diff -r 421bd394a2bd -r fb9fc18d2a52 text/ncat.tex --- a/text/ncat.tex Fri Jun 24 06:39:25 2011 -0700 +++ b/text/ncat.tex Fri Jun 24 21:41:48 2011 -0700 @@ -778,18 +778,16 @@ \label{vcone-fig} \end{figure} -\nn{maybe call this ``splittings" instead of ``V-cones"?} - -\begin{axiom}[V-cones] + +\begin{axiom}[Splittings] \label{axiom:vcones} Let $c\in \cC_k(X)$ 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. +Also, any splitting of $\bd c$ can be extended to a splitting of $c$. \end{axiom} -\nn{maybe also say that any splitting of $\bd c$ can be extended to a splitting of $c$} - It is easy to see that this axiom holds in our two motivating examples, using standard facts about transversality and general position. One starts with $q$, perturbs it so that it is in general position with respect to $c$ (in the case of string diagrams) @@ -1256,7 +1254,7 @@ for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the same as the original blob complex for $M$ with coefficients in $\cC$. -Recall that we've already anticipated this construction in the previous section, +Recall that we've already anticipated this construction Subsection \ref{ss:n-cat-def}, inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, so that we can state the boundary axiom for $\cC$ on $k+1$-balls. @@ -1283,6 +1281,10 @@ \coprod_a X_a \to W, \] which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. +We further require that $\du_a (X_a \cap \bd W) \to \bd W$ +can be completed to a (not necessarily ball) decomposition of $\bd W$. +(So, for example, in Example \ref{sin1x-example} if we take $W = B\cup C\cup D$ then $B\du C\du D \to W$ +is not allowed since $D\cap \bd W$ is not a submanifold.) Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls are glued up to yield $W$, so long as there is some (non-pathological) way to glue them. @@ -1364,10 +1366,6 @@ is given by the composition maps of $\cC$. This completes the definition of the functor $\psi_{\cC;W}$. -Note that we have constructed, at the last stage of the above procedure, -a map from $\psi_{\cC;W}(x)$ to $\cl\cC(\bd M_m) = \cl\cC(\bd W)$. -\nn{need to show at somepoint that this does not depend on choice of ball decomp} - If $k=n$ in the above definition and we are enriching in some auxiliary category, we need to say a bit more. We can rewrite the colimit as @@ -1398,8 +1396,45 @@ When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. \end{defn} -We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$ -with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. +%We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$ +%with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. + +\medskip + +We must now define restriction maps $\bd : \cl{\cC}(W) \to \cl{\cC}(\bd W)$ and gluing maps. + +Let $y\in \cl{\cC}(W)$. +Choose a representative of $y$ in the colimit: a permissible decomposition $\du_a X_a \to W$ and elements +$y_a \in \cC(X_a)$. +By assumption, $\du_a (X_a \cap \bd W) \to \bd W$ can be completed to a decomposition of $\bd W$. +Let $r(y_a) \in \cl\cC(X_a \cap \bd W)$ be the restriction. +Choose a representative of $r(y_a)$ in the colimit $\cl\cC(X_a \cap \bd W)$: a permissible decomposition +$\du_b Q_{ab} \to X_a \cap \bd W$ and elements $z_{ab} \in \cC(Q_{ab})$. +Then $\du_{ab} Q_{ab} \to \bd W$ is a permissible decomposition of $\bd W$ and $\{z_{ab}\}$ represents +an element of $\cl{\cC}(\bd W)$. Define $\bd y$ to be this element. +It is not hard to see that it is independent of the various choices involved. + +Note that since we have already (inductively) defined gluing maps for colimits of $k{-}1$-manifolds, +we can also define restriction maps from $\cl{\cC}(W)\trans{}$ to $\cl{\cC}(Y)$ where $Y$ is a codimension 0 +submanifold of $\bd W$. + +Next we define gluing maps for colimits of $k$-manifolds. +Let $W = W_1 \cup_Y W_2$. +Let $y_i \in \cl\cC(W_i)$ and assume that the restrictions of $y_1$ and $y_2$ to $\cl\cC(Y)$ agree. +We want to define $y_1\bullet y_2 \in \cl\cC(W)$. +Choose a permissible decomposition $\du_a X_{ia} \to W_i$ and elements +$y_{ia} \in \cC(X_{ia})$ 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}, +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} +shows that this is independebt of the choices of representatives of $y_i$. + + +\medskip We now give more concrete descriptions of the above colimits. @@ -1408,7 +1443,7 @@ \[ \cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) \Bigg/ \sim , \] -where $x$ runs through decomposition of $W$, and $\sim$ is the obvious equivalence relation +where $x$ runs through decompositions of $W$, and $\sim$ is the obvious equivalence relation induced by refinement and gluing. If $\cC$ is enriched over vector spaces and $W$ is an $n$-manifold, we can take @@ -1483,10 +1518,6 @@ -\nn{to do: define splittability and restrictions for colimits} - - - \begin{lem} \label{lem:colim-injective} Let $W$ be a manifold of dimension less than $n$. Then for each