have more morphisms.
% when we say ``associative only up to homotopy" above we are thinking about
% what would happen we we tried to convert to a more traditional n-cat with fewer morphisms
--- a/text/ncat.tex Tue Aug 09 23:22:07 2011 -0700
+++ b/text/ncat.tex Tue Aug 09 23:55:13 2011 -0700
@@ -3,10 +3,10 @@
\def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip}
\def\mmpar#1#2#3{\smallskip\noindent{\bf #1} (#2). {\it #3} \smallskip}
-\section{\texorpdfstring{$n$}{n}-categories and their modules}
+\section{Disk-like \texorpdfstring{$n$}{n}-categories and their modules}
\label{sec:ncats}
-\subsection{Definition of \texorpdfstring{$n$}{n}-categories}
+\subsection{Definition of disk-like \texorpdfstring{$n$}{n}-categories}
\label{ss:n-cat-def}
Before proceeding, we need more appropriate definitions of $n$-categories,
@@ -32,11 +32,11 @@
\medskip
-The axioms for an $n$-category are spread throughout this section.
-Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms},
+The axioms for a disk-like $n$-category are spread throughout this section.
+Collecting these together, a disk-like $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms},
\ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:extended-isotopies} and \ref{axiom:vcones}.
-For an enriched $n$-category we add Axiom \ref{axiom:enriched}.
-For an $A_\infty$ $n$-category, we replace
+For an enriched disk-like $n$-category we add Axiom \ref{axiom:enriched}.
+For an $A_\infty$ disk-like $n$-category, we replace
Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}.
Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms
@@ -88,7 +88,7 @@
%\nn{need to check whether this makes much difference}
(If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need
to be fussier about corners and boundaries.)
-For each flavor of manifold there is a corresponding flavor of $n$-category.
+For each flavor of manifold there is a corresponding flavor of disk-like $n$-category.
For simplicity, we will concentrate on the case of PL unoriented manifolds.
An ambitious reader may want to keep in mind two other classes of balls.
@@ -807,8 +807,8 @@
\medskip
-This completes the definition of an $n$-category.
-Next we define enriched $n$-categories.
+This completes the definition of a disk-like $n$-category.
+Next we define enriched disk-like $n$-categories.
\medskip
@@ -837,7 +837,7 @@
For convenience, we will also assume that the objects of our auxiliary category are sets with extra structure.
(Otherwise, stating the axioms for identity morphisms becomes more cumbersome.)
-Before stating the revised axioms for an $n$-category enriched in a distributive monoidal category,
+Before stating the revised axioms for a disk-like $n$-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}.
@@ -846,10 +846,10 @@
homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$.
%Let $\pi_0(\bbc)$ denote
-\begin{axiom}[Enriched $n$-categories]
+\begin{axiom}[Enriched disk-like $n$-categories]
\label{axiom:enriched}
Let $\cS$ be a distributive symmetric monoidal category.
-An $n$-category enriched in $\cS$ satisfies the above $n$-category axioms for $k=0,\ldots,n-1$,
+A disk-like $n$-category enriched in $\cS$ satisfies the above $n$-category axioms for $k=0,\ldots,n-1$,
and modifies the axioms for $k=n$ as follows:
\begin{itemize}
\item Morphisms. We have a functor $\cC_n$ from $\bbc$ ($n$-balls with boundary conditions) to $\cS$.
@@ -875,7 +875,7 @@
or more generally an appropriate sort of $\infty$-category,
we can modify the extended isotopy axiom \ref{axiom:extended-isotopies}
to require that families of homeomorphisms act
-and obtain what we shall call an $A_\infty$ $n$-category.
+and obtain what we shall call an $A_\infty$ disk-like $n$-category.
\noop{
We believe that abstract definitions should be guided by diverse collections
@@ -928,7 +928,7 @@
(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
of $C_0(\Homeo(X,c; X', c'))$ 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 $A_\infty$ disk-like $n$-category based on the blob construction.
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.
@@ -950,7 +950,7 @@
For future reference we make the following definition.
\begin{defn}
-A {\em strict $A_\infty$ $n$-category} is one in which the actions of Axiom \ref{axiom:families} are strictly associative.
+A {\em strict $A_\infty$ disk-like $n$-category} is one in which the actions of Axiom \ref{axiom:families} are strictly associative.
\end{defn}
\noop{
@@ -966,13 +966,13 @@
\medskip
-We define a $j$ times monoidal $n$-category to be an $(n{+}j)$-category $\cC$ where
+We define a $j$ times monoidal disk-like $n$-category to be a disk-like $(n{+}j)$-category $\cC$ where
$\cC(X)$ is a trivial 1-element set if $X$ is a $k$-ball with $k<j$.
See Example \ref{ex:bord-cat}.
\medskip
-The alert reader will have already noticed that our definition of an (ordinary) $n$-category
+The alert reader will have already noticed that our definition of an (ordinary) disk-like $n$-category
is extremely similar to our definition of a system of fields.
There are two differences.
First, for the $n$-category definition we restrict our attention to balls
@@ -981,7 +981,7 @@
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 system of fields and local relations $(\cF,U)$ determines a disk-like $n$-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}
@@ -995,7 +995,7 @@
In the $n$-category axioms above we have intermingled data and properties for expository reasons.
Here's a summary of the definition which segregates the data from the properties.
-An $n$-category consists of the following data:
+A disk-like $n$-category consists of the following data:
\begin{itemize}
\item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms});
\item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary});
@@ -1021,7 +1021,7 @@
\end{itemize}
-\subsection{Examples of \texorpdfstring{$n$}{n}-categories}
+\subsection{Examples of disk-like \texorpdfstring{$n$}{n}-categories}
\label{ss:ncat-examples}
@@ -1153,7 +1153,7 @@
where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary,
and $C_*$ denotes singular chains.
Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X \to T)$,
-we get an $A_\infty$ $n$-category enriched over spaces.
+we get an $A_\infty$ disk-like $n$-category enriched over spaces.
\end{example}
See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to
@@ -1163,7 +1163,7 @@
\rm
\label{ex:blob-complexes-of-balls}
Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$.
-We will define an $A_\infty$ $k$-category $\cC$.
+We will define an $A_\infty$ disk-like $k$-category $\cC$.
When $X$ is a $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$.
When $X$ is an $k$-ball,
define $\cC(X; c) = \bc^\cE_*(X\times F; c)$
@@ -1171,17 +1171,17 @@
\end{example}
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 an ordinary
-$n$-category $\cC$ into an $A_\infty$ $n$-category.
+Notice that with $F$ a point, the above example is a construction turning an ordinary disk-like
+$n$-category $\cC$ into an $A_\infty$ disk-like $n$-category.
We think of this as providing a ``free resolution"
-of the ordinary $n$-category.
+of the ordinary disk-like $n$-category.
%\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.
-Beware that the ``free resolution" of the ordinary $n$-category $\pi_{\leq n}(T)$
-is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$.
+Beware that the ``free resolution" of the ordinary disk-like $n$-category $\pi_{\leq n}(T)$
+is not the $A_\infty$ disk-like $n$-category $\pi^\infty_{\leq n}(T)$.
It's easy to see that with $n=0$, the corresponding system of fields is just
linear combinations of connected components of $T$, and the local relations are trivial.
There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$.
@@ -1225,7 +1225,7 @@
Let $A$ be an $\cE\cB_n$-algebra.
Note that this implies a $\Diff(B^n)$ action on $A$,
since $\cE\cB_n$ contains a copy of $\Diff(B^n)$.
-We will define a strict $A_\infty$ $n$-category $\cC^A$.
+We will define a strict $A_\infty$ disk-like $n$-category $\cC^A$.
If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point.
In other words, the $k$-morphisms are trivial for $k<n$.
If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction.
@@ -1237,12 +1237,12 @@
to each morphism of $J$ we associate a morphism coming from the $\cE\cB_n$ action on $A$.
Alternatively and more simply, we could define $\cC^A(X)$ to be
$\Diff(B^n\to X)\times A$ modulo the diagonal action of $\Diff(B^n)$.
-The remaining data for the $A_\infty$ $n$-category
+The remaining data for the $A_\infty$ disk-like $n$-category
--- composition and $\Diff(X\to X')$ action ---
also comes from the $\cE\cB_n$ action on $A$.
%\nn{should we spell this out?}
-Conversely, one can show that a disk-like strict $A_\infty$ $n$-category $\cC$, where the $k$-morphisms
+Conversely, one can show that a strict $A_\infty$ disk-like $n$-category $\cC$, where the $k$-morphisms
$\cC(X)$ are trivial (single point) for $k<n$, gives rise to
an $\cE\cB_n$-algebra.
%\nn{The paper is already long; is it worth giving details here?}
@@ -1257,19 +1257,19 @@
\subsection{From balls to manifolds}
\label{ss:ncat_fields} \label{ss:ncat-coend}
-In this section we show how to extend an $n$-category $\cC$ as described above
+In this section we show how to extend a disk-like $n$-category $\cC$ as described above
(of either the ordinary or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$.
This extension is a certain colimit, and the arrow in the notation is intended as a reminder of this.
-In the case of ordinary $n$-categories, this construction factors into a construction of a
+In the case of ordinary disk-like $n$-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 ordinary $n$-category $\cC$ and pass to the ``free resolution",
-an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls
+For an $A_\infty$ disk-like $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead.
+Recall that we can take a ordinary disk-like $n$-category $\cC$ and pass to the ``free resolution",
+an $A_\infty$ disk-like $n$-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
+for a manifold $M$ associated to this $A_\infty$ disk-like $n$-category is actually the
same as the original blob complex for $M$ with coefficients in $\cC$.
Recall that we've already anticipated this construction Subsection \ref{ss:n-cat-def},
@@ -1279,11 +1279,11 @@
\medskip
We will first define the {\it 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,
+A disk-like $n$-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
+In the case that the disk-like $n$-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
@@ -1334,7 +1334,7 @@
\label{partofJfig}
\end{figure}
-An $n$-category $\cC$ determines
+A disk-like $n$-category $\cC$ determines
a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets
(possibly with additional structure if $k=n$).
Let $x = \{X_a\}$ be a permissible decomposition of $W$ (i.e.\ object of $\cD(W)$).
@@ -1402,14 +1402,14 @@
\begin{defn}[System of fields functor]
\label{def:colim-fields}
-If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$.
+If $\cC$ is a disk-like $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$.
That is, for each decomposition $x$ there is a map
$\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps
above, and $\cl{\cC}(W)$ is universal with respect to these properties.
\end{defn}
\begin{defn}[System of fields functor, $A_\infty$ case]
-When $\cC$ is an $A_\infty$ $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$
+When $\cC$ is an $A_\infty$ disk-like $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$
is defined as above, as the colimit of $\psi_{\cC;W}$.
When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$.
\end{defn}
@@ -1585,16 +1585,16 @@
\subsection{Modules}
-Next we define ordinary and $A_\infty$ $n$-category modules.
-The definition will be very similar to that of $n$-categories,
+Next we define ordinary and $A_\infty$ disk-like $n$-category modules.
+The definition will be very similar to that of disk-like $n$-categories,
but with $k$-balls replaced by {\it marked $k$-balls,} defined below.
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 $\bd W$.
+Such a $W$ gives rise to a module for the disk-like $n$-category associated to $\bd W$.
This will be explained in more detail as we present the axioms.
-Throughout, we fix an $n$-category $\cC$.
+Throughout, we fix a disk-like $n$-category $\cC$.
For all but one axiom, it doesn't matter whether $\cC$ is an ordinary $n$-category or an $A_\infty$ $n$-category.
We state the final axiom, regarding actions of homeomorphisms, differently in the two cases.
@@ -1650,7 +1650,7 @@
Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$.
-If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces),
+If the disk-like $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces),
then for each marked $n$-ball $M=(B,N)$ and $c\in \cC(\bd B \setminus N)$, the set $\cM(M; c)$ should be an object in that category.
\begin{lem}[Boundary from domain and range]
@@ -1857,7 +1857,7 @@
\end{enumerate}
\end{module-axiom}
-As in the $n$-category definition, once we have product morphisms we can define
+As in the disk-like $n$-category definition, once we have product morphisms we can define
collar maps $\cM(M)\to \cM(M)$.
Note that there are two cases:
the collar could intersect the marking of the marked ball $M$, in which case
@@ -1870,7 +1870,7 @@
\medskip
There are two alternatives for the next axiom, according whether we are defining
-modules for ordinary $n$-categories or $A_\infty$ $n$-categories.
+modules for ordinary or $A_\infty$ disk-like $n$-categories.
In the ordinary case we require
\begin{module-axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$]
@@ -1903,14 +1903,14 @@
\medskip
-Note that the above axioms imply that an $n$-category module has the structure
-of an $n{-}1$-category.
+Note that the above axioms imply that a disk-like $n$-category module has the structure
+of a disk-like $n{-}1$-category.
More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$,
where $X$ is a $k$-ball and in the product $X\times J$ we pinch
above the non-marked boundary component of $J$.
(More specifically, we collapse $X\times P$ to a single point, where
$P$ is the non-marked boundary component of $J$.)
-Then $\cE$ has the structure of an $n{-}1$-category.
+Then $\cE$ has the structure of a disk-like $n{-}1$-category.
All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds
are oriented or Spin (but not unoriented or $\text{Pin}_\pm$).
@@ -1922,12 +1922,12 @@
\medskip
-We now give some examples of modules over ordinary and $A_\infty$ $n$-categories.
+We now give some examples of modules over ordinary and $A_\infty$ disk-like $n$-categories.
\begin{example}[Examples from TQFTs]
\rm
Continuing Example \ref{ex:ncats-from-tqfts}, with $\cF$ a TQFT, $W$ an $n{-}j$-manifold,
-and $\cF(W)$ the $j$-category associated to $W$.
+and $\cF(W)$ the disk-like $j$-category associated to $W$.
Let $Y$ be an $(n{-}j{+}1)$-manifold with $\bd Y = W$.
Define a $\cF(W)$ module $\cF(Y)$ as follows.
If $M = (B, N)$ is a marked $k$-ball with $k<j$ let
@@ -1940,7 +1940,7 @@
\rm
In the previous example, we can instead define
$\cF(Y)(M)\deq \bc_*((B\times W) \cup (N\times Y), c; \cF)$ (when $\dim(M) = n$)
-and get a module for the $A_\infty$ $n$-category associated to $\cF$ as in
+and get a module for the $A_\infty$ disk-like $n$-category associated to $\cF$ as in
Example \ref{ex:blob-complexes-of-balls}.
\end{example}
@@ -1965,7 +1965,7 @@
\subsection{Modules as boundary labels (colimits for decorated manifolds)}
\label{moddecss}
-Fix an ordinary $n$-category or $A_\infty$ $n$-category $\cC$.
+Fix an ordinary or $A_\infty$ disk-like $n$-category $\cC$.
Let $W$ be a $k$-manifold ($k\le n$),
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 $Y_i$.
@@ -2021,19 +2021,19 @@
$\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold
$D\times Y_i \sub \bd(D\times W)$.
It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$
-has the structure of an $n{-}k$-category.
+has the structure of a disk-like $n{-}k$-category.
\medskip
We will use a simple special case of the above
construction to define tensor products
of modules.
-Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$.
+Let $\cM_1$ and $\cM_2$ be modules for a disk-like $n$-category $\cC$.
(If $k=1$ and our manifolds are oriented, then one should be
a left module and the other a right module.)
Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$.
Define the tensor product $\cM_1 \tensor \cM_2$ to be the
-$n{-}1$-category associated as above to $J$ with its boundary labeled by $\cM_1$ and $\cM_2$.
+disk-like $n{-}1$-category associated as above to $J$ with its boundary labeled by $\cM_1$ and $\cM_2$.
This of course depends (functorially)
on the choice of 1-ball $J$.
@@ -2738,19 +2738,19 @@
\medskip
-We end this subsection with some remarks about Morita equivalence of disklike $n$-categories.
+We end this subsection with some remarks about Morita equivalence of disk-like $n$-categories.
Recall that two 1-categories $\cC$ and $\cD$ are Morita equivalent if and only if they are equivalent
objects in the 2-category of (linear) 1-categories, bimodules, and intertwiners.
-Similarly, we define two disklike $n$-categories to be Morita equivalent if they are equivalent objects in the
+Similarly, we define two disk-like $n$-categories to be Morita equivalent if they are equivalent objects in the
$n{+}1$-category of sphere modules.
-Because of the strong duality enjoyed by disklike $n$-categories, the data for such an equivalence lives only in
+Because of the strong duality enjoyed by disk-like $n$-categories, the data for such an equivalence lives only in
dimensions 1 and $n+1$ (the middle dimensions come along for free).
The $n{+}1$-dimensional part of the data must be invertible and satisfy
identities corresponding to Morse cancellations in $n$-manifolds.
We will treat this in detail for the $n=2$ case; the case for general $n$ is very similar.
-Let $\cC$ and $\cD$ be (unoriented) disklike 2-categories.
+Let $\cC$ and $\cD$ be (unoriented) disk-like 2-categories.
Let $\cS$ denote the 3-category of 2-category sphere modules.
The 1-dimensional part of the data for a Morita equivalence between $\cC$ and $\cD$ is a 0-sphere module $\cM = {}_\cC\cM_\cD$
(categorified bimodule) connecting $\cC$ and $\cD$.