--- a/pnas/pnas.tex	Tue Nov 16 16:55:55 2010 -0800
+++ b/pnas/pnas.tex	Wed Nov 17 09:17:07 2010 -0800
@@ -162,46 +162,41 @@
 
 For our purposes, an $n{+}1$-dimensional TQFT is a locally defined system of
 invariants of manifolds of dimensions 0 through $n+1$.
-
-
-\dropcap{T}opological quantum field theories (TQFTs) provide local invariants of manifolds, which are determined by the algebraic data of a higher category.
+The TQFT invariant $A(Y)$ of a closed $k$-manifold $Y$ is a linear $(n{-}k)$-category.
+If $Y$ has boundary then $A(Y)$ is a collection of $(n{-}k)$-categories which afford
+a representation of the $(n{-}k{+}1)$-category $A(\bd Y)$.
+(See \cite{1009.5025} and \cite{kw:tqft};
+for a more homotopy-theoretic point of view see \cite{0905.0465}.)
 
-An $n+1$-dimensional TQFT $\cA$ associates a vector space $\cA(M)$
-(or more generally, some object in a specified symmetric monoidal category)
-to each $n$-dimensional manifold $M$, and a linear map
-$\cA(W): \cA(M_0) \to \cA(M_1)$ to each $n+1$-dimensional manifold $W$
-with incoming boundary $M_0$ and outgoing boundary $M_1$.
-An $n+\epsilon$-dimensional TQFT provides slightly less;
-it only assigns linear maps to mapping cylinders.
-
-There is a standard formalism for constructing an $n+\epsilon$-dimensional
-TQFT from any $n$-category with sufficiently strong duality,
-and with a further finiteness condition this TQFT is in fact $n+1$-dimensional.
-\nn{not so standard, err}
+We now comment on some particular values of $k$ above.
+By convention, a linear 0-category is a vector space, and a representation
+of a vector space is an element of the dual space.
+So a TQFT assigns to each closed $n$-manifold $Y$ a vector space $A(Y)$,
+and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$.
+In fact we will be mainly be interested in so-called $(n{+}\epsilon)$-dimensional
+TQFTs which have nothing to say about $(n{+}1)$-manifolds.
+For the remainder of this paper we assume this case.
 
-These invariants are local in the following sense.
-The vector space $\cA(Y \times I)$, for $Y$ an $n-1$-manifold,
-naturally has the structure of a category, with composition given by the gluing map
-$I \sqcup I \to I$. Moreover, the vector space $\cA(Y \times I^k)$,
-for $Y$ and $n-k$-manifold, has the structure of a $k$-category.
-The original $n$-category can be recovered as $\cA(I^n)$.
-For the rest of the paragraph, we implicitly drop the factors of $I$.
-(So for example the original $n$-category is associated to the point.)
-If $Y$ contains $Z$ as a codimension $0$ submanifold of its boundary,
-then $\cA(Y)$ is natually a module over $\cA(Z)$. For any $k$-manifold
-$Y = Y_1 \cup_Z Y_2$, where $Z$ is a $k-1$-manifold, the category
-$\cA(Y)$ can be calculated via a gluing formula,
-$$\cA(Y) = \cA(Y_1) \Tensor_{\cA(Z)} \cA(Y_2).$$
+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 gluing rule for the TQFT 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 $S$.
+
+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 constructed of $A(pt)$
+via colimits (see below).
 
-In fact, recent work of Lurie on the `cobordism hypothesis' \cite{0905.0465}
-shows that all invariants of $n$-manifolds satisfying a certain related locality property
-are in a sense TQFT invariants, and in particular determined by
-a `fully dualizable object' in some $n+1$-category.
-(The discussion above begins with an object in the $n+1$-category of $n$-categories.
-The `sufficiently strong duality' mentioned above corresponds roughly to `fully dualizable'.)
+We call a TQFT semisimple if $A(S)$ is a semisimple 1-category for all $(n{-}1)$-manifolds $S$
+and $A(Y)$ is a finite-dimensional vector space for all $n$-manifolds $Y$.
+Examples of semisimple TQFTS include Witten-Reshetikhin-Turaev theories, 
+Turaev-Viro theories, and Dijkgraaf-Witten theories.
+These can all be given satisfactory accounts in the framework outlined above.
+(The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories in order to be
+extended all the way down to 0 dimensions.)
 
-This formalism successfully captures Turaev-Viro and Reshetikhin-Turaev invariants
-(and indeed invariants based on semisimple categories).
+For other TQFT-like invariants, however, the above framework seems to be inadequate.
+
 However new invariants on manifolds, particularly those coming from
 Seiberg-Witten theory and Ozsv\'{a}th-Szab\'{o} theory, do not fit the framework well.
 In particular, they have more complicated gluing formulas, involving derived or
@@ -228,6 +223,8 @@
 \nn{perhaps say something explicit about the relationship of this paper to big blob paper.
 like: in this paper we try to give a clear view of the big picture without getting bogged down in details}
 
+\nn{diff w/ lurie}
+
 \section{Definitions}
 \subsection{$n$-categories} \mbox{}
 
@@ -532,7 +529,7 @@
 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$.
 
 
-\subsubsection{Homotopy colimits}
+\subsubsection{Colimits}
 \nn{Motivation: How can we extend an $n$-category from balls to arbitrary manifolds?}
 \nn{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?}
 \nn{Explain codimension colimits here too}