# HG changeset patch # User Kevin Walker # Date 1317685216 25200 # Node ID deeff619087eee05aaaf9a0cfbef37d4b60062e0 # Parent 74ab13b63b9b22e2375ea52e40d263b7308ad40f Initial version of the new splitting axiom. This is much rougher than I would like, but it's better than nothing. diff -r 74ab13b63b9b -r deeff619087e blob to-do --- a/blob to-do Mon Sep 26 16:40:49 2011 -0600 +++ b/blob to-do Mon Oct 03 16:40:16 2011 -0700 @@ -11,13 +11,21 @@ * need to change module axioms to follow changes in n-cat axioms; search for and destroy all the "Homeo_\bd"'s, add a v-cone axiom * probably should go through and refer to new splitting axiom when we need to choose refinements etc. -** in the proof that gluing in dimension < n is injective +**** in the proof that gluing in dimension < n is injective + +* revisit splitting axiom for system of fields; check use of it in small blobs lemma * framings and duality -- work out what's going on! (alternatively, vague-ify current statement) * make sure we are clear that boundary = germ (perhaps we are already clear enough) +* places splitting axiom is used: +** in the proof that gluing in dimension < n is injective +** in the proof that D(a) is acyclic +** in the small blobs lemma + + ====== minor/optional ====== diff -r 74ab13b63b9b -r deeff619087e text/a_inf_blob.tex --- a/text/a_inf_blob.tex Mon Sep 26 16:40:49 2011 -0600 +++ b/text/a_inf_blob.tex Mon Oct 03 16:40:16 2011 -0700 @@ -120,14 +120,14 @@ (Consider the $x$-axis and the graph of $y = e^{-1/x^2} \sin(1/x)$ in $\r^2$.) However, we {\it can} find another decomposition $L$ such that $L$ shares common refinements with both $K$ and $K'$. (For instance, in the example above, $L$ can be the graph of $y=x^2-1$.) -This follows from Axiom \ref{axiom:vcones}, which in turn follows from the +This follows from Axiom \ref{axiom:splittings}, which in turn follows from the splitting axiom for the system of fields $\cE$. Let $KL$ and $K'L$ denote these two refinements. Then 1-simplices associated to the four anti-refinements $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ give the desired chain connecting $(a, K)$ and $(a, K')$ (see Figure \ref{zzz4}). -(In the language of Axiom \ref{axiom:vcones}, this is $\vcone(K \du K')$.) +(In the language of Lemma \ref{lemma:vcones}, this is $\vcone(K \du K')$.) \begin{figure}[t] \centering \begin{tikzpicture} @@ -147,7 +147,7 @@ Consider next a 1-cycle in $E(b, b')$, such as one arising from a different choice of decomposition $L'$ in place of $L$ above. %We want to find 2-simplices which fill in this cycle. -By Axiom \ref{axiom:vcones} we can fill in this 1-cycle with 2-simplices. +By Lemma \ref{lemma:vcones} we can fill in this 1-cycle with 2-simplices. Choose a decomposition $M$ which has common refinements with each of $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. (We also require that $KLM$ antirefines to $KM$, etc.) @@ -190,7 +190,7 @@ \end{figure} Continuing in this way we see that $D(a)$ is acyclic. -By Axiom \ref{axiom:vcones} we can fill in any cycle with a V-Cone. +By Lemma \ref{lemma:vcones} we can fill in any cycle with a V-Cone. \end{proof} We are now in a position to apply the method of acyclic models to get a map diff -r 74ab13b63b9b -r deeff619087e text/ncat.tex --- a/text/ncat.tex Mon Sep 26 16:40:49 2011 -0600 +++ b/text/ncat.tex Mon Oct 03 16:40:16 2011 -0700 @@ -34,7 +34,7 @@ The axioms for an $n$-category are spread throughout this section. Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, -\ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:extended-isotopies} and \ref{axiom:vcones}. +\ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:extended-isotopies} and \ref{axiom:splittings}. For an enriched $n$-category we add Axiom \ref{axiom:enriched}. For an $A_\infty$ $n$-category, we replace Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. @@ -672,6 +672,159 @@ \medskip +We need one additional axiom. +It says, roughly, that given a $k$-ball $X$, $k=.06cm, shorten <=.06cm}, + kw label/.style={cca}, + ] + + \definecolor{cca}{rgb}{.1,.4,.3}; + + \node at (0,0) { + \begin{tikzpicture} + \draw + (0,0) node[kw node](p1){} + (1,.5) node[kw node](p2){} + (2,0) node[kw node](p3){}; + + \draw[kw arrow] (p1) -- (p3); + \draw[kw arrow] (p2) -- (p3); + \draw[kw arrow] (p1) -- (p2); + + \draw[kw label] (1,-.6) node{(a)}; + \end{tikzpicture} + }; + + \node at (7,0) { + \begin{tikzpicture} + \draw + (0,0) node[kw node](p1){} + ++(0,2.5) node[kw node](q1){} + (1,.5) node[kw node](p2){} + ++(0,2.5) node[kw node](q2){} + (2,0) node[kw node](p3){} + ++(0,2.5) node[kw node](q3){} + ; + + \draw[kw arrow] (p1) -- (p3); + \draw[kw arrow] (p2) -- (p3); + \draw[kw arrow] (p1) -- (p2); + \draw[kw arrow] (q1) -- (q3); + \draw[kw arrow] (q2) -- (q3); + \draw[kw arrow] (q1) -- (q2); + \draw[kw arrow] (p1) -- (q1); + \draw[kw arrow] (p2) -- (q2); + \draw[kw arrow] (p3) -- (q3); + + \draw[kw label] (1,-.6) node{(b)}; + \end{tikzpicture} + }; + + \node at (0,-5) { + \begin{tikzpicture} + \draw + (0,0) node[kw node](p1){} + (1,.5) node[kw node](p2){} + ++(0,2.5) node[kw node](v){} + (2,0) node[kw node](p3){} + ; + + \draw[kw arrow] (p1) -- (p3); + \draw[kw arrow] (p2) -- (p3); + \draw[kw arrow] (p1) -- (p2); + \draw[kw arrow] (p1) -- (v); + \draw[kw arrow] (p2) -- (v); + \draw[kw arrow] (p3) -- (v); + + \draw[kw label] (1,-.6) node{(c)}; + \end{tikzpicture} + }; + + \node at (7,-5) { + \begin{tikzpicture} + \draw + (0,0) node[kw node](p1){} + ++(-2,2.5) node[kw node](q1){} + (1,.5) node[kw node](p2){} + ++(-2,2.5) node[kw node](q2){} + ++(4,0) node[kw node](v){} + (2,0) node[kw node](p3){} + ++(-2,2.5) node[kw node](q3){} + ; + + \draw[kw arrow] (p1) -- (p3); + \draw[kw arrow] (p2) -- (p3); + \draw[kw arrow] (p1) -- (p2); + \draw[kw arrow] (p1) -- (v); + \draw[kw arrow] (p2) -- (v); + \draw[kw arrow] (p3) -- (v); + \draw[kw arrow] (q1) -- (q3); + \draw[kw arrow] (q2) -- (q3); + \draw[kw arrow] (q1) -- (q2); + \draw[kw arrow] (p1) -- (q1); + \draw[kw arrow] (p2) -- (q2); + \draw[kw arrow] (p3) -- (q3); + + \draw[kw label] (1,-.6) node{(d)}; + \end{tikzpicture} + }; + +\end{tikzpicture} +\caption{(a) $P$, (b) $P\times I$, (c) $\Cone(P)$, (d) $\vcone(P)$} +\label{vcone-fig} +\end{figure} + + +\begin{lem} +\label{lemma:vcones} +Let $c\in \cC_k(X)$, with $0\le k < n$, and +let $P$ be a finite poset of splittings of $c$. +Then we can embed $\vcone(P)$ into the splittings of $c$, with $P$ corresponding to the base of $\vcone(P)$. +Furthermore, if $q$ is any decomposition of $X$, then we can take the vertex of $\vcone(P)$ to be $q$ up to a small perturbation. +\end{lem} + +\begin{proof} +After a small perturbation, we may assume that $q$ is simultaneously transverse to all the splittings in $P$, and +(by Axiom \ref{axiom:splittings}) that $c$ splits along $q$. +We can now choose, for each splitting $p$ in $P$, a common refinement $p'$ of $p$ and $q$. +This constitutes the middle part of $\vcone(P)$. +\end{proof} + + +\noop{ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + We need one additional axiom, in order to constrain the poset of decompositions of a given morphism. We will soon want to take colimits (and homotopy colimits) indexed by such posets, and we want to require that these colimits are in some sense locally acyclic. @@ -811,6 +964,7 @@ Two decompositions of $X$ might intersect in a very messy way, but one can always find a third decomposition which has common refinements with each of the original two decompositions. +} %%%%%% end \noop %%%%%%%%%%%%%%%%%%%%%%%%%%%% \medskip @@ -1024,7 +1178,7 @@ \item The product maps are associative and also compatible with homeomorphism actions, gluing and restriction (Axiom \ref{axiom:product}). \item If enriching in an auxiliary category, all of the data should be compatible with the auxiliary category structure on $\cC_n(X; c)$ (Axiom \ref{axiom:enriched}). -\item The possible splittings of a morphism satisfy various conditions (Axiom \ref{axiom:vcones}). +\item The possible splittings of a morphism satisfy various conditions (Axiom \ref{axiom:splittings}). \item For ordinary categories, invariance of $n$-morphisms under extended isotopies and collar maps (Axiom \ref{axiom:extended-isotopies}). \end{itemize} @@ -1464,11 +1618,11 @@ $y_{ia} \in \cC(X_{ia})$ representing $y_i$. It might not be the case that $\du_{ia} X_{ia} \to W$ is a permissible decomposition of $W$, since intersections of the pieces with $\bd W$ might not be well-behaved. -However, using the fact that $\bd y_i$ splits along $\bd Y$ and applying Axiom \ref{axiom:vcones}, +However, using the fact that $\bd y_i$ splits along $\bd Y$ and applying Axiom \ref{axiom:splittings}, we can choose the decomposition $\du_{a} X_{ia}$ so that its restriction to $\bd W_i$ is a refinement of the splitting along $\bd Y$, and this implies that the combined decomposition $\du_{ia} X_{ia}$ is permissible. -We can now define the gluing $y_1\bullet y_2$ in the obvious way, and a further application of Axiom \ref{axiom:vcones} +We can now define the gluing $y_1\bullet y_2$ in the obvious way, and a further application of Axiom \ref{axiom:splittings} shows that this is independent of the choices of representatives of $y_i$. diff -r 74ab13b63b9b -r deeff619087e text/tqftreview.tex --- a/text/tqftreview.tex Mon Sep 26 16:40:49 2011 -0600 +++ b/text/tqftreview.tex Mon Oct 03 16:40:16 2011 -0700 @@ -196,8 +196,9 @@ are transverse to $Y$ or splittable along $Y$. \item Splittings. Let $c\in \cC_k(X)$ and let $Y\sub X$ be a codimension 1 properly embedded submanifold of $X$. -Then for most small perturbations of $Y$ (i.e.\ for an open dense -subset of such perturbations) $c$ splits along $Y$. +Then for most small perturbations of $Y$ (e.g.\ for an open dense +subset of such perturbations, or for all perturbations satisfying +a transversality condition) $c$ splits along $Y$. (In Example \ref{ex:maps-to-a-space(fields)}, $c$ splits along all such $Y$. In Example \ref{ex:traditional-n-categories(fields)}, $c$ splits along $Y$ so long as $Y$ is in general position with respect to the cell decomposition