--- a/text/intro.tex Fri Jun 04 08:15:08 2010 -0700
+++ b/text/intro.tex Fri Jun 04 11:42:07 2010 -0700
@@ -2,34 +2,69 @@
\section{Introduction}
-We construct the ``blob complex'' $\bc_*(M; \cC)$ associated to an $n$-manifold $M$ and a ``linear $n$-category with strong duality'' $\cC$. This blob complex provides a simultaneous generalisation of several well-understood constructions:
+We construct the ``blob complex'' $\bc_*(M; \cC)$ associated to an $n$-manifold $M$ and a ``linear $n$-category with strong duality'' $\cC$.
+This blob complex provides a simultaneous generalisation of several well-understood constructions:
\begin{itemize}
-\item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. (See Property \ref{property:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.)
-\item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. (See Property \ref{property:hochschild} and \S \ref{sec:hochschild}.)
+\item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$.
+(See Property \ref{property:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.)
+\item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra),
+the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$.
+(See Property \ref{property:hochschild} and \S \ref{sec:hochschild}.)
\item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have
that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains
on the configuration space of unlabeled points in $M$.
%$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$
\end{itemize}
-The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space (replacing quotient of fields by local relations with some sort of resolution),
-and for a generalization of Hochschild homology to higher $n$-categories. We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. The blob complex gives us all of these! More detailed motivations are described in \S \ref{sec:motivations}.
+The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space
+(replacing quotient of fields by local relations with some sort of resolution),
+and for a generalization of Hochschild homology to higher $n$-categories.
+We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold.
+The blob complex gives us all of these! More detailed motivations are described in \S \ref{sec:motivations}.
-The blob complex has good formal properties, summarized in \S \ref{sec:properties}. These include an action of $\CH{M}$,
-extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Property \ref{property:evaluation}) and a gluing formula allowing calculations by cutting manifolds into smaller parts (see Property \ref{property:gluing}).
+The blob complex has good formal properties, summarized in \S \ref{sec:properties}.
+These include an action of $\CH{M}$,
+extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Property \ref{property:evaluation}) and a gluing
+formula allowing calculations by cutting manifolds into smaller parts (see Property \ref{property:gluing}).
-We expect applications of the blob complex to contact topology and Khovanov homology but do not address these in this paper. See \S \ref{sec:future} for slightly more detail.
+We expect applications of the blob complex to contact topology and Khovanov homology but do not address these in this paper.
+See \S \ref{sec:future} for slightly more detail.
\subsubsection{Structure of the paper}
-The three subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), summarise the formal properties of the blob complex (see \S \ref{sec:properties}) and outline anticipated future directions and applications (see \S \ref{sec:future}).
+The three subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}),
+summarise the formal properties of the blob complex (see \S \ref{sec:properties})
+and outline anticipated future directions and applications (see \S \ref{sec:future}).
-The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex, and establishes some of its properties. There are many alternative definitions of $n$-categories, and part of our difficulty defining the blob complex is simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. At first we entirely avoid this problem by introducing the notion of a `system of fields', and define the blob complex associated to an $n$-manifold and an $n$-dimensional system of fields. We sketch the construction of a system of fields from a $1$-category or from a pivotal $2$-category.
+The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex,
+and establishes some of its properties.
+There are many alternative definitions of $n$-categories, and part of our difficulty defining the blob complex is
+simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs.
+At first we entirely avoid this problem by introducing the notion of a `system of fields', and define the blob complex
+associated to an $n$-manifold and an $n$-dimensional system of fields.
+We sketch the construction of a system of fields from a $1$-category or from a pivotal $2$-category.
-Nevertheless, when we attempt to establish all of the observed properties of the blob complex, we find this situation unsatisfactory. Thus, in the second part of the paper (\S\S \ref{sec:ncats}-\ref{sec:ainfblob}) we give yet another definition of an $n$-category, or rather a definition of an $n$-category with strong duality. (It appears that removing the duality conditions from our definition would make it more complicated rather than less.) We call these ``topological $n$-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.
+Nevertheless, when we attempt to establish all of the observed properties of the blob complex,
+we find this situation unsatisfactory.
+Thus, in the second part of the paper (\S\S \ref{sec:ncats}-\ref{sec:ainfblob}) we give yet another
+definition of an $n$-category, or rather a definition of an $n$-category with strong duality.
+(It appears that removing the duality conditions from our definition would make it more complicated rather than less.)
+We call these ``topological $n$-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.
-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 topological $n$-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 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 blob complexes of $n$-balls labelled by a topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$.
+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 topological $n$-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 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 blob complexes of $n$-balls labelled by a
+topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$.
-In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category (using a colimit along cellulations of a manifold), and in \S \ref{sec:ainfblob} 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 topological $n$-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), in particular the `gluing formula' of Property \ref{property:gluing} below.
+In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category
+(using a colimit along cellulations of a manifold), and in \S \ref{sec:ainfblob} 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 topological $n$-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),
+in particular the `gluing formula' of Property \ref{property:gluing} below.
The relationship between all these ideas is sketched in Figure \ref{fig:outline}.
@@ -77,7 +112,14 @@
\label{fig:outline}
\end{figure}
-Finally, later sections address other topics. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, thought of as a topological $n$-category, in terms of the topology of $M$. Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. The appendixes prove technical results about $\CH{M}$ and the `small blob complex', 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.
+Finally, later sections address other topics.
+Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra,
+thought of as a topological $n$-category, in terms of the topology of $M$.
+Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves)
+a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex.
+The appendixes prove technical results about $\CH{M}$ and the `small blob complex',
+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.
\nn{some more things to cover in the intro}
@@ -165,16 +207,20 @@
\begin{property}[Functoriality]
\label{property:functoriality}%
-The blob complex is functorial with respect to homeomorphisms. That is,
+The blob complex is functorial with respect to homeomorphisms.
+That is,
for a fixed $n$-dimensional system of fields $\cC$, the association
\begin{equation*}
X \mapsto \bc_*^{\cC}(X)
\end{equation*}
-is a functor from $n$-manifolds and homeomorphisms between them to chain complexes and isomorphisms between them.
+is a functor from $n$-manifolds and homeomorphisms between them to chain
+complexes and isomorphisms between them.
\end{property}
-As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*^\cC(X)$; this action is extended to all of $C_*(\Homeo(X))$ in Property \ref{property:evaluation} below.
+As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*^\cC(X)$;
+this action is extended to all of $C_*(\Homeo(X))$ in Property \ref{property:evaluation} below.
-The blob complex is also functorial (indeed, exact) with respect to $\cC$, although we will not address this in detail here.
+The blob complex is also functorial (indeed, exact) with respect to $\cC$,
+although we will not address this in detail here.
\begin{property}[Disjoint union]
\label{property:disjoint-union}
@@ -184,7 +230,9 @@
\end{equation*}
\end{property}
-If an $n$-manifold $X_\text{cut}$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, write $X_\text{glued} = X_\text{cut} \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. Note that this includes the case of gluing two disjoint manifolds together.
+If an $n$-manifold $X_\text{cut}$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary,
+write $X_\text{glued} = X_\text{cut} \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$.
+Note that this includes the case of gluing two disjoint manifolds together.
\begin{property}[Gluing map]
\label{property:gluing-map}%
%If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map
@@ -201,7 +249,8 @@
\begin{property}[Contractibility]
\label{property:contractibility}%
-With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology. Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cC$ to balls.
+With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology.
+Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cC$ to balls.
\begin{equation}
\xymatrix{\bc_*^{\cC}(B^n) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*^{\cC}(B^n)) \ar[r]^(0.6)\iso & \cC(B^n)}
\end{equation}
@@ -211,13 +260,15 @@
\label{property:skein-modules}%
The $0$-th blob homology of $X$ is the usual
(dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$
-by $\cC$. (See \S \ref{sec:local-relations}.)
+by $\cC$.
+(See \S \ref{sec:local-relations}.)
\begin{equation*}
H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X)
\end{equation*}
\end{property}
-\todo{Somehow, the Hochschild homology thing isn't a "property". Let's move it and call it a theorem? -S}
+\todo{Somehow, the Hochschild homology thing isn't a "property".
+Let's move it and call it a theorem? -S}
\begin{property}[Hochschild homology when $X=S^1$]
\label{property:hochschild}%
The blob complex for a $1$-category $\cC$ on the circle is
@@ -250,7 +301,8 @@
\bc_*(X) \ar[u]_{\gl_Y}
}
\end{equation*}
-\item Any such chain map satisfying points 2. and 3. above is unique, up to an iterated homotopy. (That is, any pair of homotopies have a homotopy between them, and so on.)
+\item Any such chain map satisfying points 2. and 3. above is unique, up to an iterated homotopy.
+(That is, any pair of homotopies have a homotopy between them, and so on.)
\item This map is associative, in the sense that the following diagram commutes (up to homotopy).
\begin{equation*}
\xymatrix{
@@ -266,23 +318,34 @@
for any homeomorphic pair $X$ and $Y$,
satisfying corresponding conditions.
-In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields. Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category.
+In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields.
+Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields.
+Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category.
\begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category]
\label{property:blobs-ainfty}
-Let $\cC$ be a topological $n$-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 $0 \leq m < k$, 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 Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Property \ref{property:evaluation}.
+Let $\cC$ be a topological $n$-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 $0 \leq m < k$,
+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
+Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Property \ref{property:evaluation}.
\end{property}
\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 a topological $n$-category. We think of this $A_\infty$ $n$-category as a free resolution.
+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 a topological $n$-category.
+We think of this $A_\infty$ $n$-category as a free resolution.
\end{rem}
There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category
-instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$.
+instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}.
+The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$.
\begin{property}[Product formula]
\label{property:product}
-Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. Let $\cC$ be an $n$-category.
+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 (see Property \ref{property:blobs-ainfty}).
Then
\[
@@ -291,7 +354,9 @@
\end{property}
We also give a generalization of this statement for arbitrary fibre bundles, in \S \ref{moddecss}, and a sketch of a statement for arbitrary maps.
-Fix a topological $n$-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 topological $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.)
+Fix a topological $n$-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 topological $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.)
\begin{property}[Gluing formula]
\label{property:gluing}%
@@ -329,18 +394,31 @@
Properties \ref{property:functoriality} and \ref{property:skein-modules} will be immediate from the definition given in
\S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there.
Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.
-Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, Property \ref{property:blobs-ainfty} as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats},
+Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation}
+in \S \ref{sec:evaluation}, Property \ref{property:blobs-ainfty} as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats},
and Properties \ref{property:product} and \ref{property:gluing} in \S \ref{sec:ainfblob} as consequences of Theorem \ref{product_thm}.
\subsection{Future directions}
\label{sec:future}
Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard).
-In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do. We could presumably also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories), and likely it will prove useful to think about the connections between what we do here and $(\infty,k)$-categories.
-More could be said about finite characteristic (there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$, for example). Much more could be said about other types of manifolds, in particular oriented, $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; there may be some differences for topological manifolds and smooth manifolds.
+In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do.
+We could presumably also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories),
+and likely it will prove useful to think about the connections between what we do here and $(\infty,k)$-categories.
+More could be said about finite characteristic
+(there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$, for example).
+Much more could be said about other types of manifolds, in particular oriented,
+$\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated.
+(We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.)
+We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds;
+there may be some differences for topological manifolds and smooth manifolds.
-The paper ``Skein homology'' \cite{MR1624157} has similar motivations, and it may be interesting to investigate if there is a connection with the material here.
+The paper ``Skein homology'' \cite{MR1624157} has similar motivations, and it may be
+interesting to investigate if there is a connection with the material here.
-Many results in Hochschild homology can be understood `topologically' via the blob complex. For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ (see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$, but haven't investigated the details.
+Many results in Hochschild homology can be understood `topologically' via the blob complex.
+For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$
+(see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$,
+but haven't investigated the details.
Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh}
--- a/text/ncat.tex Fri Jun 04 08:15:08 2010 -0700
+++ b/text/ncat.tex Fri Jun 04 11:42:07 2010 -0700
@@ -24,7 +24,9 @@
\medskip
-There are many existing definitions of $n$-categories, with various intended uses. In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. Generally, these sets are indexed by instances of a certain typical shape.
+There are many existing definitions of $n$-categories, with various intended uses.
+In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$.
+Generally, these sets are indexed by instances of a certain typical shape.
Some $n$-category definitions model $k$-morphisms on the standard bihedrons (interval, bigon, and so on).
Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$,
a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$,
@@ -33,8 +35,10 @@
Still other definitions (see, for example, \cite{MR2094071})
model the $k$-morphisms on more complicated combinatorial polyhedra.
-For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball. Thus we expect to associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic
-to the standard $k$-ball. By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the
+For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball.
+Thus we expect to associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic
+to the standard $k$-ball.
+By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the
standard $k$-ball.
We {\it do not} assume that it is equipped with a
preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below.
@@ -79,7 +83,10 @@
Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range.
(Actually, this is only true in the oriented case, with 1-morphisms parameterized
by oriented 1-balls.)
-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.
+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.
Instead, we will combine the domain and range into a single entity which we call the
boundary of a morphism.
@@ -98,7 +105,9 @@
homeomorphisms to the category of sets and bijections.
\end{lem}
-We postpone the proof \todo{} of this result until after we've actually given all the axioms. Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, along with the data described in the other Axioms at lower levels.
+We postpone the proof \todo{} of this result until after we've actually given all the axioms.
+Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$,
+along with the data described in the other Axioms at lower levels.
%In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point.
@@ -284,7 +293,9 @@
The next axiom is related to identity morphisms, though that might not be immediately obvious.
\begin{axiom}[Product (identity) morphisms]
-For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. These maps must satisfy the following conditions.
+For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$,
+usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$.
+These maps must satisfy the following conditions.
\begin{enumerate}
\item
If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram
@@ -478,7 +489,8 @@
(and their boundaries), while for fields we consider all manifolds.
Second, in category definition we directly impose isotopy
invariance in dimension $n$, while in the fields definition we have do not expect isotopy invariance on fields
-but 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.)
+but 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
balls and, at level $n$, quotienting out by the local relations:
\begin{align*}
@@ -497,7 +509,8 @@
\begin{example}[Maps to a space]
\rm
\label{ex:maps-to-a-space}%
-Fix a `target space' $T$, any topological space. We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows.
+Fix a `target space' $T$, any topological space.
+We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows.
For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of
all continuous maps from $X$ to $T$.
For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo
@@ -506,14 +519,17 @@
For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to
be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection.
-Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above. Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example.
+Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above.
+Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example.
\end{example}
\begin{example}[Maps to a space, with a fiber]
\rm
\label{ex:maps-to-a-space-with-a-fiber}%
We can modify the example above, by fixing a
-closed $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$, otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged. Taking $F$ to be a point recovers the previous case.
+closed $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$,
+otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged.
+Taking $F$ to be a point recovers the previous case.
\end{example}
\begin{example}[Linearized, twisted, maps to a space]
@@ -530,7 +546,8 @@
\nn{need to say something about fundamental classes, or choose $\alpha$ carefully}
\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. Further, most of these definitions don't even have an agreed-upon notion of `strong duality', which we assume here.
+The next example is only intended to be illustrative, as we don't specify which definition of a `traditional $n$-category' we intend.
+Further, most of these definitions don't even have an agreed-upon notion of `strong duality', which we assume here.
\begin{example}[Traditional $n$-categories]
\rm
\label{ex:traditional-n-categories}
@@ -550,7 +567,10 @@
\nn{refer elsewhere for details?}
-Recall we described a system of fields and local relations based on a `traditional $n$-category' $C$ in Example \ref{ex:traditional-n-categories(fields)} above. Constructing a system of fields from $\cC$ recovers that example. \todo{Except that it doesn't: pasting diagrams v.s. string diagrams.}
+Recall we described a system of fields and local relations based on a `traditional $n$-category'
+$C$ in Example \ref{ex:traditional-n-categories(fields)} above.
+Constructing a system of fields from $\cC$ recovers that example.
+\todo{Except that it doesn't: pasting diagrams v.s. string diagrams.}
\end{example}
Finally, we describe a version of the bordism $n$-category suitable to our definitions.
@@ -593,7 +613,8 @@
\nn{maybe should also mention version where we enrich over spaces rather than chain complexes}
\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 $\pi^\infty_{\le n}(T)$.
+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 $\pi^\infty_{\le n}(T)$.
\begin{example}[Blob complexes of balls (with a fiber)]
\rm
@@ -606,9 +627,19 @@
where $\bc^\cE_*$ denotes the blob complex based on $\cE$.
\end{example}
-This example will be essential for Theorem \ref{product_thm} 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 which we'll denote by $\bc_*(\cC)$. We think of this as providing a `free resolution' of the topological $n$-category. \todo{Say more here!} 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.
+This example will be essential for Theorem \ref{product_thm} 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 which we'll denote by $\bc_*(\cC)$.
+We think of this as providing a `free resolution' of the topological $n$-category.
+\todo{Say more here!}
+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 $\pi^\infty_{\leq n}(T)$. 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 $\pi^\infty_{\leq 0}(T) \iso C_* T$.
+Be careful that the `free resolution' of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$.
+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 $\pi^\infty_{\leq 0}(T) \iso C_* T$.
\begin{example}[The bordism $n$-category, $A_\infty$ version]
\rm
@@ -639,15 +670,30 @@
%\subsection{From $n$-categories to systems of fields}
\subsection{From balls to manifolds}
\label{ss:ncat_fields} \label{ss:ncat-coend}
-In this section we describe how to extend an $n$-category $\cC$ as described above (of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. This extension is a certain colimit, and we've chosen the notation to remind you of this.
+In this section we describe how to extend an $n$-category $\cC$ as described above
+(of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$.
+This extension is a certain colimit, and we've chosen the notation to remind you of this.
That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension
-from $k$-balls to arbitrary $k$-manifolds. Recall that we've already anticipated this construction in the previous section, inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, so that we can state the boundary axiom for $\cC$ on $k+1$-balls.
-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). 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 $\cC$.
+from $k$-balls to arbitrary $k$-manifolds.
+Recall that we've already anticipated this construction in the previous section,
+inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls,
+so that we can state the boundary axiom for $\cC$ on $k+1$-balls.
+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).
+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 $\cC$.
We will first define the `cell-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 $\cC(W)$ 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 $n$-manifolds 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).
+An $n$-category $\cC$ provides a functor from this poset to the category of sets,
+and we will define $\cC(W)$ 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 $n$-manifolds 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).
\begin{defn}
Say that a `permissible decomposition' of $W$ is a cell decomposition
@@ -659,7 +705,8 @@
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 $x$.
-The category $\cell(W)$ has objects the permissible decompositions of $W$, and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.
+The category $\cell(W)$ 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.
\end{defn}
@@ -695,7 +742,8 @@
When the $n$-category $\cC$ is enriched in some symmetric monoidal category $(A,\boxtimes)$, and $W$ is a
closed $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$ and
-we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$. (Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.)
+we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$.
+(Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.)
Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we
fix a field on $\bd W$
(i.e. fix an element of the colimit associated to $\bd W$).
@@ -710,12 +758,17 @@
\end{defn}
\begin{defn}[System of fields functor, $A_\infty$ case]
-When $\cC$ is an $A_\infty$ $n$-category, $\cC(W)$ 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 $\cC(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$.
+When $\cC$ is an $A_\infty$ $n$-category, $\cC(W)$ 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 $\cC(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$.
\end{defn}
-We can specify boundary data $c \in \cC(\bdy W)$, and define functors $\psi_{\cC;W,c}$ with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$.
+We can specify boundary data $c \in \cC(\bdy W)$, and define functors $\psi_{\cC;W,c}$
+with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$.
-We now give a more concrete description of the colimit in each case. If $\cC$ is enriched over vector spaces, and $W$ is an $n$-manifold, we can take the vector space $\cC(W,c)$ to be the direct sum over all permissible decompositions of $W$
+We now give a more concrete description of the colimit in each case.
+If $\cC$ is enriched over vector spaces, and $W$ is an $n$-manifold,
+we can take the vector space $\cC(W,c)$ to be the direct sum over all permissible decompositions of $W$
\begin{equation*}
\cC(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K
\end{equation*}
@@ -732,7 +785,9 @@
\[
V = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] ,
\]
-where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. (Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$, the complex $U[m]$ is concentrated in degree $m$.)
+where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$.
+(Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$,
+the complex $U[m]$ is concentrated in degree $m$.)
We endow $V$ 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.
More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$
@@ -752,7 +807,8 @@
permissible decomposition (filtration degree 0).
Then we glue these together with mapping cylinders coming from gluing maps
(filtration degree 1).
-Then we kill the extra homology we just introduced with mapping cylinders between the mapping cylinders (filtration degree 2), and so on.
+Then we kill the extra homology we just introduced with mapping
+cylinders between the mapping cylinders (filtration degree 2), and so on.
$\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds.
@@ -781,7 +837,9 @@
\nn{should also develop $\pi_{\le n}(T, S)$ as a module for $\pi_{\le n}(T)$, where $S\sub T$.}
-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.
+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.
Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair
$$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$
@@ -819,7 +877,8 @@
the category of marked $k$-hemispheres and
homeomorphisms to the category of sets and bijections.}
\end{lem}
-The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. We use the same type of colimit construction.
+The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details.
+We use the same type of colimit construction.
In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$.
@@ -1040,13 +1099,22 @@
\end{example}
\begin{example}
-Suppose $S$ is a topological space, with a subspace $T$. We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs $(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all such maps modulo homotopies fixed on $\bdy B \setminus N$. This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and \ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains.
+Suppose $S$ is a topological space, with a subspace $T$.
+We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$
+for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs
+$(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all
+such maps modulo homotopies fixed on $\bdy B \setminus N$.
+This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}.
+Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and
+\ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to
+Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains.
\end{example}
\subsection{Modules as boundary labels (colimits for decorated manifolds)}
\label{moddecss}
-Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. Let $W$ be a $k$-manifold ($k\le n$),
+Fix a topological $n$-category or $A_\infty$ $n$-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 $Y_i$.
@@ -1055,7 +1123,8 @@
%component $\bd_i W$ of $W$.
%(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.)
-We will define a set $\cC(W, \cN)$ using a colimit construction similar to the one appearing in \S \ref{ss:ncat_fields} above.
+We will define a set $\cC(W, \cN)$ using a colimit construction 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.)
@@ -1070,7 +1139,8 @@
\begin{figure}[!ht]\begin{equation*}
\mathfig{.4}{ncat/mblabel}
\end{equation*}\caption{A permissible decomposition of a manifold
-whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure}
+whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$.
+Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure}
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$.
This defines a partial ordering $\cell(W)$, which we will think of as a category.
@@ -1096,7 +1166,8 @@
If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define
$\cC(D\times W, \cN)$, 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 $D \mapsto \cC(D\times W, \cN)$
+$D\times Y_i \sub \bd(D\times W)$.
+It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$
has the structure of an $n{-}k$-category, which we call $\cT(W, \cN)(D)$.
\medskip
@@ -1110,7 +1181,8 @@
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 $\cT(J, \{\cM_1, \cM_2\})$. This of course depends (functorially)
+$n{-}1$-category $\cT(J, \{\cM_1, \cM_2\})$.
+This of course depends (functorially)
on the choice of 1-ball $J$.
We will define a more general self tensor product (categorified coend) below.
@@ -1132,7 +1204,8 @@
we need to define morphisms of $A_\infty$ $1$-category modules and establish
some of their elementary properties.
-To motivate the definitions which follow, consider algebras $A$ and $B$, right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction
+To motivate the definitions which follow, consider algebras $A$ and $B$,
+right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction
\begin{eqnarray*}
\hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\
f &\mapsto& [x \mapsto f(x\ot -)] \\
@@ -1260,7 +1333,9 @@
\[
\olD\ot x \ot \cbar \mapsto g(\olD\ot x \ot \cbar) \in \cY(I_1\cup\cdots\cup I_{p-1}) .
\]
-For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$, where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and $\cbar''$ corresponds to the subintervals
+For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$,
+where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and
+$\cbar''$ corresponds to the subintervals
which are dropped off the right side.
(Either $\cbar'$ or $\cbar''$ might be empty.)
\nn{surely $\cbar'$ can't be empy: we don't allow $D_1$ to be empty.}
@@ -1389,8 +1464,10 @@
\label{feb21a}
\end{figure}
-The $0$-marked balls can be cut into smaller balls in various ways. We only consider those decompositions in which the smaller balls are either
- $0$-marked (i.e. intersect the $0$-marking of the large ball in a disc) or plain (don't intersect the $0$-marking of the large ball).
+The $0$-marked balls can be cut into smaller balls in various ways.
+We only consider those decompositions in which the smaller balls are either
+$0$-marked (i.e. intersect the $0$-marking of the large ball in a disc)
+or plain (don't intersect the $0$-marking of the large ball).
We can also take the boundary of a $0$-marked ball, which is $0$-marked sphere.
Fix $n$-categories $\cA$ and $\cB$.
--- a/text/tqftreview.tex Fri Jun 04 08:15:08 2010 -0700
+++ b/text/tqftreview.tex Fri Jun 04 11:42:07 2010 -0700
@@ -5,7 +5,15 @@
\label{sec:tqftsviafields}
In this section we review the notion of a ``system of fields and local relations".
-For more details see \cite{kw:tqft}. From a system of fields and local relations we can readily construct TQFT invariants of manifolds. This is described in \S \ref{sec:constructing-a-tqft}. A system of fields is very closely related to an $n$-category. In Example \ref{ex:traditional-n-categories(fields)}, which runs throughout this section, we sketch the construction of a system of fields from an $n$-category. We make this more precise for $n=1$ or $2$ in \S \ref{sec:example:traditional-n-categories(fields)}, and much later, after we've have given our own definition of a `topological $n$-category' in \S \ref{sec:ncats}, we explain precisely how to go back and forth between a topological $n$-category and a system of fields and local relations.
+For more details see \cite{kw:tqft}.
+From a system of fields and local relations we can readily construct TQFT invariants of manifolds.
+This is described in \S \ref{sec:constructing-a-tqft}.
+A system of fields is very closely related to an $n$-category.
+In Example \ref{ex:traditional-n-categories(fields)}, which runs throughout this section,
+we sketch the construction of a system of fields from an $n$-category.
+We make this more precise for $n=1$ or $2$ in \S \ref{sec:example:traditional-n-categories(fields)},
+and much later, after we've have given our own definition of a `topological $n$-category' in \S \ref{sec:ncats},
+we explain precisely how to go back and forth between a topological $n$-category and a system of fields and local relations.
We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0
submanifold of $X$, then $X \setmin Y$ implicitly means the closure
@@ -21,7 +29,9 @@
oriented, topological, smooth, spin, etc. --- but for definiteness we
will stick with unoriented PL.)
-Fix a symmetric monoidal category $\cS$. While reading the definition, you should just think about the cases $\cS = \Set$ or $\cS = \Vect$. The presentation here requires that the objects of $\cS$ have an underlying set, but this could probably be avoided if desired.
+Fix a symmetric monoidal category $\cS$.
+While reading the definition, you should just think about the cases $\cS = \Set$ or $\cS = \Vect$.
+The presentation here requires that the objects of $\cS$ have an underlying set, but this could probably be avoided if desired.
A $n$-dimensional {\it system of fields} in $\cS$
is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$
@@ -54,7 +64,10 @@
For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of
$\cC(X)$ which restricts to $c$.
In this context, we will call $c$ a boundary condition.
-\item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields must be morphisms in $\cS$.
+\item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$.
+(This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$),
+then this extra structure is considered part of the definition of $\cC_n$.
+Any maps mentioned below between top level fields must be morphisms in $\cS$.
\item $\cC_k$ is compatible with the symmetric monoidal
structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$,
compatibly with homeomorphisms and restriction to boundary.
@@ -185,11 +198,12 @@
\subsection{Systems of fields from $n$-categories}
\label{sec:example:traditional-n-categories(fields)}
-We now describe in more detail Example \ref{ex:traditional-n-categories(fields)}, systems of fields coming from sub-cell-complexes labeled
+We now describe in more detail Example \ref{ex:traditional-n-categories(fields)},
+systems of fields coming from sub-cell-complexes labeled
by $n$-category morphisms.
Given an $n$-category $C$ with the right sort of duality
-(e.g. a pivotal 2-category, 1-category with duals, star 1-category),
+(e.g. a pivotal 2-category, *-1-category),
we can construct a system of fields as follows.
Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$
with codimension $i$ cells labeled by $i$-morphisms of $C$.
@@ -197,7 +211,8 @@
If $X$ has boundary, we require that the cell decompositions are in general
position with respect to the boundary --- the boundary intersects each cell
-transversely, so cells meeting the boundary are mere half-cells. Put another way, the cell decompositions we consider are dual to standard cell
+transversely, so cells meeting the boundary are mere half-cells.
+Put another way, the cell decompositions we consider are dual to standard cell
decompositions of $X$.
We will always assume that our $n$-categories have linear $n$-morphisms.
@@ -270,7 +285,8 @@
\subsection{Local relations}
\label{sec:local-relations}
-Local relations are certain subspaces of the fields on balls, which form an ideal under gluing. Again, we give the examples first.
+Local relations are certain subspaces of the fields on balls, which form an ideal under gluing.
+Again, we give the examples first.
\addtocounter{prop}{-2}
\begin{example}[contd.]
@@ -353,7 +369,8 @@
Let $Y$ be an $n{-}1$-manifold.
Define a (linear) 1-category $A(Y)$ as follows.
The objects of $A(Y)$ are $\cC(Y)$.
-The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$, where $a$ and $b$ label the two boundary components of the cylinder $Y\times I$.
+The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$,
+where $a$ and $b$ label the two boundary components of the cylinder $Y\times I$.
Composition is given by gluing of cylinders.
Let $X$ be an $n$-manifold with boundary and consider the collection of vector spaces