776 \end{tikzpicture} |
776 \end{tikzpicture} |
777 \caption{(a) $P$, (b) $P\times I$, (c) $\Cone(P)$, (d) $\vcone(P)$} |
777 \caption{(a) $P$, (b) $P\times I$, (c) $\Cone(P)$, (d) $\vcone(P)$} |
778 \label{vcone-fig} |
778 \label{vcone-fig} |
779 \end{figure} |
779 \end{figure} |
780 |
780 |
781 \nn{maybe call this ``splittings" instead of ``V-cones"?} |
781 |
782 |
782 \begin{axiom}[Splittings] |
783 \begin{axiom}[V-cones] |
|
784 \label{axiom:vcones} |
783 \label{axiom:vcones} |
785 Let $c\in \cC_k(X)$ and |
784 Let $c\in \cC_k(X)$ and |
786 let $P$ be a finite poset of splittings of $c$. |
785 let $P$ be a finite poset of splittings of $c$. |
787 Then we can embed $\vcone(P)$ into the splittings of $c$, with $P$ corresponding to the base of $\vcone(P)$. |
786 Then we can embed $\vcone(P)$ into the splittings of $c$, with $P$ corresponding to the base of $\vcone(P)$. |
788 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. |
787 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. |
|
788 Also, any splitting of $\bd c$ can be extended to a splitting of $c$. |
789 \end{axiom} |
789 \end{axiom} |
790 |
|
791 \nn{maybe also say that any splitting of $\bd c$ can be extended to a splitting of $c$} |
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792 |
790 |
793 It is easy to see that this axiom holds in our two motivating examples, |
791 It is easy to see that this axiom holds in our two motivating examples, |
794 using standard facts about transversality and general position. |
792 using standard facts about transversality and general position. |
795 One starts with $q$, perturbs it so that it is in general position with respect to $c$ (in the case of string diagrams) |
793 One starts with $q$, perturbs it so that it is in general position with respect to $c$ (in the case of string diagrams) |
796 and also with respect to each of the decompositions of $P$, then chooses common refinements of each decomposition of $P$ |
794 and also with respect to each of the decompositions of $P$, then chooses common refinements of each decomposition of $P$ |
1254 (recall Example \ref{ex:blob-complexes-of-balls} above). |
1252 (recall Example \ref{ex:blob-complexes-of-balls} above). |
1255 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
1253 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
1256 for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the |
1254 for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the |
1257 same as the original blob complex for $M$ with coefficients in $\cC$. |
1255 same as the original blob complex for $M$ with coefficients in $\cC$. |
1258 |
1256 |
1259 Recall that we've already anticipated this construction in the previous section, |
1257 Recall that we've already anticipated this construction Subsection \ref{ss:n-cat-def}, |
1260 inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
1258 inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
1261 so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
1259 so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
1262 |
1260 |
1263 \medskip |
1261 \medskip |
1264 |
1262 |
1281 Define a {\it permissible decomposition} of $W$ to be a map |
1279 Define a {\it permissible decomposition} of $W$ to be a map |
1282 \[ |
1280 \[ |
1283 \coprod_a X_a \to W, |
1281 \coprod_a X_a \to W, |
1284 \] |
1282 \] |
1285 which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. |
1283 which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. |
|
1284 We further require that $\du_a (X_a \cap \bd W) \to \bd W$ |
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1285 can be completed to a (not necessarily ball) decomposition of $\bd W$. |
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1286 (So, for example, in Example \ref{sin1x-example} if we take $W = B\cup C\cup D$ then $B\du C\du D \to W$ |
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1287 is not allowed since $D\cap \bd W$ is not a submanifold.) |
1286 Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls |
1288 Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls |
1287 are glued up to yield $W$, so long as there is some (non-pathological) way to glue them. |
1289 are glued up to yield $W$, so long as there is some (non-pathological) way to glue them. |
1288 |
1290 |
1289 (Every smooth or PL manifold has a ball decomposition, but certain topological manifolds (e.g.\ non-smoothable |
1291 (Every smooth or PL manifold has a ball decomposition, but certain topological manifolds (e.g.\ non-smoothable |
1290 topological 4-manifolds) do not have ball decompositions. |
1292 topological 4-manifolds) do not have ball decompositions. |
1361 permissible decompositions, we instead require compatibility with all ways of gluing up the decomposition.) |
1363 permissible decompositions, we instead require compatibility with all ways of gluing up the decomposition.) |
1362 |
1364 |
1363 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ |
1365 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ |
1364 is given by the composition maps of $\cC$. |
1366 is given by the composition maps of $\cC$. |
1365 This completes the definition of the functor $\psi_{\cC;W}$. |
1367 This completes the definition of the functor $\psi_{\cC;W}$. |
1366 |
|
1367 Note that we have constructed, at the last stage of the above procedure, |
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1368 a map from $\psi_{\cC;W}(x)$ to $\cl\cC(\bd M_m) = \cl\cC(\bd W)$. |
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1369 \nn{need to show at somepoint that this does not depend on choice of ball decomp} |
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1370 |
1368 |
1371 If $k=n$ in the above definition and we are enriching in some auxiliary category, |
1369 If $k=n$ in the above definition and we are enriching in some auxiliary category, |
1372 we need to say a bit more. |
1370 we need to say a bit more. |
1373 We can rewrite the colimit as |
1371 We can rewrite the colimit as |
1374 \[ % \begin{equation} \label{eq:psi-CC} |
1372 \[ % \begin{equation} \label{eq:psi-CC} |
1396 When $\cC$ is an $A_\infty$ $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$ |
1394 When $\cC$ is an $A_\infty$ $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$ |
1397 is defined as above, as the colimit of $\psi_{\cC;W}$. |
1395 is defined as above, as the colimit of $\psi_{\cC;W}$. |
1398 When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
1396 When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
1399 \end{defn} |
1397 \end{defn} |
1400 |
1398 |
1401 We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$ |
1399 %We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$ |
1402 with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
1400 %with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
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1401 |
|
1402 \medskip |
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1403 |
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1404 We must now define restriction maps $\bd : \cl{\cC}(W) \to \cl{\cC}(\bd W)$ and gluing maps. |
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1405 |
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1406 Let $y\in \cl{\cC}(W)$. |
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1407 Choose a representative of $y$ in the colimit: a permissible decomposition $\du_a X_a \to W$ and elements |
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1408 $y_a \in \cC(X_a)$. |
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1409 By assumption, $\du_a (X_a \cap \bd W) \to \bd W$ can be completed to a decomposition of $\bd W$. |
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1410 Let $r(y_a) \in \cl\cC(X_a \cap \bd W)$ be the restriction. |
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1411 Choose a representative of $r(y_a)$ in the colimit $\cl\cC(X_a \cap \bd W)$: a permissible decomposition |
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1412 $\du_b Q_{ab} \to X_a \cap \bd W$ and elements $z_{ab} \in \cC(Q_{ab})$. |
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1413 Then $\du_{ab} Q_{ab} \to \bd W$ is a permissible decomposition of $\bd W$ and $\{z_{ab}\}$ represents |
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1414 an element of $\cl{\cC}(\bd W)$. Define $\bd y$ to be this element. |
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1415 It is not hard to see that it is independent of the various choices involved. |
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1416 |
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1417 Note that since we have already (inductively) defined gluing maps for colimits of $k{-}1$-manifolds, |
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1418 we can also define restriction maps from $\cl{\cC}(W)\trans{}$ to $\cl{\cC}(Y)$ where $Y$ is a codimension 0 |
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1419 submanifold of $\bd W$. |
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1420 |
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1421 Next we define gluing maps for colimits of $k$-manifolds. |
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1422 Let $W = W_1 \cup_Y W_2$. |
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1423 Let $y_i \in \cl\cC(W_i)$ and assume that the restrictions of $y_1$ and $y_2$ to $\cl\cC(Y)$ agree. |
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1424 We want to define $y_1\bullet y_2 \in \cl\cC(W)$. |
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1425 Choose a permissible decomposition $\du_a X_{ia} \to W_i$ and elements |
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1426 $y_{ia} \in \cC(X_{ia})$ representing $y_i$. |
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1427 It might not be the case that $\du_{ia} X_{ia} \to W$ is a permissible decomposition of $W$, |
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1428 since intersections of the pieces with $\bd W$ might not be well-behaved. |
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1429 However, using the fact that $\bd y_i$ splits along $\bd Y$ and applying Axiom \ref{axiom:vcones}, |
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1430 we can choose the decomposition $\du_{a} X_{ia}$ so that its restriction to $\bd W_i$ is a refinement |
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1431 of the splitting along $\bd Y$, and this implies that the combined decomposition $\du_{ia} X_{ia}$ |
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1432 is permissible. |
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1433 We can now define the gluing $y_1\bullet y_2$ in the obvious way, and a further application of Axiom \ref{axiom:vcones} |
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1434 shows that this is independebt of the choices of representatives of $y_i$. |
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1435 |
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1436 |
|
1437 \medskip |
1403 |
1438 |
1404 We now give more concrete descriptions of the above colimits. |
1439 We now give more concrete descriptions of the above colimits. |
1405 |
1440 |
1406 In the non-enriched case (e.g.\ $k<n$), where each $\cC(X_a; \beta)$ is just a set, |
1441 In the non-enriched case (e.g.\ $k<n$), where each $\cC(X_a; \beta)$ is just a set, |
1407 the colimit is |
1442 the colimit is |
1408 \[ |
1443 \[ |
1409 \cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) \Bigg/ \sim , |
1444 \cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) \Bigg/ \sim , |
1410 \] |
1445 \] |
1411 where $x$ runs through decomposition of $W$, and $\sim$ is the obvious equivalence relation |
1446 where $x$ runs through decompositions of $W$, and $\sim$ is the obvious equivalence relation |
1412 induced by refinement and gluing. |
1447 induced by refinement and gluing. |
1413 If $\cC$ is enriched over vector spaces and $W$ is an $n$-manifold, |
1448 If $\cC$ is enriched over vector spaces and $W$ is an $n$-manifold, |
1414 we can take |
1449 we can take |
1415 \begin{equation*} |
1450 \begin{equation*} |
1416 \cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K, |
1451 \cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K, |