diff -r fdb1cd651fd2 -r 71b4e45f47f6 blob1.tex --- a/blob1.tex Tue Mar 03 16:40:59 2009 +0000 +++ b/blob1.tex Tue Mar 03 23:26:11 2009 +0000 @@ -585,7 +585,7 @@ \label{sec:blob-definition} Let $X$ be an $n$-manifold. -Assume a fixed system of fields. +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 $\lf(X; c)$). @@ -593,36 +593,37 @@ 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. +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 the space of all local relations that can be imposed on $\bc_0(X)$. -More specifically, define a 1-blob diagram to consist of +$\bc_1(X)$ 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") $B \sub X$. -%\nn{Does $B$ need a homeo to the standard $B^n$? I don't think so. -%(See note in previous subsection.)} -%\item A field (boundary condition) $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$. \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} -%(Note that the field $c$ is determined (implicitly) as the boundary of $u$ and/or $r$, -%so we will omit $c$ from the notation.) -Define $\bc_1(X)$ to be the space of all finite linear combinations of -1-blob diagrams, modulo the simple relations relating labels of 0-cells and -also the label ($u$ above) of the blob. -\nn{maybe spell this out in more detail} -(See xxxx above.) -\nn{maybe restate this in terms of direct sums of tensor products.} +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 $(B, u, r)$. -There is a map $\bd : \bc_1(X) \to \bc_0(X)$ which sends $(B, r, u)$ to $ru$, the linear -combination of fields on $X$ obtained by gluing $r$ to $u$. +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 $u$). @@ -630,23 +631,22 @@ 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 $\bc_1(X)$. +$\bc_2(X)$ 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 (defined below), modulo the usual linear label relations. -\nn{and also modulo blob reordering relations?} +2-blob diagrams, of which there are two types, disjoint and nested. -\nn{maybe include longer discussion to motivate the two sorts of 2-blob diagrams} - -There are two types of 2-blob diagram: disjoint and nested. A disjoint 2-blob diagram consists of \begin{itemize} -\item A pair of disjoint closed balls (blobs) $B_0, B_1 \sub X$. -%\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. +\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 $u_i \in U(B_i; c_i)$. +\item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. \end{itemize} -Define $\bd(B_0, B_1, r, u_0, u_1) = (B_1, ru_0, u_1) - (B_0, ru_1, u_0) \in \bc_1(X)$. +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. @@ -656,74 +656,86 @@ \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 $c_0 \in \cC(\bd B_0)$). +(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 $r' \in \cC(X \setmin B_1; c_1)$. -\item A local relation field $u_0 \in U(B_0; c_0)$. -\end{itemize} -Define $\bd(B_0, B_1, r, u_0) = (B_1, r', r_1u_0) - (B_0, r, u_0)$. -Note that xxxx above guarantees that $r_1u_0 \in U(B_1)$. +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 $r_1u_0$. +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 $B_i \cap B_j$ is empty or +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 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 $c^t \in \cC(\bd B^t)$. +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} -We define $\bc_k(X)$ to be the vector space of all finite linear combinations -of $k$-blob diagrams, modulo the linear label relations and -blob reordering relations defined in the remainder of this paragraph. -Let $x$ be a blob diagram with one undetermined $n$-morphism label. -The unlabeled entity is either a blob or a 0-cell outside of the twig blobs. -Let $a$ and $b$ be two possible $n$-morphism labels for -the unlabeled blob or 0-cell. -Let $c = \lambda a + b$. -Let $x_a$ be the blob diagram with label $a$, and define $x_b$ and $x_c$ similarly. -Then we impose the relation -\eq{ - x_c = \lambda x_a + x_b . -} -\nn{should do this in terms of direct sums of tensor products} -Let $x$ and $x'$ be two blob diagrams which differ only by a permutation $\pi$ -of their blob labelings. -Then we impose the relation -\eq{ - x = \sign(\pi) x' . -} +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 $D_2$. -(Alert readers will have noticed that for $k=2$ our definition -of $\bc_k(X)$ is slightly different from the previous definition -of $\bc_2(X)$ --- we did not impose the reordering relations. -The general definition takes precedence; -the earlier definition was simplified for purposes of exposition.) +$\bc_k(X)$ 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 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. +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 $u_l = r_lu_j$, +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{ @@ -734,10 +746,7 @@ \nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} - -\nn{TO DO: -expand definition to handle DGA and $A_\infty$ versions of $n$-categories; -relations to Chas-Sullivan string stuff} +\nn{?? remark about dendroidal sets}