1341 |
1341 |
1342 \subsection{The $n{+}1$-category of sphere modules} |
1342 \subsection{The $n{+}1$-category of sphere modules} |
1343 \label{ssec:spherecat} |
1343 \label{ssec:spherecat} |
1344 |
1344 |
1345 In this subsection we define an $n{+}1$-category $\cS$ of ``sphere modules" |
1345 In this subsection we define an $n{+}1$-category $\cS$ of ``sphere modules" |
1346 whose objects correspond to $n$-categories. |
1346 whose objects are $n$-categories. |
1347 When $n=2$ |
1347 When $n=2$ |
1348 this is a version of the familiar algebras-bimodules-intertwiners 2-category. |
1348 this is a version of the familiar algebras-bimodules-intertwiners $2$-category. |
1349 (Terminology: It is clearly appropriate to call an $S^0$ module a bimodule, |
1349 While it is clearly appropriate to call an $S^0$ module a bimodule, |
1350 but this is much less true for higher dimensional spheres, |
1350 but this is much less true for higher dimensional spheres, |
1351 so we prefer the term ``sphere module" for the general case.) |
1351 so we prefer the term ``sphere module" for the general case. |
1352 |
1352 |
1353 The $0$- through $n$-dimensional parts of $\cC$ are various sorts of modules, and we describe |
1353 The $0$- through $n$-dimensional parts of $\cC$ are various sorts of modules, and we describe |
1354 these first. |
1354 these first. |
1355 The $n{+}1$-dimensional part of $\cS$ consists of intertwiners |
1355 The $n{+}1$-dimensional part of $\cS$ consists of intertwiners |
1356 (of garden-variety $1$-category modules associated to decorated $n$-balls). |
1356 of (garden-variety) $1$-category modules associated to decorated $n$-balls. |
1357 We will see below that in order for these $n{+}1$-morphisms to satisfy all of |
1357 We will see below that in order for these $n{+}1$-morphisms to satisfy all of |
1358 the duality requirements of an $n{+}1$-category, we will have to assume |
1358 the duality requirements of an $n{+}1$-category, we will have to assume |
1359 that our $n$-categories and modules have non-degenerate inner products. |
1359 that our $n$-categories and modules have non-degenerate inner products. |
1360 (In other words, we need to assume some extra duality on the $n$-categories and modules.) |
1360 (In other words, we need to assume some extra duality on the $n$-categories and modules.) |
1361 |
1361 |
1365 These will be defined in terms of certain classes of marked balls, very similarly |
1365 These will be defined in terms of certain classes of marked balls, very similarly |
1366 to the definition of $n$-category modules above. |
1366 to the definition of $n$-category modules above. |
1367 (This, in turn, is very similar to our definition of $n$-category.) |
1367 (This, in turn, is very similar to our definition of $n$-category.) |
1368 Because of this similarity, we only sketch the definitions below. |
1368 Because of this similarity, we only sketch the definitions below. |
1369 |
1369 |
1370 We start with 0-sphere modules, which also could reasonably be called (categorified) bimodules. |
1370 We start with $0$-sphere modules, which also could reasonably be called (categorified) bimodules. |
1371 (For $n=1$ they are precisely bimodules in the usual, uncategorified sense.) |
1371 (For $n=1$ they are precisely bimodules in the usual, uncategorified sense.) |
1372 Define a 0-marked $k$-ball $(X, M)$, $1\le k \le n$, to be a pair homeomorphic to the standard |
1372 Define a $0$-marked $k$-ball $(X, M)$, $1\le k \le n$, to be a pair homeomorphic to the standard |
1373 $(B^k, B^{k-1})$, where $B^{k-1}$ is properly embedded in $B^k$. |
1373 $(B^k, B^{k-1})$. |
1374 See Figure \ref{feb21a}. |
1374 See Figure \ref{feb21a}. |
1375 Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
1375 Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
1376 |
1376 |
1377 \begin{figure}[!ht] |
1377 \begin{figure}[!ht] |
1378 \begin{equation*} |
1378 \begin{equation*} |
1380 \end{equation*} |
1380 \end{equation*} |
1381 \caption{0-marked 1-ball and 0-marked 2-ball} |
1381 \caption{0-marked 1-ball and 0-marked 2-ball} |
1382 \label{feb21a} |
1382 \label{feb21a} |
1383 \end{figure} |
1383 \end{figure} |
1384 |
1384 |
1385 0-marked balls can be cut into smaller balls in various ways. |
1385 The $0$-marked balls can be cut into smaller balls in various ways. We only consider those decompositions in which the smaller balls are either |
1386 These smaller balls could be 0-marked or plain. |
1386 $0$-marked (i.e. intersect the $0$-marking of the large ball in a disc) or plain (don't intersect the $0$-marking of the large ball). |
1387 We can also take the boundary of a 0-marked ball, which is 0-marked sphere. |
1387 We can also take the boundary of a $0$-marked ball, which is $0$-marked sphere. |
1388 |
1388 |
1389 Fix $n$-categories $\cA$ and $\cB$. |
1389 Fix $n$-categories $\cA$ and $\cB$. |
1390 These will label the two halves of a 0-marked $k$-ball. |
1390 These will label the two halves of a $0$-marked $k$-ball. |
1391 The 0-sphere module we define next will depend on $\cA$ and $\cB$ |
1391 The $0$-sphere module we define next will depend on $\cA$ and $\cB$ |
1392 (it's an $\cA$-$\cB$ bimodule), but we will suppress that from the notation. |
1392 (it's an $\cA$-$\cB$ bimodule), but we will suppress that from the notation. |
1393 |
1393 |
1394 An $n$-category 0-sphere module $\cM$ is a collection of functors $\cM_k$ from the category |
1394 An $n$-category $0$-sphere module $\cM$ is a collection of functors $\cM_k$ from the category |
1395 of 0-marked $k$-balls, $1\le k \le n$, |
1395 of $0$-marked $k$-balls, $1\le k \le n$, |
1396 (with the two halves labeled by $\cA$ and $\cB$) to the category of sets. |
1396 (with the two halves labeled by $\cA$ and $\cB$) to the category of sets. |
1397 If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are. |
1397 If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are. |
1398 Given a decomposition of a 0-marked $k$-ball $X$ into smaller balls $X_i$, we have |
1398 Given a decomposition of a $0$-marked $k$-ball $X$ into smaller balls $X_i$, we have |
1399 morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) |
1399 morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) |
1400 or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side) |
1400 or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side) |
1401 or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball). |
1401 or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball). |
1402 Corresponding to this decomposition we have an action and/or composition map |
1402 Corresponding to this decomposition we have an action and/or composition map |
1403 from the product of these various sets into $\cM(X)$. |
1403 from the product of these various sets into $\cM(X)$. |
1404 |
1404 |
1405 \medskip |
1405 \medskip |
1406 |
1406 |
1407 Part of the structure of an $n$-category 0-sphere module is captured by saying it is |
1407 Part of the structure of an $n$-category 0-sphere module $\cM$ is captured by saying it is |
1408 a collection $\cD^{ab}$ of $n{-}1$-categories, indexed by pairs $(a, b)$ of objects (0-morphisms) |
1408 a collection $\cD^{ab}$ of $n{-}1$-categories, indexed by pairs $(a, b)$ of objects (0-morphisms) |
1409 of $\cA$ and $\cB$. |
1409 of $\cA$ and $\cB$. |
1410 Let $J$ be some standard 0-marked 1-ball (i.e.\ an interval with a marked point in its interior). |
1410 Let $J$ be some standard 0-marked 1-ball (i.e.\ an interval with a marked point in its interior). |
1411 Given a $j$-ball $X$, $0\le j\le n-1$, we define |
1411 Given a $j$-ball $X$, $0\le j\le n-1$, we define |
1412 \[ |
1412 \[ |
1413 \cD(X) \deq \cM(X\times J) . |
1413 \cD(X) \deq \cM(X\times J) . |
1414 \] |
1414 \] |
1415 The product is pinched over the boundary of $J$. |
1415 The product is pinched over the boundary of $J$. |
1416 $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
1416 The set $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
1417 (see Figure \ref{feb21b}). |
1417 (see Figure \ref{feb21b}). |
1418 These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
1418 These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
1419 |
1419 |
1420 \begin{figure}[!ht] |
1420 \begin{figure}[!ht] |
1421 \begin{equation*} |
1421 \begin{equation*} |
1427 |
1427 |
1428 More generally, consider an interval with interior marked points, and with the complements |
1428 More generally, consider an interval with interior marked points, and with the complements |
1429 of these points labeled by $n$-categories $\cA_i$ ($0\le i\le l$) and the marked points labeled |
1429 of these points labeled by $n$-categories $\cA_i$ ($0\le i\le l$) and the marked points labeled |
1430 by $\cA_i$-$\cA_{i+1}$ bimodules $\cM_i$. |
1430 by $\cA_i$-$\cA_{i+1}$ bimodules $\cM_i$. |
1431 (See Figure \ref{feb21c}.) |
1431 (See Figure \ref{feb21c}.) |
1432 To this data we can apply to coend construction as in Subsection \ref{moddecss} above |
1432 To this data we can apply the coend construction as in Subsection \ref{moddecss} above |
1433 to obtain an $\cA_0$-$\cA_l$ bimodule and, forgetfully, an $n{-}1$-category. |
1433 to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category. |
1434 This amounts to a definition of taking tensor products of bimodules over $n$-categories. |
1434 This amounts to a definition of taking tensor products of $0$-sphere module over $n$-categories. |
1435 |
1435 |
1436 \begin{figure}[!ht] |
1436 \begin{figure}[!ht] |
1437 \begin{equation*} |
1437 \begin{equation*} |
1438 \mathfig{1}{tempkw/feb21c} |
1438 \mathfig{1}{tempkw/feb21c} |
1439 \end{equation*} |
1439 \end{equation*} |
1443 |
1443 |
1444 We could also similarly mark and label a circle, obtaining an $n{-}1$-category |
1444 We could also similarly mark and label a circle, obtaining an $n{-}1$-category |
1445 associated to the marked and labeled circle. |
1445 associated to the marked and labeled circle. |
1446 (See Figure \ref{feb21c}.) |
1446 (See Figure \ref{feb21c}.) |
1447 If the circle is divided into two intervals, we can think of this $n{-}1$-category |
1447 If the circle is divided into two intervals, we can think of this $n{-}1$-category |
1448 as the 2-ended tensor product of the two bimodules associated to the two intervals. |
1448 as the 2-sided tensor product of the two bimodules associated to the two intervals. |
1449 |
1449 |
1450 \medskip |
1450 \medskip |
1451 |
1451 |
1452 Next we define $n$-category 1-sphere modules. |
1452 Next we define $n$-category 1-sphere modules. |
1453 These are just representations of (modules for) $n{-}1$-categories associated to marked and labeled |
1453 These are just representations of (modules for) $n{-}1$-categories associated to marked and labeled |
1556 \item morphisms of modules; show that it's adjoint to tensor product |
1556 \item morphisms of modules; show that it's adjoint to tensor product |
1557 (need to define dual module for this) |
1557 (need to define dual module for this) |
1558 \item functors |
1558 \item functors |
1559 \end{itemize} |
1559 \end{itemize} |
1560 |
1560 |
1561 \bigskip |
1561 |
1562 |
|
1563 \hrule |
|
1564 \nn{Some salvaged paragraphs that we might want to work back in:} |
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1565 \bigskip |
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1566 |
|
1567 Appendix \ref{sec:comparing-A-infty} explains the translation between this definition of an $A_\infty$ $1$-category and the usual one expressed in terms of `associativity up to higher homotopy', as in \cite{MR1854636}. (In this version of the paper, that appendix is incomplete, however.) |
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1568 |
|
1569 The motivating example is `chains of maps to $M$' for some fixed target space $M$. This is a topological $A_\infty$ category $\Xi_M$ with $\Xi_M(J) = C_*(\Maps(J \to M))$. The gluing maps $\Xi_M(J) \tensor \Xi_M(J') \to \Xi_M(J \cup J')$ takes the product of singular chains, then glues maps to $M$ together; the associativity condition is automatically satisfied. The evaluation map $\ev_{J,J'} : \CD{J \to J'} \tensor \Xi_M(J) \to \Xi_M(J')$ is the composition |
|
1570 \begin{align*} |
|
1571 \CD{J \to J'} \tensor C_*(\Maps(J \to M)) & \to C_*(\Diff(J \to J') \times \Maps(J \to M)) \\ & \to C_*(\Maps(J' \to M)), |
|
1572 \end{align*} |
|
1573 where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism. |
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1574 |
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1575 \hrule |
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