--- a/pnas/pnas.tex Tue Nov 23 09:28:41 2010 -0800
+++ b/pnas/pnas.tex Tue Nov 23 09:28:45 2010 -0800
@@ -136,8 +136,9 @@
\begin{article}
\begin{abstract}
+\nn{needs revision}
We explain the need for new axioms for topological quantum field theories that include ideas from derived
-categories and homotopy theory. We summarize our axioms for higher categories, and describe the `blob complex'.
+categories and homotopy theory. We summarize our axioms for higher categories, and describe the ``blob complex".
Fixing an $n$-category $\cC$, the blob complex associates a chain complex $\bc_*(W;\cC)$ to any $n$-manifold $W$.
The $0$-th homology of this chain complex recovers the usual TQFT invariants of $W$.
The higher homology groups should be viewed as generalizations of Hochschild homology.
@@ -184,14 +185,14 @@
Thus a TQFT assigns to each closed $n$-manifold $Y$ a vector space $A(Y)$,
and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$.
For the remainder of this paper we will in fact be interested in so-called $(n{+}\epsilon)$-dimensional
-TQFTs, which are slightly weaker structures and do not assign anything to general $(n{+}1)$-manifolds,
-but only to mapping cylinders.
+TQFTs, which are slightly weaker structures in that they assign
+invariants to mapping cylinders of homeomorphisms between $n$-manifolds, but not to general $(n{+}1)$-manifolds.
-When $k=n-1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$,
+When $k=n{-}1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$,
and a representation of $A(\bd Y)$ for each $n$-manifold $Y$.
The TQFT gluing rule in dimension $n$ states that
$A(Y_1\cup_S Y_2) \cong A(Y_1) \ot_{A(S)} A(Y_2)$,
-where $Y_1$ and $Y_2$ and $n$-manifolds with common boundary $S$.
+where $Y_1$ and $Y_2$ are $n$-manifolds with common boundary $S$.
When $k=0$ we have an $n$-category $A(pt)$.
This can be thought of as the local part of the TQFT, and the full TQFT can be reconstructed from $A(pt)$
@@ -207,7 +208,7 @@
extended all the way down to dimension 0.)
For other non-semisimple TQFT-like invariants, however, the above framework seems to be inadequate.
-For example, the gluing rule for 3-manifolds in Ozsv\'{a}th-Szab\'{o}/Seiberg-Witten theory
+For example, the gluing rule for 3-manifolds in Ozsv\'ath-Szab\'o/Seiberg-Witten theory
involves a tensor product over an $A_\infty$ 1-category associated to 2-manifolds \cite{1003.0598,1005.1248}.
Long exact sequences are important computational tools in these theories,
and also in Khovanov homology, but the colimit construction breaks exactness.
@@ -236,13 +237,14 @@
yields a higher categorical and higher dimensional generalization of Deligne's
conjecture on Hochschild cochains and the little 2-disks operad.
+\nn{needs revision}
Of course, there are currently many interesting alternative notions of $n$-category and of TQFT.
We note that our $n$-categories are both more and less general
than the ``fully dualizable" ones which play a prominent role in \cite{0905.0465}.
They are more general in that we make no duality assumptions in the top dimension $n{+}1$.
They are less general in that we impose stronger duality requirements in dimensions 0 through $n$.
-Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional unoriented or oriented TQFTs, while
-Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional framed TQFTs.
+Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional {\it unoriented} or {\it oriented} TQFTs, while
+Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional {\it framed} TQFTs.
At several points we only sketch an argument briefly; full details can be found in \cite{1009.5025}.
In this paper we attempt to give a clear view of the big picture without getting
@@ -271,10 +273,10 @@
%More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.}
We will define two variations simultaneously, as all but one of the axioms are identical in the two cases.
-These variations are `isotopy $n$-categories', where homeomorphisms fixing the boundary
+These variations are ``plain $n$-categories", where homeomorphisms fixing the boundary
act trivially on the sets associated to $n$-balls
(and these sets are usually vector spaces or more generally modules over a commutative ring)
-and `$A_\infty$ $n$-categories', where there is a homotopy action of
+and ``$A_\infty$ $n$-categories", where there is a homotopy action of
$k$-parameter families of homeomorphisms on these sets
(which are usually chain complexes or topological spaces).
@@ -318,12 +320,12 @@
Note that the functoriality in the above axiom allows us to operate via
homeomorphisms which are not the identity on the boundary of the $k$-ball.
The action of these homeomorphisms gives the ``strong duality" structure.
-As such, we don't subdivide the boundary of a morphism
-into domain and range --- the duality operations can convert between domain and range.
+For this reason we don't subdivide the boundary of a morphism
+into domain and range in the next axiom --- the duality operations can convert between domain and range.
Later we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$
-from arbitrary manifolds to sets. We need these functors for $k$-spheres,
-for $k<n$, for the next axiom.
+defined on arbitrary manifolds.
+We need these functors for $k$-spheres, for $k<n$, for the next axiom.
\begin{axiom}[Boundaries]\label{nca-boundary}
For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$.
@@ -339,7 +341,7 @@
compatible with the $\cS$ structure on $\cC_n(X; c)$.
-Given two hemispheres (a `domain' and `range') that agree on the equator, we need to be able to
+Given two hemispheres (a ``domain" and ``range") that agree on the equator, we need to be able to
assemble them into a boundary value of the entire sphere.
\begin{lem}
@@ -374,9 +376,9 @@
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions
to the intersection of the boundaries of $B$ and $B_i$.
If $k < n$,
-or if $k=n$ and we are in the $A_\infty$ case \nn{Kevin: remind me why we ask this?},
+or if $k=n$ and we are in the $A_\infty$ case,
we require that $\gl_Y$ is injective.
-(For $k=n$ in the isotopy $n$-category case, see below. \nn{where?})
+(For $k=n$ in the plain $n$-category case, see Axiom \ref{axiom:extended-isotopies}.)
\end{axiom}
\begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity}
@@ -385,7 +387,12 @@
$$\bigsqcup B_i \to B,$$
any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result.
\end{axiom}
-This axiom is only reasonable because the definition assigns a set to every ball; any identifications would limit the extent to which we can demand associativity.
+%This axiom is only reasonable because the definition assigns a set to every ball;
+%any identifications would limit the extent to which we can demand associativity.
+%%%% KW: It took me quite a while figure out what you [or I??] meant by the above, so I'm attempting a rewrite.
+Note that even though our $n$-categories are ``weak" in the traditional sense, we can require
+strict associativity because we have more morphisms (cf.\ discussion of Moore loops above).
+
For the next axiom, a \emph{pinched product} is a map locally modeled on a degeneracy map between simplices.
\begin{axiom}[Product (identity) morphisms]
\label{axiom:product}
@@ -457,7 +464,7 @@
to the identity on the boundary.
-\begin{axiom}[\textup{\textbf{[for isotopy $n$-categories]}} Extended isotopy invariance in dimension $n$.]
+\begin{axiom}[\textup{\textbf{[for plain $n$-categories]}} Extended isotopy invariance in dimension $n$.]
\label{axiom:extended-isotopies}
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
to the identity on $\bd X$ and isotopic (rel boundary) to the identity.
@@ -485,7 +492,7 @@
a diagram like the one in Theorem \ref{thm:CH} commutes.
\end{axiom}
-\subsection{Example (the fundamental $n$-groupoid)}
+\subsection{Example (the fundamental $n$-groupoid)} \mbox{}
We will define $\pi_{\le n}(T)$, the fundamental $n$-groupoid of a topological space $T$.
When $X$ is a $k$-ball with $k<n$, define $\pi_{\le n}(T)(X)$
to be the set of continuous maps from $X$ to $T$.
@@ -499,15 +506,15 @@
(if we are enriching over spaces) or the singular chains on that space (if we are enriching over chain complexes).
-\subsection{Example (string diagrams)}
-Fix a `traditional' $n$-category $C$ with strong duality (e.g.\ a pivotal 2-category).
+\subsection{Example (string diagrams)} \mbox{}
+Fix a ``traditional" $n$-category $C$ with strong duality (e.g.\ a pivotal 2-category).
Let $X$ be a $k$-ball and define $\cS_C(X)$ to be the set of $C$ string diagrams drawn on $X$;
that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$.
If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$.
Boundary restrictions and gluing are again straightforward to define.
Define product morphisms via product cell decompositions.
-\subsection{Example (bordism)}
+\subsection{Example (bordism)} \mbox{}
When $X$ is a $k$-ball with $k<n$, $\Bord^n(X)$ is the set of all $k$-dimensional
submanifolds $W$ in $X\times \bbR^\infty$ which project to $X$ transversely
to $\bd X$.
@@ -561,37 +568,38 @@
\psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl
\end{equation*}
where the restrictions to the various pieces of shared boundaries amongst the balls
-$X_a$ all agree (this is a fibered product of all the labels of $k$-balls over the labels of $k-1$-manifolds).
-When $k=n$, the `subset' and `product' in the above formula should be
+$X_a$ all agree (similar to a fibered product).
+When $k=n$, the ``subset" and ``product" in the above formula should be
interpreted in the appropriate enriching category.
If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$.
\end{defn}
-We will use the term `field on $W$' to refer to a point of this functor,
-that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$.
+%We will use the term ``field on $W$" to refer to a point of this functor,
+%that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$.
\subsubsection{Colimits}
Recall that our definition of an $n$-category is essentially a collection of functors
-defined on the categories of homeomorphisms $k$-balls
+defined on the categories of homeomorphisms of $k$-balls
for $k \leq n$ satisfying certain axioms.
It is natural to hope to extend such functors to the
larger categories of all $k$-manifolds (again, with homeomorphisms).
In fact, the axioms stated above already require such an extension to $k$-spheres for $k<n$.
The natural construction achieving this is a colimit along the poset of permissible decompositions.
-For an isotopy $n$-category $\cC$,
-we will denote the extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$,
+Given a plain $n$-category $\cC$,
+we will denote its extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$,
this is defined to be the colimit along $\cell(W)$ of the functor $\psi_{\cC;W}$.
Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity}
imply that $\cl{\cC}(X) \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$.
-Recall that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball,
-the set $\cC(X;c)$ is a vector space. Using this, we note that for $c \in \cl{\cC}(\bdy W)$,
-for $W$ an arbitrary $n$-manifold, the set $\cl{\cC}(W;c) = \bdy^{-1} (c)$ inherits the structure of a vector space.
+Suppose that $\cC$ is enriched in vector spaces: this means that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball,
+the set $\cC(X;c)$ is a vector space.
+In this case, for $W$ an arbitrary $n$-manifold and $c \in \cl{\cC}(\bdy W)$,
+the set $\cl{\cC}(W;c) = \bdy^{-1} (c)$ inherits the structure of a vector space.
These are the usual TQFT skein module invariants on $n$-manifolds.
We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$
-with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$
+with coefficients in the $n$-category $\cC$ as the {\it homotopy} colimit along $\cell(W)$
of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$.
An explicit realization of the homotopy colimit is provided by the simplices of the
@@ -607,15 +615,16 @@
product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs,
cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root,
and taking product identifies the roots of several trees.
-The `local homotopy colimit' is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$.
+The ``local homotopy colimit" is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$.
+We further require that all (compositions of) morphisms in a directed tree are not expressible as a product.
The differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required.
-A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. In the local homotopy colimit we can further require that any composition of morphisms in a directed tree is not expressible as a product.
+A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization.
%When $\cC$ is a topological $n$-category,
%the flexibility available in the construction of a homotopy colimit allows
%us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$.
%\todo{either need to explain why this is the same, or significantly rewrite this section}
-When $\cC$ is the isotopy $n$-category based on string diagrams for a traditional
+When $\cC$ is the plain $n$-category based on string diagrams for a traditional
$n$-category $C$,
one can show \cite{1009.5025} that the above two constructions of the homotopy colimit
are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; \cC)$.
@@ -628,21 +637,36 @@
We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible}
if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that
each $B_i$ appears as a connected component of one of the $M_j$.
-Note that this allows the balls to be pairwise either disjoint or nested.
+Note that this forces the balls to be pairwise either disjoint or nested.
Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$.
-These pieces need not be manifolds, but they do automatically have permissible decompositions.
+These pieces need not be manifolds,
+but they can be further subdivided into pieces which are manifolds
+and which fit into a permissible decomposition of $W$.
-The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of
+The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams.
+A $k$-blob diagram consists of
\begin{itemize}
-\item a permissible collection of $k$ embedded balls, and
-\item for each resulting piece of $W$, a field,
+ \item a permissible collection of $k$ embedded balls, and
+ \item a linear combination $s$ of string diagrams on $W$,
\end{itemize}
-such that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$.
-We call such a field a `null field on $B$'.
+such that
+\begin{itemize}
+ \item there is a permissible decomposition of $W$, compatible with the $k$ blobs, such that
+ $s$ is the product of linear combinations of string diagrams $s_i$ on the initial pieces $X_i$ of the decomposition
+ (for fixed restrictions to the boundaries of the pieces),
+ \item the $s_i$'s corresponding to innermost blobs evaluate to zero in $\cC$, and
+ \item the $s_i$'s corresponding to the other pieces are single string diagrams (linear combinations with only one term).
+\end{itemize}
+%that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$.
+\nn{yech}
+We call such linear combinations which evaluate to zero on a blob $B$ a ``null field on $B$".
The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs.
-We now spell this out for some small values of $k$. For $k=0$, the $0$-blob group is simply fields on $W$.
+\nn{KW: I have not finished changng terminology from ``field" to ``string diagram"}
+
+We now spell this out for some small values of $k$.
+For $k=0$, the $0$-blob group is simply linear combinations of string diagrams on $W$.
For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field to that ball is a null field.
The differential simply forgets the ball.
Thus we see that $H_0$ of the blob complex is the quotient of fields by fields which are null on some ball.
@@ -723,7 +747,7 @@
\begin{thm}[Skein modules]
\label{thm:skein-modules}
-Suppose $\cC$ is an isotopy $n$-category
+Suppose $\cC$ is a plain $n$-category.
The $0$-th blob homology of $X$ is the usual
(dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$
by $\cC$.
@@ -757,7 +781,7 @@
This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data.
Note that there is no restriction on the connectivity of $T$ as there is for
the corresponding result in topological chiral homology \cite[Theorem 3.8.6]{0911.0018}.
-The result was proved in \cite[\S 7.3]{1009.5025}.
+The result is proved in \cite[\S 7.3]{1009.5025}.
\subsection{Structure of the blob complex}
\label{sec:structure}
@@ -808,7 +832,7 @@
family of homeomorphisms can be localized to at most $k$ small sets.
With this alternate version in hand, the theorem is straightforward.
-By functoriality (Property \cite{property:functoriality}) $Homeo(X)$ acts on the set $BD_j(X)$ of $j$-blob diagrams, and this
+By functoriality (Property \ref{property:functoriality}) $\Homeo(X)$ acts on the set $BD_j(X)$ of $j$-blob diagrams, and this
induces a chain map $\CH{X}\tensor C_*(BD_j(X))\to C_*(BD_j(X))$
and hence a map $e_X: \CH{X} \tensor \cB\cT_*(X) \to \cB\cT_*(X)$.
It is easy to check that $e_X$ thus defined has the desired properties.
@@ -832,7 +856,7 @@
This result is described in more detail as Example 6.2.8 of \cite{1009.5025}.
Fix a topological $n$-category $\cC$, which we'll now omit from notation.
-Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category.
+From the above, associated to any $(n{-}1)$-manifold $Y$ is an $A_\infty$ category $\bc_*(Y)$.
\begin{thm}[Gluing formula]
\label{thm:gluing}
@@ -843,7 +867,7 @@
$A_\infty$ module for $\bc_*(Y)$.
\item The blob complex of a glued manifold $X\bigcup_Y \selfarrow$ is the $A_\infty$ self-tensor product of
-$\bc_*(X)$ as an $\bc_*(Y)$-bimodule:
+$\bc_*(X)$ as a $\bc_*(Y)$-bimodule:
\begin{equation*}
\bc_*(X\bigcup_Y \selfarrow) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow
\end{equation*}
@@ -858,7 +882,7 @@
There is a tautological map from the 0-simplices of $T_*$ to $\bc_*(X\bigcup_Y \selfarrow)$,
and this map can be extended to a chain map on all of $T_*$ by sending the higher simplices to zero.
Constructing a homotopy inverse to this natural map involves making various choices, but one can show that the
-choices form contractible subcomplexes and apply the theory of acyclic models.
+choices form contractible subcomplexes and apply the acyclic models theorem.
\end{proof}
We next describe the blob complex for product manifolds, in terms of the
@@ -867,12 +891,14 @@
\begin{thm}[Product formula]
\label{thm:product}
Let $W$ be a $k$-manifold and $Y$ be an $n{-}k$ manifold.
-Let $\cC$ be an isotopy $n$-category.
+Let $\cC$ be a plain $n$-category.
Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above.
Then
\[
\bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W).
\]
+That is, the blob complex of $Y\times W$ with coefficients in $\cC$ is homotopy equivalent
+to the blob complex of $W$ with coefficients in $\bc_*(Y;\cC)$.
\end{thm}
The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps
(see \cite[\S7.1]{1009.5025}).
@@ -895,7 +921,7 @@
%\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.}
-\section{Deligne conjecture for $n$-categories}
+\section{Extending Deligne's conjecture to $n$-categories}
\label{sec:applications}
Let $M$ and $N$ be $n$-manifolds with common boundary $E$.
@@ -905,12 +931,12 @@
from $\bc_*(M)$ to $\bc_*(N)$.
Let $R$ be another $n$-manifold with boundary $E^\text{op}$.
There is a chain map
-\[
+\begin{equation*}
\hom_A(\bc_*(M), \bc_*(N)) \ot \bc_*(M) \ot_A \bc_*(R) \to \bc_*(N) \ot_A \bc_*(R) .
-\]
+\end{equation*}
We think of this map as being associated to a surgery which cuts $M$ out of $M\cup_E R$ and
replaces it with $N$, yielding $N\cup_E R$.
-(This is a more general notion of surgery that usual --- $M$ and $N$ can be any manifolds
+(This is a more general notion of surgery that usual: $M$ and $N$ can be any manifolds
which share a common boundary.)
In analogy to Hochschild cochains, we will call elements of $\hom_A(\bc_*(M), \bc_*(N))$ ``blob cochains".
@@ -919,7 +945,7 @@
An $n$-dimensional surgery cylinder is
defined to be a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}),
modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism.
-One can associated to this data an $(n{+}1)$-manifold with a foliation by intervals,
+One can associate to this data an $(n{+}1)$-manifold with a foliation by intervals,
and the relations we impose correspond to homeomorphisms of the $(n{+}1)$-manifolds
which preserve the foliation.
@@ -940,10 +966,10 @@
\end{multline*}
which satisfy the operad compatibility conditions.
-\begin{proof}
+\begin{proof} (Sketch.)
We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation},
and the action of surgeries is just composition of maps of $A_\infty$-modules.
-We only need to check that the relations of the $n$-SC operad are satisfied.
+We only need to check that the relations of the surgery cylinder operad are satisfied.
This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity.
\end{proof}