# HG changeset patch # User Scott Morrison # Date 1276578765 25200 # Node ID 6c624cd07bebb25da3c2705971ef850943c46313 # Parent 92f0dac39ce30e7248f6b7533da2af0c2f87252b# Parent dd441d7439163ec5b26893989d01572ca3cbf970 Automated merge with https://tqft.net/hg/blob/ diff -r dd441d743916 -r 6c624cd07beb blob1.tex --- a/blob1.tex Mon Jun 07 22:02:40 2010 -0700 +++ b/blob1.tex Mon Jun 14 22:12:45 2010 -0700 @@ -16,9 +16,12 @@ \maketitle -[revision $\ge$ 320; $\ge$ 2 June 2010] +[revision $\ge$ 360; $\ge$ 14 June 2010] -\textbf{Draft version, read with caution.} +{\color[rgb]{.9,.5,.2} \large \textbf{Draft version, read with caution.}} +We're in the midst of revising this, and hope to have a version on the arXiv soon. + +\noop{ \paragraph{To do list} \begin{itemize} @@ -59,6 +62,8 @@ \end{itemize} +} % end \noop + \tableofcontents diff -r dd441d743916 -r 6c624cd07beb text/evmap.tex --- a/text/evmap.tex Mon Jun 07 22:02:40 2010 -0700 +++ b/text/evmap.tex Mon Jun 14 22:12:45 2010 -0700 @@ -35,12 +35,6 @@ satisfying the above two conditions. \end{prop} - -\nn{Also need to say something about associativity. -Put it in the above prop or make it a separate prop? -I lean toward the latter.} -\medskip - Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, and then give an outline of the method of proof. @@ -509,6 +503,7 @@ Let $R_*$ be the chain complex with a generating 0-chain for each non-negative integer and a generating 1-chain connecting each adjacent pair $(j, j+1)$. +(So $R_*$ is a simplicial version of the non-negative reals.) Denote the 0-chains by $j$ (for $j$ a non-negative integer) and the 1-chain connecting $j$ and $j+1$ by $\iota_j$. Define a map (homotopy equivalence) @@ -591,33 +586,71 @@ but we have come very close} \nn{better: change statement of thm} +\medskip +Next we show that the action maps are compatible with gluing. +Let $G^m_*$ and $\ol{G}^m_*$ be the complexes, as above, used for defining +the action maps $e_{X\sgl}$ and $e_X$. +The gluing map $X\sgl\to X$ induces a map +\[ + \gl: R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) \to R_*\ot CH_*(X, X) \otimes \bc_*(X) , +\] +and it is easy to see that $\gl(G^m_*)\sub \ol{G}^m_*$. +From this it follows that the diagram in the statement of Proposition \ref{CHprop} commutes. + +\medskip -\nn{...} - - +Finally we show that the action maps defined above are independent of +the choice of metric (up to iterated homotopy). +The arguments are very similar to ones given above, so we only sketch them. +Let $g$ and $g'$ be two metrics on $X$, and let $e$ and $e'$ be the corresponding +actions $CH_*(X, X) \ot \bc_*(X)\to\bc_*(X)$. +We must show that $e$ and $e'$ are homotopic. +As outlined in the discussion preceding this proof, +this follows from the facts that both $e$ and $e'$ are compatible +with gluing and that $\bc_*(B^n)$ is contractible. +As above, we define a subcomplex $F_*\sub CH_*(X, X) \ot \bc_*(X)$ generated +by $p\ot b$ such that $|p|\cup|b|$ is contained in a disjoint union of balls. +Using acyclic models, we can construct a homotopy from $e$ to $e'$ on $F_*$. +We now observe that $CH_*(X, X) \ot \bc_*(X)$ retracts to $F_*$. +Similar arguments show that this homotopy from $e$ to $e'$ is well-defined +up to second order homotopy, and so on. +\end{proof} -\medskip\hrule\medskip\hrule\medskip - -\nn{outline of what remains to be done:} +\begin{prop} +The $CH_*(X, Y)$ actions defined above are associative. +That is, the following diagram commutes up to homotopy: +\[ \xymatrix{ +& CH_*(Y, Z) \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\ +CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\ +& CH_*(X, Z) \ot \bc_*(X) \ar[ur]_{e_{XZ}} & +} \] +Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition +of homeomorphisms. +\end{prop} -\begin{itemize} -\item Independence of metric, $\ep_i$, $\delta_i$: -For a different metric etc. let $\hat{G}^{i,m}$ denote the alternate subcomplexes -and $\hat{N}_{i,l}$ the alternate neighborhoods. -Main idea is that for all $i$ there exists sufficiently large $k$ such that -$\hat{N}_{k,l} \sub N_{i,l}$, and similarly with the roles of $N$ and $\hat{N}$ reversed. -\item prove gluing compatibility, as in statement of main thm (this is relatively easy) -\item Also need to prove associativity. -\end{itemize} +\begin{proof} +The strategy of the proof is similar to that of Proposition \ref{CHprop}. +We will identify a subcomplex +\[ + G_* \sub CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) +\] +where it is easy to see that the two sides of the diagram are homotopic, then +show that there is a deformation retraction of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$ into $G_*$. - -\end{proof} +Let $p\ot q\ot b$ be a generator of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$. +By definition, $p\ot q\ot b\in G_*$ if there is a disjoint union of balls in $X$ which +contains $|p| \cup p\inv(|q|) \cup |b|$. +(If $p:P\times X\to Y$, then $p\inv(|q|)$ means the union over all $x\in P$ of +$p(x, \cdot)\inv(|q|)$.) -\nn{to be continued....} - +As in the proof of Proposition \ref{CHprop}, we can construct a homotopy +between the upper and lower maps restricted to $G_*$. +This uses the facts that the maps agree on $CH_0(X, Y) \ot CH_0(Y, Z) \ot \bc_*(X)$, +that they are compatible with gluing, and the contractibility of $\bc_*(X)$. - - +We can now apply Lemma \ref{extension_lemma_c}, using a series of increasingly fine covers, +to construct a deformation retraction of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$ into $G_*$. +\end{proof} diff -r dd441d743916 -r 6c624cd07beb text/ncat.tex --- a/text/ncat.tex Mon Jun 07 22:02:40 2010 -0700 +++ b/text/ncat.tex Mon Jun 14 22:12:45 2010 -0700 @@ -82,7 +82,7 @@ The 0-sphere is unusual among spheres in that it is disconnected. Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. (Actually, this is only true in the oriented case, with 1-morphisms parameterized -by oriented 1-balls.) +by {\it oriented} 1-balls.) For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. (sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary.