# HG changeset patch # User Scott Morrison # Date 1275622476 25200 # Node ID 7b632b53eb45bca97847ec17d412f44991ce9375 # Parent 956f373f6ff6d6ca8a16dae8177879b1114cb59a# Parent 3e61a91976133b37f10e889aa6ea04d6f23c3f20 Automated merge with https://tqft.net/hg/blob/ diff -r 956f373f6ff6 -r 7b632b53eb45 text/basic_properties.tex --- a/text/basic_properties.tex Thu Jun 03 18:42:39 2010 -0700 +++ b/text/basic_properties.tex Thu Jun 03 20:34:36 2010 -0700 @@ -106,5 +106,3 @@ This map is very far from being an isomorphism, even on homology. We fix this deficit in Section \ref{sec:gluing} below. - -As we pointed out earlier, Property \ref{property:skein-modules} is immediate from the definitions. diff -r 956f373f6ff6 -r 7b632b53eb45 text/blobdef.tex --- a/text/blobdef.tex Thu Jun 03 18:42:39 2010 -0700 +++ b/text/blobdef.tex Thu Jun 03 20:34:36 2010 -0700 @@ -180,6 +180,8 @@ The $(-1)^{j+1}$ factors imply that the terms of $\bd^2(b)$ all cancel. Thus we have a chain complex. +Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, is immediately obvious from the definition. A homeomorphism acts in an obvious on blobs and on fields. + We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, to be the union of the blobs of $b$. For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), diff -r 956f373f6ff6 -r 7b632b53eb45 text/intro.tex --- a/text/intro.tex Thu Jun 03 18:42:39 2010 -0700 +++ b/text/intro.tex Thu Jun 03 20:34:36 2010 -0700 @@ -328,9 +328,9 @@ \end{property} See \S \ref{sec:deligne} for an explanation of the terms appearing here. The proof will appear elsewhere. -Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in -\S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} -Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. +Properties \ref{property:functoriality} and \ref{property:skein-modules} will be immediate from the definition given in +\S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. +Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, Property \ref{property:blobs-ainfty} as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats}, and Properties \ref{property:product} and \ref{property:gluing} in \S \ref{sec:ainfblob} as consequences of Theorem \ref{product_thm}. diff -r 956f373f6ff6 -r 7b632b53eb45 text/ncat.tex --- a/text/ncat.tex Thu Jun 03 18:42:39 2010 -0700 +++ b/text/ncat.tex Thu Jun 03 20:34:36 2010 -0700 @@ -86,13 +86,13 @@ In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for $1\le k \le n$. At first it might seem that we need another axiom for this, but in fact once we have -all the axioms in the subsection for $0$ through $k-1$ we can use a coend +all the axioms in the subsection for $0$ through $k-1$ we can use a colimit construction, as described in Subsection \ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ to spheres (and any other manifolds): \begin{prop} \label{axiom:spheres} -For each $1 \le k \le n$, we have a functor $\cC_{k-1}$ from +For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from the category of $k{-}1$-spheres and homeomorphisms to the category of sets and bijections. \end{prop} @@ -102,18 +102,18 @@ %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. \begin{axiom}[Boundaries]\label{nca-boundary} -For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cC_{k-1}(\bd X)$. +For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. These maps, for various $X$, comprise a natural transformation of functors. \end{axiom} (Note that the first ``$\bd$" above is part of the data for the category, while the second is the ordinary boundary of manifolds.) -Given $c\in\cC(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. +Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. Most of the examples of $n$-categories we are interested in are enriched in the following sense. The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and -all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category +all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category (e.g.\ vector spaces, or modules over some ring, or chain complexes), and all the structure maps of the $n$-category should be compatible with the auxiliary category structure. @@ -142,27 +142,27 @@ domain and range, but the converse meets with our approval. That is, given compatible domain and range, we should be able to combine them into the full boundary of a morphism. -The following proposition follows from the coend construction used to define $\cC_{k-1}$ +The following proposition follows from the colimit construction used to define $\cl{\cC}_{k-1}$ on spheres. \begin{prop}[Boundary from domain and range] Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, $B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). -Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the -two maps $\bd: \cC(B_i)\to \cC(E)$. +Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the +two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$. Then we have an injective map \[ - \gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \into \cC(S) + \gl_E : \cC(B_1) \times_{\\cl{cC}(E)} \cC(B_2) \into \cl{\cC}(S) \] which is natural with respect to the actions of homeomorphisms. -(When $k=1$ we stipulate that $\cC(E)$ is a point, so that the above fibered product +(When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product becomes a normal product.) \end{prop} \begin{figure}[!ht] $$ \begin{tikzpicture}[%every label/.style={green} - ] +] \node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {}; \node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {}; \draw (S) arc (-90:90:1); @@ -175,15 +175,15 @@ Note that we insist on injectivity above. -Let $\cC(S)_E$ denote the image of $\gl_E$. -We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$". +Let $\cl{\cC}(S)_E$ denote the image of $\gl_E$. +We will refer to elements of $\\cl{cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$". If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ -as above, then we define $\cC(X)_E = \bd^{-1}(\cC(\bd X)_E)$. +as above, then we define $\cC(X)_E = \bd^{-1}(\cl{\cC}(\bd X)_E)$. -We will call the projection $\cC(S)_E \to \cC(B_i)$ +We will call the projection $\cl{\cC}(S)_E \to \cC(B_i)$ a {\it restriction} map and write $\res_{B_i}(a)$ -(or simply $\res(a)$ when there is no ambiguity), for $a\in \cC(S)_E$. +(or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)_E$. More generally, we also include under the rubric ``restriction map" the the boundary maps of Axiom \ref{nca-boundary} above, another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition