5 \section{$n$-categories (maybe)} |
5 \section{$n$-categories (maybe)} |
6 \label{sec:ncats} |
6 \label{sec:ncats} |
7 |
7 |
8 \nn{experimental section. maybe this should be rolled into other sections. |
8 \nn{experimental section. maybe this should be rolled into other sections. |
9 maybe it should be split off into a separate paper.} |
9 maybe it should be split off into a separate paper.} |
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10 |
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11 \subsection{Definition of $n$-categories} |
10 |
12 |
11 Before proceeding, we need more appropriate definitions of $n$-categories, |
13 Before proceeding, we need more appropriate definitions of $n$-categories, |
12 $A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
14 $A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
13 (As is the case throughout this paper, by ``$n$-category" we mean |
15 (As is the case throughout this paper, by ``$n$-category" we mean |
14 a weak $n$-category with strong duality.) |
16 a weak $n$-category with strong duality.) |
22 and so on. |
24 and so on. |
23 (This allows for strict associativity.) |
25 (This allows for strict associativity.) |
24 Still other definitions \nn{need refs for all these; maybe the Leinster book} |
26 Still other definitions \nn{need refs for all these; maybe the Leinster book} |
25 model the $k$-morphisms on more complicated combinatorial polyhedra. |
27 model the $k$-morphisms on more complicated combinatorial polyhedra. |
26 |
28 |
27 We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to a $k$-ball. |
29 We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to |
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30 the standard $k$-ball. |
28 In other words, |
31 In other words, |
29 |
32 |
30 \xxpar{Morphisms (preliminary version):} |
33 \xxpar{Morphisms (preliminary version):} |
31 {For any $k$-manifold $X$ homeomorphic |
34 {For any $k$-manifold $X$ homeomorphic |
32 to the standard $k$-ball, we have a set of $k$-morphisms |
35 to the standard $k$-ball, we have a set of $k$-morphisms |
349 define $\cC(X; c) = \bc^C_*(X\times F; c)$, where $X$ is an $n$-ball |
352 define $\cC(X; c) = \bc^C_*(X\times F; c)$, where $X$ is an $n$-ball |
350 and $\bc^C_*$ denotes the blob complex based on $C$. |
353 and $\bc^C_*$ denotes the blob complex based on $C$. |
351 |
354 |
352 \end{itemize} |
355 \end{itemize} |
353 |
356 |
354 \medskip |
357 |
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358 |
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359 |
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360 |
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361 |
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362 \subsection{From $n$-categories to systems of fields} |
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363 |
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364 We can extend the functors $\cC$ above from $k$-balls to arbitrary $k$-manifolds as follows. |
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365 |
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366 Let $W$ be a $k$-manifold, $1\le k \le n$. |
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367 We will define a set $\cC(W)$. |
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368 (If $k = n$ and our $k$-categories are enriched, then |
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369 $\cC(W)$ will have additional structure; see below.) |
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370 $\cC(W)$ will be the colimit of a functor defined on a category $\cJ(W)$, |
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371 which we define next. |
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372 |
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373 Define a permissible decomposition of $W$ to be a decomposition |
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374 \[ |
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375 W = \bigcup_a X_a , |
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376 \] |
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377 where each $X_a$ is a $k$-ball. |
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378 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
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379 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
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380 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
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381 (The objects of $\cJ(W)$ are permissible decompositions of $W$, and there is a unique |
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382 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
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383 \nn{need figures} |
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384 |
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385 $\cC$ determines |
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386 a functor $\psi_\cC$ from $\cJ(W)$ to the category of sets |
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387 (possibly with additional structure if $k=n$). |
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388 For a decomposition $x = (X_a)$ in $\cJ(W)$, define $\psi_\cC(x)$ to be the subset |
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389 \[ |
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390 \psi_\cC(x) \sub \prod_a \cC(X_a) |
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391 \] |
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392 such that the restrictions to the various pieces of shared boundaries amongst the |
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393 $X_a$ all agree. |
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394 (Think fibered product.) |
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395 If $x$ is a refinement of $y$, define a map $\psi_\cC(x)\to\psi_\cC(y)$ |
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396 via the composition maps of $\cC$. |
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397 |
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398 Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$. |
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399 In other words, for each decomposition $x$ there is a map |
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400 $\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps |
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401 above, and $\cC(W)$ is universal with respect to these properties. |
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402 |
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403 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
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404 |
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405 It is easy to see that |
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406 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
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407 comprise a natural transformation of functors. |
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408 |
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409 \nn{need to finish explaining why we have a system of fields; |
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410 need to say more about ``homological" fields? |
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411 (actions of homeomorphisms); |
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412 define $k$-cat $\cC(\cdot\times W)$} |
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413 |
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414 |
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415 |
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416 \subsection{Modules} |
355 |
417 |
356 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
418 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
357 a.k.a.\ actions). |
419 a.k.a.\ actions). |
358 The definition will be very similar to that of $n$-categories. |
420 The definition will be very similar to that of $n$-categories. |
359 |
421 |
557 of marked 1-balls, call them left-marked and right-marked, |
619 of marked 1-balls, call them left-marked and right-marked, |
558 and hence there are two types of modules, call them right modules and left modules. |
620 and hence there are two types of modules, call them right modules and left modules. |
559 In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
621 In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
560 there is no left/right module distinction. |
622 there is no left/right module distinction. |
561 |
623 |
562 \medskip |
624 |
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625 \subsection{Modules as boundary labels} |
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626 |
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627 Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
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628 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |
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629 component $\bd_i W$ of $W$. |
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630 |
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631 We will define a set $\cC(W, \cN)$ using a colimit construction similar to above. |
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632 \nn{give ref} |
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633 (If $k = n$ and our $k$-categories are enriched, then |
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634 $\cC(W, \cN)$ will have additional structure; see below.) |
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635 |
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636 Define a permissible decomposition of $W$ to be a decomposition |
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637 \[ |
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638 W = (\bigcup_a X_a) \cup (\bigcup_{i,b} M_{ib}) , |
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639 \] |
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640 where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and |
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641 each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, |
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642 with $M_{ib}\cap\bd_i W$ being the marking. |
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643 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
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644 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
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645 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
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646 (The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
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647 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
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648 \nn{need figures} |
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649 |
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650 $\cN$ determines |
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651 a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets |
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652 (possibly with additional structure if $k=n$). |
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653 For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset |
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654 \[ |
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655 \psi_\cN(x) \sub (\prod_a \cC(X_a)) \prod (\prod_{ib} \cN_i(M_{ib})) |
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656 \] |
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657 such that the restrictions to the various pieces of shared boundaries amongst the |
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658 $X_a$ and $M_{ib}$ all agree. |
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659 (Think fibered product.) |
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660 If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
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661 via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
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662 |
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663 Finally, define $\cC(W, \cN)$ to be the colimit of $\psi_\cN$. |
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664 In other words, for each decomposition $x$ there is a map |
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665 $\psi(x)\to \cC(W, \cN)$, these maps are compatible with the refinement maps |
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666 above, and $\cC(W, \cN)$ is universal with respect to these properties. |
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667 |
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668 \nn{boundary restrictions, $k$-cat $\cC(\cdot\times W; N)$ etc.} |
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669 |
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670 \subsection{Tensor products} |
563 |
671 |
564 Next we consider tensor products (or, more generally, self tensor products |
672 Next we consider tensor products (or, more generally, self tensor products |
565 or coends). |
673 or coends). |
566 |
674 |
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675 \nn{maybe ``tensor product" is not the best name?} |
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676 |
567 \nn{start with (less general) tensor products; maybe change this later} |
677 \nn{start with (less general) tensor products; maybe change this later} |
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678 |
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679 ** \nn{stuff below needs to be rewritten (shortened), because of new subsections above} |
568 |
680 |
569 Let $\cM$ and $\cM'$ be modules for an $n$-category $\cC$. |
681 Let $\cM$ and $\cM'$ be modules for an $n$-category $\cC$. |
570 (If $k=1$ and manifolds are oriented, then one should be |
682 (If $k=1$ and manifolds are oriented, then one should be |
571 a left module and the other a right module.) |
683 a left module and the other a right module.) |
572 We will define an $n{-}1$-category $\cM\ot_\cC\cM'$, which depend (functorially) |
684 We will define an $n{-}1$-category $\cM\ot_\cC\cM'$, which depend (functorially) |