blob: comparison text/tqftreview.tex

equal deleted inserted replaced

-:adc0780aa5e7
+:9698f584e732
 One can think of such sub-cell-complexes as dual to pasting diagrams for $C$.
 This is described in more detail in \S \ref{sec:example:traditional-n-categories(fields)}.
 \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 $c$.
 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 $\cC_k$ 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 $\bd X = Y \du Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds.
-Let $X\sgl$ denote $X$ glued to itself along $\pm Y$.
+Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$.
-Using the boundary restriction, disjoint union, and (in one case) orientation reversal
+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.
 Then (here's the axiom/definition part) there is an injective ``gluing" map
 \[
 	\Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) ,
 \]
 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
+We say that fields on $X\sgl$ in the image of the gluing map
-gluing surface, we say that fields 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 $\bd X = Y \cup Y \cup W$, where the two copies of $Y$ and
-Let $X\sgl$ denote $X$ glued to itself along $\pm Y$.
+$W$ 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
 $c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$.
 Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$.
 \]
 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,
 the gluing map is surjective.
-From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the
+We say that fields in the image of the gluing map
-gluing surface, 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
 $c \mapsto c\times I$.
 These maps comprise a natural transformation of functors, and commute appropriately
 with all the structure maps above (disjoint union, boundary restriction, etc.).
 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 $x$).
-\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(?)}
 For top dimensional ($n$-dimensional) manifolds, we're actually interested
 \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
 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 $C$.
 We'll spell this out for $n=1,2$ and then describe the general case.

changeset 339	9698f584e732
parent 329	eb03c4a92f98
child 340	f7da004e1f14