1432 Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. |
1432 Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. |
1433 Let $W$ be a $k$-manifold ($k\le n$), |
1433 Let $W$ be a $k$-manifold ($k\le n$), |
1434 let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
1434 let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
1435 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
1435 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
1436 |
1436 |
1437 %Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
|
1438 %and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |
|
1439 %component $\bd_i W$ of $W$. |
|
1440 %(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.) |
|
1441 |
|
1442 We will define a set $\cC(W, \cN)$ using a colimit construction similar to |
1437 We will define a set $\cC(W, \cN)$ using a colimit construction similar to |
1443 the one appearing in \S \ref{ss:ncat_fields} above. |
1438 the one appearing in \S \ref{ss:ncat_fields} above. |
1444 (If $k = n$ and our $n$-categories are enriched, then |
1439 (If $k = n$ and our $n$-categories are enriched, then |
1445 $\cC(W, \cN)$ will have additional structure; see below.) |
1440 $\cC(W, \cN)$ will have additional structure; see below.) |
1446 |
1441 |
1447 Define a permissible decomposition of $W$ to be a decomposition |
1442 Define a permissible decomposition of $W$ to be a decomposition |
1448 \[ |
1443 \[ |
1449 W = \left(\bigcup_a X_a\right) \cup \left(\bigcup_{i,b} M_{ib}\right) , |
1444 W = \left(\bigcup_a X_a\right) \cup \left(\bigcup_{i,b} M_{ib}\right) , |
1450 \] |
1445 \] |
1451 where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and |
1446 where each $X_a$ is a plain $k$-ball (disjoint from $\cup Y_i$) and |
1452 each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, |
1447 each $M_{ib}$ is a marked $k$-ball intersecting $Y_i$, |
1453 with $M_{ib}\cap Y_i$ being the marking. |
1448 with $M_{ib}\cap Y_i$ being the marking. |
1454 (See Figure \ref{mblabel}.) |
1449 (See Figure \ref{mblabel}.) |
1455 \begin{figure}[!ht]\begin{equation*} |
1450 \begin{figure}[t] |
|
1451 \begin{equation*} |
1456 \mathfig{.4}{ncat/mblabel} |
1452 \mathfig{.4}{ncat/mblabel} |
1457 \end{equation*}\caption{A permissible decomposition of a manifold |
1453 \end{equation*} |
|
1454 \caption{A permissible decomposition of a manifold |
1458 whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. |
1455 whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. |
1459 Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure} |
1456 Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel} |
|
1457 \end{figure} |
1460 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
1458 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
1461 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
1459 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
1462 This defines a partial ordering $\cell(W)$, which we will think of as a category. |
1460 This defines a partial ordering $\cell(W)$, which we will think of as a category. |
1463 (The objects of $\cell(D)$ are permissible decompositions of $W$, and there is a unique |
1461 (The objects of $\cell(D)$ are permissible decompositions of $W$, and there is a unique |
1464 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
1462 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
1470 \[ |
1468 \[ |
1471 \psi_\cN(x) \sub \left(\prod_a \cC(X_a)\right) \times \left(\prod_{ib} \cN_i(M_{ib})\right) |
1469 \psi_\cN(x) \sub \left(\prod_a \cC(X_a)\right) \times \left(\prod_{ib} \cN_i(M_{ib})\right) |
1472 \] |
1470 \] |
1473 such that the restrictions to the various pieces of shared boundaries amongst the |
1471 such that the restrictions to the various pieces of shared boundaries amongst the |
1474 $X_a$ and $M_{ib}$ all agree. |
1472 $X_a$ and $M_{ib}$ all agree. |
1475 (That is, the fibered product over the boundary maps.) |
1473 (That is, the fibered product over the boundary restriction maps.) |
1476 If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
1474 If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
1477 via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
1475 via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
1478 |
1476 |
1479 We now define the set $\cC(W, \cN)$ to be the colimit of the functor $\psi_\cN$. |
1477 We now define the set $\cC(W, \cN)$ to be the colimit of the functor $\psi_\cN$. |
1480 (As usual, if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means |
1478 (As in \S\ref{ss:ncat-coend}, if $k=n$ we take a colimit in whatever |
1481 homotopy colimit.) |
1479 category we are enriching over, and if additionally we are in the $A_\infty$ case, |
|
1480 then we use a homotopy colimit.) |
|
1481 |
|
1482 \medskip |
1482 |
1483 |
1483 If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
1484 If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
1484 $\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
1485 $\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
1485 $D\times Y_i \sub \bd(D\times W)$. |
1486 $D\times Y_i \sub \bd(D\times W)$. |
1486 It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$ |
1487 It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$ |
1487 has the structure of an $n{-}k$-category, which we call $\cT(W, \cN)(D)$. |
1488 has the structure of an $n{-}k$-category. |
1488 |
1489 |
1489 \medskip |
1490 \medskip |
1490 |
|
1491 |
1491 |
1492 We will use a simple special case of the above |
1492 We will use a simple special case of the above |
1493 construction to define tensor products |
1493 construction to define tensor products |
1494 of modules. |
1494 of modules. |
1495 Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
1495 Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
1496 (If $k=1$ and our manifolds are oriented, then one should be |
1496 (If $k=1$ and our manifolds are oriented, then one should be |
1497 a left module and the other a right module.) |
1497 a left module and the other a right module.) |
1498 Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
1498 Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
1499 Define the tensor product $\cM_1 \tensor \cM_2$ to be the |
1499 Define the tensor product $\cM_1 \tensor \cM_2$ to be the |
1500 $n{-}1$-category $\cT(J, \{\cM_1, \cM_2\})$. |
1500 $n{-}1$-category associated as above to $J$ with its boundary labeled by $\cM_1$ and $\cM_2$. |
1501 This of course depends (functorially) |
1501 This of course depends (functorially) |
1502 on the choice of 1-ball $J$. |
1502 on the choice of 1-ball $J$. |
1503 |
1503 |
1504 We will define a more general self tensor product (categorified coend) below. |
1504 We will define a more general self tensor product (categorified coend) below. |
1505 |
1505 |