blob: comparison text/ncat.tex

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-:421bd394a2bd
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 \end{tikzpicture}
 \caption{(a) $P$, (b) $P\times I$, (c) $\Cone(P)$, (d) $\vcone(P)$}
 \label{vcone-fig}
 \end{figure}
-\nn{maybe call this ``splittings" instead of ``V-cones"?}
+\begin{axiom}[Splittings]
-\begin{axiom}[V-cones]
 \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)
 and also with respect to each of the decompositions of $P$, then chooses common refinements of each decomposition of $P$
 (recall Example \ref{ex:blob-complexes-of-balls} above).
 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant
 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.
 \medskip
 Define a {\it permissible decomposition} of $W$ to be a map
 \[
 	\coprod_a X_a \to W,
 \]
 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.
 (Every smooth or PL manifold has a ball decomposition, but certain topological manifolds (e.g.\ non-smoothable
 topological 4-manifolds) do not have ball decompositions.
 permissible decompositions, we instead require compatibility with all ways of gluing up the decomposition.)
 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$
 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
 \[  % \begin{equation} \label{eq:psi-CC}
 When $\cC$ is an $A_\infty$ $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$
 is defined as above, as the colimit of $\psi_{\cC;W}$.
 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 $\psi_{\cC;W,c}$
+%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 $W$.
+%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.
 In the non-enriched case (e.g.\ $k<n$), where each $\cC(X_a; \beta)$ is just a set,
 the colimit is
 \[
 	\cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) \Bigg/ \sim ,
 \]
-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
 \begin{equation*}
 	\cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K,
 Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}.
 It is easy to see that
 there are well-defined maps $\cl{\cC}(W)\to\cl{\cC}(\bd W)$, and that these maps
 comprise a natural transformation of functors.
-\nn{to do: define splittability and restrictions for colimits}
 \begin{lem}
 \label{lem:colim-injective}

changeset 818	fb9fc18d2a52
parent 817	421bd394a2bd
child 820	57425531f564