# HG changeset patch # User Kevin Walker # Date 1278184755 21600 # Node ID 98b8559b0b7ae0dbcbf8e789dd8f878e4927c594 # Parent 14e3124a48e8ad5b723b395ebaec86071fe25159 starting to work on tqdftreview.tex diff -r 14e3124a48e8 -r 98b8559b0b7a text/intro.tex --- a/text/intro.tex Wed Jun 30 08:55:46 2010 -0700 +++ b/text/intro.tex Sat Jul 03 13:19:15 2010 -0600 @@ -445,7 +445,19 @@ \subsection{Thanks and acknowledgements} -We'd like to thank David Ben-Zvi, Kevin Costello, Chris Douglas, -Michael Freedman, Vaughan Jones, Alexander Kirillov, Justin Roberts, Chris Schommer-Pries, Peter Teichner, Thomas Tradler \nn{and who else?} for many interesting and useful conversations. +% attempting to make this chronological rather than alphabetical +We'd like to thank +Justin Roberts, +Michael Freedman, +Peter Teichner, +David Ben-Zvi, +Vaughan Jones, +Chris Schommer-Pries, +Thomas Tradler, +Kevin Costello, +Chris Douglas, +and +Alexander Kirillov +for many interesting and useful conversations. During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. diff -r 14e3124a48e8 -r 98b8559b0b7a text/ncat.tex --- a/text/ncat.tex Wed Jun 30 08:55:46 2010 -0700 +++ b/text/ncat.tex Sat Jul 03 13:19:15 2010 -0600 @@ -2073,6 +2073,7 @@ \end{figure} Invariance under this movie move follows from the compatibility of the inner product for $B_1\cup B_2$ with the inner products for $B_1$ and $B_2$. +\nn{should also say something about locality/distant-commutativity} If $n\ge 2$, these two movie move suffice: diff -r 14e3124a48e8 -r 98b8559b0b7a text/tqftreview.tex --- a/text/tqftreview.tex Wed Jun 30 08:55:46 2010 -0700 +++ b/text/tqftreview.tex Sat Jul 03 13:19:15 2010 -0600 @@ -4,16 +4,31 @@ \label{sec:fields} \label{sec:tqftsviafields} -In this section we review the notion of a ``system of fields and local relations". +In this section we review the construction of TQFTs from 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}. +For our purposes, a TQFT is {\it defined} to be something which arises +from this construction. +This is an alternative to the more common definition of a TQFT +as a functor on cobordism categories satisfying various conditions. +A fully local (``down to points") version of the cobordism-functor TQFT definition +should be equivalent to the fields-and-local-relations definition. + 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. +In one direction, Example \ref{ex:traditional-n-categories(fields)} +shows how to construct a system of fields from a (traditional) $n$-category. +We do this in detail for $n=1,2$ (Subsection \ref{sec:example:traditional-n-categories(fields)}) +and more informally for general $n$. +In the other direction, +our preferred definition of an $n$-category in Section \ref{sec:ncats} is essentially +just a system of fields restricted to balls of dimensions 0 through $n$; +one could call this the ``local" part of a system of fields. + +Since this section is intended primarily to motivate +the blob complex construction of Section \ref{sec:blob-definition}, +we suppress some technical details. +In Section \ref{sec:ncats} the analogous details are treated more carefully. + +\medskip 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 @@ -26,11 +41,12 @@ unoriented PL manifolds of dimension $k$ and morphisms homeomorphisms. (We could equally well work with a different category of manifolds --- -oriented, topological, smooth, spin, etc. --- but for definiteness we +oriented, topological, smooth, spin, etc. --- but for simplicity 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 $\cS = \Vect$. +Fields on $n$-manifolds will be enriched over $\cS$. +Good examples to keep in mind are $\cS = \Set$ 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 $\cS$ @@ -64,10 +80,12 @@ 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 $\Kom$), +\item The subset $\cC_n(X;c)$ of top-dimensional 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 $\cS$. +Any maps mentioned below between fields on $n$-manifolds 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. @@ -86,22 +104,24 @@ \] and this gluing map is compatible with all of the above structure (actions of homeomorphisms, boundary restrictions, disjoint union). -Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, +Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity +and collaring maps, the gluing map is surjective. 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 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 $Y$. +Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$ +(Figure xxxx). 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 $c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. (This restriction map uses the gluing without corners map above.) -Using the boundary restriction, gluing without corners, and (in one case) orientation reversal -maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two +Using the boundary restriction and gluing without corners maps, +we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two copies of $Y$ in $\bd X$. Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps. Then (here's the axiom/definition part) there is an injective ``gluing" map @@ -109,12 +129,14 @@ \Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) , \] and this gluing map is compatible with all of the above structure (actions -of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). -Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, +of homeomorphisms, boundary restrictions, disjoint union). +Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity +and collaring maps, the gluing map is surjective. 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 +\item Product fields. +There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted $c \mapsto c\times I$. These maps comprise a natural transformation of functors, and commute appropriately with all the structure maps above (disjoint union, boundary restriction, etc.). @@ -136,9 +158,9 @@ \medskip -Using the functoriality and $\cdot\times I$ properties above, together -with boundary collar homeomorphisms of manifolds, we can define the notion of -{\it extended isotopy}. +Using the functoriality and product field properties above, together +with boundary collar homeomorphisms of manifolds, we can define +{\it collar maps} $\cC(M)\to \cC(M)$. Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold of $\bd M$. Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is splittable along $\bd Y$. @@ -146,10 +168,16 @@ Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$. Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. -Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$. -More generally, we define extended isotopy to be the equivalence relation on fields -on $M$ generated by isotopy plus all instance of the above construction -(for all appropriate $Y$ and $x$). +Then we call the map $x \mapsto f(x \bullet (c\times I))$ a {\it collar map}. +We call the equivalence relation generated by collar maps and +homeomorphisms isotopic to the identity {\it extended isotopy}, since the collar maps +can be thought of (informally) as the limit of homeomorphisms +which expand an infinitesimally thin collar neighborhood of $Y$ to a thicker +collar neighborhood. + + +% all this linearizing stuff is unnecessary, I think +\noop{ \nn{the following discussion of linearizing fields is kind of lame. maybe just assume things are already linearized.} @@ -195,6 +223,8 @@ We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the above tensor products. +} % end \noop + \subsection{Systems of fields from $n$-categories} \label{sec:example:traditional-n-categories(fields)}