diff -r a3311a926113 -r c5a43be00ed4 text/definitions.tex --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/text/definitions.tex Wed Jul 22 03:38:13 2009 +0000 @@ -0,0 +1,471 @@ +%!TEX root = ../blob1.tex + +\section{Definitions} +\label{sec:definitions} + +\subsection{Systems of fields} +\label{sec:fields} + +Let $\cM_k$ denote the category (groupoid, in fact) with objects +oriented PL manifolds of dimension +$k$ and morphisms homeomorphisms. +(We could equally well work with a different category of manifolds --- +unoriented, topological, smooth, spin, etc. --- but for definiteness we +will stick with oriented PL.) + +Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$. + +A $n$-dimensional {\it system of fields} in $\cS$ +is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ +together with some additional data and satisfying some additional conditions, all specified below. + +\nn{refer somewhere to my TQFT notes \cite{kw:tqft}, and possibly also to paper with Chris} + +Before finishing the definition of fields, we give two motivating examples +(actually, families of examples) of systems of fields. + +The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps +from X to $B$. + +The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be +the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by +$j$-morphisms of $C$. +One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. +This is described in more detail below. + +Now for the rest of the definition of system of fields. +\begin{enumerate} +\item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, +and these maps are 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 There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps +again comprise a natural transformation of functors. +In addition, the orientation reversal maps are compatible with the boundary restriction maps. +\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. +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 +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). +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 +are transverse to $Y$ or cuttable 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 $\pm Y$. +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 cuttable 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 +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 +\[ + \Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) , +\] +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, +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 +are transverse to $Y$ or cuttable 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.). +Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism +covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. +\end{enumerate} + +\nn{need to introduce two notations for glued fields --- $x\bullet y$ and $x\sgl$} + +\bigskip +Using the functoriality and $\bullet\times I$ properties above, together +with boundary collar homeomorphisms of manifolds, we can define the notion of +{\it extended isotopy}. +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 cuttable along $\bd Y$. +Let $c$ be $x$ restricted to $Y$. +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 $x$). + +\nn{should also say something about pseudo-isotopy} + +%\bigskip +%\hrule +%\bigskip +% +%\input{text/fields.tex} +% +% +%\bigskip +%\hrule +%\bigskip + +\nn{note: probably will suppress from notation the distinction +between fields and their (orientation-reversal) duals} + +\nn{remark that if top dimensional fields are not already linear +then we will soon linearize them(?)} + +We now describe in more detail 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. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-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. + +If $X$ has boundary, we require that the cell decompositions are in general +position with respect to the boundary --- the boundary intersects each cell +transversely, so cells meeting the boundary are mere half-cells. + +Put another way, the cell decompositions we consider are dual to standard cell +decompositions of $X$. + +We will always assume that our $n$-categories have linear $n$-morphisms. + +For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with +an object (0-morphism) of the 1-category $C$. +A field on a 1-manifold $S$ consists of +\begin{itemize} + \item A cell decomposition of $S$ (equivalently, a finite collection +of points in the interior of $S$); + \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) +by an object (0-morphism) of $C$; + \item a transverse orientation of each 0-cell, thought of as a choice of +``domain" and ``range" for the two adjacent 1-cells; and + \item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with +domain and range determined by the transverse orientation and the labelings of the 1-cells. +\end{itemize} + +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. + +\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 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 +and the labelings of the 2-cells; + \item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood +of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped +to $\pm 1 \in S^1$; and + \item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range +determined by the labelings of the 1-cells and the parameterizations of the previous +bullet. +\end{itemize} +\nn{need to say this better; don't try to fit everything into the bulleted list} + +For general $n$, a field on a $k$-manifold $X^k$ consists of +\begin{itemize} + \item A cell decomposition of $X$; + \item an explicit general position homeomorphism from the link of each $j$-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 +%be treated differently (segregated) from other local relations, but I'm not sure +%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 $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is +the vector space of finite +linear combinations of fields on $X$. +If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$. +Thus the restriction (to boundary) maps are well defined because we never +take linear combinations of fields with differing boundary conditions. + +In some cases we don't linearize the default way; instead we take the +spaces $\lf(X; a)$ to be part of the data for the system of fields. +In particular, for fields based on linear $n$-category pictures we linearize as follows. +Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by +obvious relations on 0-cell labels. +More specifically, let $L$ be a cell decomposition of $X$ +and let $p$ be a 0-cell of $L$. +Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that +$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. +Then the subspace $K$ is generated by things of the form +$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader +to infer the meaning of $\alpha_{\lambda c + d}$. +Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. + +\nn{Maybe comment further: if there's a natural basis of morphisms, then no need; +will do something similar below; in general, whenever a label lives in a linear +space we do something like this; ? say something about tensor +product of all the linear label spaces? Yes:} + +For top dimensional ($n$-dimensional) manifolds, we linearize as follows. +Define an ``almost-field" to be a field without labels on the 0-cells. +(Recall that 0-cells are labeled by $n$-morphisms.) +To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism +space determined by the labeling of the link of the 0-cell. +(If the 0-cell were labeled, the label would live in this space.) +We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). +We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the +above tensor products. + + + +\subsection{Local relations} +\label{sec:local-relations} + + +A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$, +for all $n$-manifolds $B$ which are +homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, +satisfying the following properties. +\begin{enumerate} +\item functoriality: +$f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$ +\item local relations imply extended isotopy: +if $x, y \in \cC(B; c)$ and $x$ is extended isotopic +to $y$, then $x-y \in U(B; c)$. +\item ideal with respect to gluing: +if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$ +\end{enumerate} +See \cite{kw:tqft} for details. + + +For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$, +where $a$ and $b$ are maps (fields) which are homotopic rel boundary. + +For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map +$\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into +domain and range. + +\nn{maybe examples of local relations before general def?} + +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 +is defined to be the dual of $A(Y; c)$. +(See \cite{kw:tqft} or xxxx for details.) + +\nn{should expand above paragraph} + +The blob complex is in some sense the derived version of $A(Y; c)$. + + + +\subsection{The blob complex} +\label{sec:blob-definition} + +Let $X$ be an $n$-manifold. +Assume a fixed system of fields and local relations. +In this section we will usually suppress boundary conditions on $X$ from the notation +(e.g. write $\lf(X)$ instead of $\lf(X; c)$). + +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 +$\overline{X \setmin Y}$. + +We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. + +Define $\bc_0(X) = \lf(X)$. +(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. +We'll omit this sort of detail in the rest of this section.) +In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. + +$\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$. +Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear +combinations of 1-blob diagrams, where a 1-blob diagram to consists of +\begin{itemize} +\item An embedded closed ball (``blob") $B \sub X$. +\item A field $r \in \cC(X \setmin B; c)$ +(for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). +\item A local relation field $u \in U(B; c)$ +(same $c$ as previous bullet). +\end{itemize} +In order to get the linear structure correct, we (officially) define +\[ + \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . +\] +The first direct sum is indexed by all blobs $B\subset X$, and the second +by all boundary conditions $c \in \cC(\bd B)$. +Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$. + +Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by +\[ + (B, u, r) \mapsto u\bullet r, +\] +where $u\bullet r$ denotes the linear +combination of fields on $X$ obtained by gluing $u$ to $r$. +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 space $A(X)$ +is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. + +$\bc_2(X)$ is, roughly, the space of all relations (redundancies) among the +local relations encoded in $\bc_1(X)$. +More specifically, $\bc_2(X)$ is the space of all finite linear combinations of +2-blob diagrams, of which there are two types, disjoint and nested. + +A disjoint 2-blob diagram consists of +\begin{itemize} +\item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors. +\item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ +(where $c_i \in \cC(\bd B_i)$). +\item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. +\end{itemize} +We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; +reversing the order of the blobs changes the sign. +Define $\bd(B_0, B_1, u_0, u_1, r) = +(B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$. +In other words, the boundary of a disjoint 2-blob diagram +is the sum (with alternating signs) +of the two ways of erasing one of the blobs. +It's easy to check that $\bd^2 = 0$. + +A nested 2-blob diagram consists of +\begin{itemize} +\item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. +\item A field $r \in \cC(X \setmin B_0; c_0)$ +(for some $c_0 \in \cC(\bd B_0)$), which is cuttable along $\bd B_1$. +\item A local relation field $u_0 \in U(B_0; c_0)$. +\end{itemize} +Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ +(for some $c_1 \in \cC(B_1)$) and +$r' \in \cC(X \setmin B_1; c_1)$. +Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$. +Note that the requirement that +local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$. +As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating +sum of the two ways of erasing one of the blobs. +If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$. +It is again easy to check that $\bd^2 = 0$. + +\nn{should draw figures for 1, 2 and $k$-blob diagrams} + +As with the 1-blob diagrams, in order to get the linear structure correct it is better to define +(officially) +\begin{eqnarray*} + \bc_2(X) & \deq & + \left( + \bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1} + U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1) + \right) \\ + && \bigoplus \left( + \bigoplus_{B_0 \subset B_1} \bigoplus_{c_0} + U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0) + \right) . +\end{eqnarray*} +The final $\lf(X\setmin B_0; c_0)$ above really means fields cuttable along $\bd B_1$, +but we didn't feel like introducing a notation for that. +For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign +(rather than a new, linearly independent 2-blob diagram). + +Now for the general case. +A $k$-blob diagram consists of +\begin{itemize} +\item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. +For each $i$ and $j$, we require that either $B_i$ and $B_j$have disjoint interiors or +$B_i \sub B_j$ or $B_j \sub B_i$. +(The case $B_i = B_j$ is allowed. +If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) +If a blob has no other blobs strictly contained in it, we call it a twig blob. +\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. +(These are implied by the data in the next bullets, so we usually +suppress them from the notation.) +$c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ +if the latter space is not empty. +\item A field $r \in \cC(X \setmin B^t; c^t)$, +where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ +is determined by the $c_i$'s. +$r$ is required to be cuttable along the boundaries of all blobs, twigs or not. +\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, +where $c_j$ is the restriction of $c^t$ to $\bd B_j$. +If $B_i = B_j$ then $u_i = u_j$. +\end{itemize} + +If two blob diagrams $D_1$ and $D_2$ +differ only by a reordering of the blobs, then we identify +$D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. + +$\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams. +As before, the official definition is in terms of direct sums +of tensor products: +\[ + \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} + \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . +\] +Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. +$\overline{c}$ runs over all boundary conditions, again as described above. +$j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are cuttable along all of the blobs in $\overline{B}$. + +The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. +Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. +Let $E_j(b)$ denote the result of erasing the $j$-th blob. +If $B_j$ is not a twig blob, this involves only decrementing +the indices of blobs $B_{j+1},\ldots,B_{k-1}$. +If $B_j$ is a twig blob, we have to assign new local relation labels +if removing $B_j$ creates new twig blobs. +If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, +where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. +Finally, define +\eq{ + \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). +} +The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. +Thus we have a chain complex. + +\nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} + +\nn{?? remark about dendroidal sets} + +