# HG changeset patch # User Kevin Walker # Date 1275664508 25200 # Node ID 9698f584e732cfb4e6621d5d0a007a7b32275cdd # Parent adc0780aa5e7ecbf1560bc218247b0c3146f42df starting to revise the ancient TQFTs-from-fields section; other minor stuff diff -r adc0780aa5e7 -r 9698f584e732 blob1.tex --- a/blob1.tex Thu Jun 03 23:08:47 2010 -0700 +++ b/blob1.tex Fri Jun 04 08:15:08 2010 -0700 @@ -44,8 +44,13 @@ \item[B] (S) look at this, decide what to keep \item Work in the references Chris Douglas gave us on the classification of local field theories, \cite{BDH-seminar,DSP-seminar,schommer-pries-thesis,0905.0465}. -\item Make clear exactly what counts as a "blob diagram", and search for -"blob diagram" +\nn{KW: Do we need to do this? We don't really classify field theories. +I suppose our work could be interpreted as a alternative proof of cobordism hypothesis, but we +don't emphasize that at the moment. +On the other hand, I'm happy to do Chris a favor by citing this stuff.} + +\item Make clear exactly what counts as a ``blob diagram", and search for +``blob diagram" \item Say something about stabilizing an $n$-category (centre), taking the top $k$ levels of a category, and the stabilization hypothesis? diff -r adc0780aa5e7 -r 9698f584e732 text/ncat.tex --- a/text/ncat.tex Thu Jun 03 23:08:47 2010 -0700 +++ b/text/ncat.tex Fri Jun 04 08:15:08 2010 -0700 @@ -7,6 +7,7 @@ \label{sec:ncats} \subsection{Definition of $n$-categories} +\label{ss:n-cat-def} Before proceeding, we need more appropriate definitions of $n$-categories, $A_\infty$ $n$-categories, modules for these, and tensor products of these modules. @@ -536,7 +537,7 @@ Given a `traditional $n$-category with strong duality' $C$ define $\cC(X)$, for $X$ a $k$-ball with $k < n$, to be the set of all $C$-labeled sub cell complexes of $X$ (c.f. \S \ref{sec:fields}). -For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X)$ to finite linear +For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X; c)$ to be finite linear combinations of $C$-labeled sub cell complexes of $X$ modulo the kernel of the evaluation map. Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, diff -r adc0780aa5e7 -r 9698f584e732 text/tqftreview.tex --- a/text/tqftreview.tex Thu Jun 03 23:08:47 2010 -0700 +++ b/text/tqftreview.tex Fri Jun 04 08:15:08 2010 -0700 @@ -45,9 +45,11 @@ \end{example} Now for the rest of the definition of system of fields. +(Readers desiring a more precise definition should refer to Subsection \ref{ss:n-cat-def} +and replace $k$-balls with $k$-manifolds.) \begin{enumerate} \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, -and these maps are a natural +and these maps comprise a natural transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of $\cC(X)$ which restricts to $c$. @@ -55,13 +57,13 @@ \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, restriction to boundary, and orientation reversal. +compatibly with homeomorphisms and restriction to boundary. We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ restriction maps. \item Gluing without corners. -Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. -Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. -Using the boundary restriction, disjoint union, and (in one case) orientation reversal +Let $\bd X = Y \du Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. +Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$. +Using the boundary restriction and disjoint union maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two copies of $Y$ in $\bd X$. Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps. @@ -70,15 +72,15 @@ \Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) , \] and this gluing map is compatible with all of the above structure (actions -of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). +of homeomorphisms, boundary restrictions, disjoint union). Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, the gluing map is surjective. -From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the -gluing surface, we say that fields in the image of the gluing map +We say that fields on $X\sgl$ in the image of the gluing map are transverse to $Y$ or splittable along $Y$. \item Gluing with corners. -Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries. -Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. +Let $\bd X = Y \cup Y \cup W$, where the two copies of $Y$ and +$W$ might intersect along their boundaries. +Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$. Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself (without corners) along two copies of $\bd Y$. Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a splittable field on $W\sgl$ and let @@ -97,8 +99,7 @@ of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, the gluing map is surjective. -From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the -gluing surface, we say that fields in the image of the gluing map +We say that fields in the image of the gluing map are transverse to $Y$ or splittable along $Y$. \item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted $c \mapsto c\times I$. @@ -137,8 +138,8 @@ on $M$ generated by isotopy plus all instance of the above construction (for all appropriate $Y$ and $x$). -\nn{should also say something about pseudo-isotopy} - +\nn{the following discussion of linearizing fields is kind of lame. +maybe just assume things are already linearized.} \nn{remark that if top dimensional fields are not already linear then we will soon linearize them(?)} @@ -188,7 +189,7 @@ 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, disklike $n$-category), +(e.g. a pivotal 2-category, 1-category with duals, star 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 $C$.