blob: comparison blob1.tex

equal deleted inserted replaced

-:1b441b3595c9
+:8599e156a169
 \usepackage{amsmath,amssymb,amsthm}
 \usepackage[all]{xy}
-% test edit #2
+% test edit #3
 %%%%% excerpts from my include file of standard macros
 \def\bc{{\cal B}}
 If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels
 of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the
 interior of $S$, each transversely oriented and each labeled by an element (1-morphism)
 of the algebra.
-For $n=2$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with
+\medskip
+For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories)
+that are common in the literature.
+We describe these carefully here.
+A field on a 0-manifold $P$ is a labeling of each point of $P$ with
 an object of the 2-category $C$.
 A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$.
 A field on a 2-manifold $Y$ consists of
 \begin{itemize}
 	\item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such
 that each component of the complement is homeomorphic to a disk);
-	\item a labeling of each 2-cell (and each half 2-cell adjacent to $\bd Y$)
+	\item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$)
 by a 0-morphism of $C$;
 	\item a transverse orientation of each 1-cell, thought of as a choice of
 ``domain" and ``range" for the two adjacent 2-cells;
 	\item a labeling of each 1-cell by a 1-morphism of $C$, with
 domain and range determined by the transverse orientation of the 1-cell
 to the boundary of the standard $(k-j)$-dimensional bihedron; and
 	\item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with
 domain and range determined by the labelings of the link of $j$-cell.
 \end{itemize}
-\nn{next definition might need some work; I think linearity relations should
+%\nn{next definition might need some work; I think linearity relations should
-be treated differently (segregated) from other local relations, but I'm not sure
+%be treated differently (segregated) from other local relations, but I'm not sure
-the next definition is the best way to do it}
+%the next definition is the best way to do it}
+\medskip
 For top dimensional ($n$-dimensional) manifolds, we're actually interested
 in the linearized space of fields.
 By default, define $\cC_l(X) = \c[\cC(X)]$; that is, $\cC_l(X)$ is
 the vector space of finite
 Let $B^n$ denote the standard $n$-ball.
 A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$
 (for all $c \in \cC(\bd B^n)$) satisfying the following (three?) properties.
-\nn{implies (extended?) isotopy; stable under gluing; open covers?; ...}
+\nn{Roughly, these are (1) the local relations imply (extended) isotopy;
+(2) $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing; and
+(3) this ideal is generated by ``small" generators (contained in an open cover of $B^n$).
+See KW TQFT notes for details.  Need to transfer details to here.}
 For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$,
 where $a$ and $b$ are maps (fields) which are homotopic rel boundary.
 For $n$-category pictures, $U(B^n; c)$ is equal to the kernel of the evaluation map
 \nn{maybe examples of local relations before general def?}
 Note that the $Y$ is an $n$-manifold which is merely homeomorphic to the standard $B^n$,
 then any homeomorphism $B^n \to Y$ induces the same local subspaces for $Y$.
 We'll denote these by $U(Y; c) \sub \cC_l(Y; c)$, $c \in \cC(\bd Y)$.
+\nn{Is this true in high (smooth) dimensions?  Self-diffeomorphisms of $B^n$
+rel boundary might not be isotopic to the identity.  OK for PL and TOP?}
 Given a system of fields and local relations, we define the skein space
 $A(Y^n; c)$ to be the space of all finite linear combinations of fields on
 the $n$-manifold $Y$ modulo local relations.
 The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations
 combination of fields on $X$ obtained by gluing $r$ to $u$.
 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by
 just erasing the blob from the picture
 (but keeping the blob label $u$).
-Note that the skein module $A(X)$
+Note that the skein space $A(X)$
 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.
 $\bc_2(X)$ is the space of all relations (redundancies) among the relations of $\bc_1(X)$.
 More specifically, $\bc_2(X)$ is the space of all finite linear combinations of
 2-blob diagrams (defined below), modulo the usual linear label relations.
 	x = \sign(\pi) x' .
 }
 (Alert readers will have noticed that for $k=2$ our definition
 of $\bc_k(X)$ is slightly different from the previous definition
-of $\bc_2(X)$.
+of $\bc_2(X)$ --- we did not impose the reordering relations.
 The general definition takes precedence;
 the earlier definition was simplified for purposes of exposition.)
 The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows.
 Let $b = (\{B_i\}, r, \{u_j\})$ be a $k$-blob diagram.
 Thus we have a chain complex.
 \nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)}
-\nn{TO DO: ((?)) allow $n$-morphisms to be chain complex instead of just
+\nn{TO DO:
-a vector space; relations to Chas-Sullivan string stuff}
+expand definition to handle DGA and $A_\infty$ versions of $n$-categories;
+relations to Chas-Sullivan string stuff}
 \section{Basic properties of the blob complex}
 There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$.
 \qed
 \end{prop}
+% oops -- duplicate
+%\begin{prop} \label{functorialprop}
+%The assignment $X \mapsto \bc_*(X)$ extends to a functor from the category of
+%$n$-manifolds and homeomorphisms to the category of chain complexes and linear isomorphisms.
+%\end{prop}
+%\begin{proof}
+%Obvious.
+%\end{proof}
+%\nn{need to same something about boundaries and boundary conditions above.
+%maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.}
 \begin{prop}
 For fixed fields ($n$-cat), $\bc_*$ is a functor from the category
 of $n$-manifolds and diffeomorphisms to the category of chain complexes and
 (chain map) isomorphisms.
 \qed
 \end{prop}
+\nn{need to same something about boundaries and boundary conditions above.
+maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.}
 In particular,
 \begin{prop}  \label{diff0prop}
 There is an action of $\Diff(X)$ on $\bc_*(X)$.

changeset 4	8599e156a169
parent 3	1b441b3595c9
child 5	61751866cf69