442 such that the restrictions to the various pieces of shared boundaries amongst the |
442 such that the restrictions to the various pieces of shared boundaries amongst the |
443 $X_a$ all agree. |
443 $X_a$ all agree. |
444 (Think fibered product.) |
444 (Think fibered product.) |
445 If $x$ is a refinement of $y$, define a map $\psi_\cC(x)\to\psi_\cC(y)$ |
445 If $x$ is a refinement of $y$, define a map $\psi_\cC(x)\to\psi_\cC(y)$ |
446 via the composition maps of $\cC$. |
446 via the composition maps of $\cC$. |
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447 (If $\dim(W) = n$ then we need to also make use of the monoidal |
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448 product in the enriching category. |
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449 \nn{should probably be more explicit here}) |
447 |
450 |
448 Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$. |
451 Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$. |
449 In other words, for each decomposition $x$ there is a map |
452 In the plain (non-$A_\infty$) case, this means that |
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453 for each decomposition $x$ there is a map |
450 $\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps |
454 $\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps |
451 above, and $\cC(W)$ is universal with respect to these properties. |
455 above, and $\cC(W)$ is universal with respect to these properties. |
452 \nn{in A-inf case, need to say more} |
456 In the $A_\infty$ case, it means |
453 |
457 \nn{.... need to check if there is a def in the literature before writing this down} |
454 \nn{should give more concrete description (two cases)} |
458 |
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459 More concretely, in the plain case enriched over vector spaces, and with $\dim(W) = n$, we can take |
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460 \[ |
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461 \cC(W) = \left( \oplus_x \psi_\cC(x)\right) \big/ K |
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462 \] |
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463 where $K$ is generated by all things of the form $a - g(a)$, where |
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464 $a\in \psi_\cC(x)$ for some decomposition $x$, $x\le y$, and $g: \psi_\cC(x) |
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465 \to \psi_\cC(y)$ is value of $\psi_\cC$ on the antirefinement $x\to y$. |
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466 |
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467 In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit |
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468 is as follows. |
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469 Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_{m-1}$ of permissible decompositions. |
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470 Such sequences (for all $m$) form a simplicial set. |
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471 Let |
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472 \[ |
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473 V = \bigoplus_{(x_i)} \psi_\cC(x_0) , |
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474 \] |
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475 where the sum is over all $m$-sequences and all $m$. |
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476 We endow $V$ with a differential which is the sum of the differential of the $\psi_\cC(x_0)$ |
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477 summands plus another term using the differential of the simplicial set of $m$-sequences. |
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478 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
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479 summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
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480 \[ |
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481 \bd (a, \bar{x}) = (\bd a, \bar{x}) \pm (g(a), d_0(\bar{x})) + \sum_{j=1}^k \pm (a, d_j(\bar{x})) , |
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482 \] |
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483 where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
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484 is the usual map. |
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485 \nn{need to say this better} |
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486 \nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
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487 combine only two balls at a time; for $n=1$ this version will lead to usual definition |
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488 of $A_\infty$ category} |
455 |
489 |
456 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
490 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
457 |
491 |
458 It is easy to see that |
492 It is easy to see that |
459 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
493 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
734 Finally, define $\cC(W, \cN)$ to be the colimit of $\psi_\cN$. |
767 Finally, define $\cC(W, \cN)$ to be the colimit of $\psi_\cN$. |
735 In other words, for each decomposition $x$ there is a map |
768 In other words, for each decomposition $x$ there is a map |
736 $\psi(x)\to \cC(W, \cN)$, these maps are compatible with the refinement maps |
769 $\psi(x)\to \cC(W, \cN)$, these maps are compatible with the refinement maps |
737 above, and $\cC(W, \cN)$ is universal with respect to these properties. |
770 above, and $\cC(W, \cN)$ is universal with respect to these properties. |
738 |
771 |
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772 More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$. |
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773 \nn{need to say more?} |
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774 |
739 \nn{boundary restrictions, $k$-cat $\cC(\cdot\times W; N)$ etc.} |
775 \nn{boundary restrictions, $k$-cat $\cC(\cdot\times W; N)$ etc.} |
740 |
776 |
741 \subsection{Tensor products} |
777 \subsection{Tensor products} |
742 |
778 |
743 Next we consider tensor products (or, more generally, self tensor products |
779 Next we consider tensor products. |
744 or coends). |
780 |
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781 \nn{what about self tensor products /coends ?} |
745 |
782 |
746 \nn{maybe ``tensor product" is not the best name?} |
783 \nn{maybe ``tensor product" is not the best name?} |
747 |
784 |
748 \nn{start with (less general) tensor products; maybe change this later} |
785 \nn{start with (less general) tensor products; maybe change this later} |
749 |
786 |
750 ** \nn{stuff below needs to be rewritten (shortened), because of new subsections above} |
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751 |
787 |
752 Let $\cM$ and $\cM'$ be modules for an $n$-category $\cC$. |
788 Let $\cM$ and $\cM'$ be modules for an $n$-category $\cC$. |
753 (If $k=1$ and manifolds are oriented, then one should be |
789 (If $k=1$ and manifolds are oriented, then one should be |
754 a left module and the other a right module.) |
790 a left module and the other a right module.) |
755 We will define an $n{-}1$-category $\cM\ot_\cC\cM'$, which depends (functorially) |
791 We will define an $n{-}1$-category $\cM\ot_\cC\cM'$, which depends (functorially) |
756 on a choice of 1-ball (interval) $J$. |
792 on a choice of 1-ball (interval) $J$. |
757 |
793 |
758 |
794 Let $p$ and $p'$ be the boundary points of $J$. |
759 |
795 Given a $k$-ball $X$, let $(X\times J, \cM, \cM')$ denote $X\times J$ with |
760 |
796 $X\times\{p\}$ labeled by $\cM$ and $X\times\{p'\}$ labeled by $\cM'$, as in Subsection \ref{moddecss}. |
761 |
797 Let |
762 |
798 \[ |
763 Define a {\it doubly marked $k$-ball} to be a triple $(B, N, N')$, where $B$ is a $k$-ball |
799 \cT(X) \deq \cC(X\times J, \cM, \cM') , |
764 and $N$ and $N'$ are disjoint $k{-}1$-balls in $\bd B$. |
800 \] |
765 |
801 where the right hand side is the colimit construction defined in Subsection \ref{moddecss}. |
766 Let $D = (B, N, N')$ be a doubly marked $k$-ball, $1\le k \le n$. |
802 It is not hard to see that $\cT$ becomes an $n{-}1$-category. |
767 We will define a set $\cM\ot_\cC\cM'(D)$. |
803 \nn{maybe follows from stuff (not yet written) in previous subsection?} |
768 (If $k = n$ and our $k$-categories are enriched, then |
804 |
769 $\cM\ot_\cC\cM'(D)$ will have additional structure; see below.) |
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770 $\cM\ot_\cC\cM'(D)$ will be the colimit of a functor defined on a category $\cJ(D)$, |
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771 which we define next. |
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772 |
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773 Define a permissible decomposition of $D$ to be a decomposition |
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774 \[ |
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775 D = (\cup_a X_a) \cup (\cup_b M_b) \cup (\cup_c M'_c) , |
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776 \] |
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777 Where each $X_a$ is a plain $k$-ball (disjoint from the markings $N$ and $N'$ of $D$), |
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778 each $M_b$ is a marked $k$-ball intersecting $N$, and |
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779 each $M'_b$ is a marked $k$-ball intersecting $N'$. |
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780 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
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781 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
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782 This defines a partial ordering $\cJ(D)$, which we will think of as a category. |
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783 (The objects of $\cJ(D)$ are permissible decompositions of $D$, and there is a unique |
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784 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
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785 \nn{need figures} |
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786 |
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787 $\cC$, $\cM$ and $\cM'$ determine |
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788 a functor $\psi$ from $\cJ(D)$ to the category of sets |
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789 (possibly with additional structure if $k=n$). |
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790 For a decomposition $x = (X_a, M_b, M'_c)$ in $\cJ(D)$, define $\psi(x)$ to be the subset |
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791 \[ |
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792 \psi(x) \sub (\prod_a \cC(X_a)) \times (\prod_b \cM(M_b)) \times (\prod_c \cM'(M'_c)) |
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793 \] |
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794 such that the restrictions to the various pieces of shared boundaries amongst the |
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795 $X_a$, $M_b$ and $M'_c$ all agree. |
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796 (Think fibered product.) |
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797 If $x$ is a refinement of $y$, define a map $\psi(x)\to\psi(y)$ |
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798 via the gluing (composition or action) maps from $\cC$, $\cM$ and $\cM'$. |
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799 |
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800 Finally, define $\cM\ot_\cC\cM'(D)$ to be the colimit of $\psi$. |
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801 In other words, for each decomposition $x$ there is a map |
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802 $\psi(x)\to \cM\ot_\cC\cM'(D)$, these maps are compatible with the refinement maps |
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803 above, and $\cM\ot_\cC\cM'(D)$ is universal with respect to these properties. |
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804 |
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805 Define a {\it marked $k$-annulus} to be a manifold homeomorphic |
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806 to $S^{k-1}\times I$, with its entire boundary ``marked". |
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807 Define the boundary of a doubly marked $k$-ball $(B, N, N')$ to be the marked |
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808 $k{-}1$-annulus $\bd B \setmin(N\cup N')$. |
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809 |
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810 Using a colimit construction similar to the one above, we can define a set |
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811 $\cM\ot_\cC\cM'(A)$ for any marked $k$-annulus $A$ (for $k < n$). |
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812 |
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813 $\cM\ot_\cC\cM'$ is (among other things) a functor from the category of |
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814 doubly marked $k$-balls ($k\le n$) and homeomorphisms to the category of sets. |
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815 We have other functors, also denoted $\cM\ot_\cC\cM'$, from the category of |
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816 marked $k$-annuli ($k < n$) and homeomorphisms to the category of sets. |
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817 |
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818 For each marked $k$-ball $D$ there is a restriction map |
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819 \[ |
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820 \bd : \cM\ot_\cC\cM(D) \to \cM\ot_\cC\cM(\bd D) . |
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821 \] |
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822 These maps comprise a natural transformation of functors. |
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823 \nn{possible small problem: might need to define $\cM$ of a singly marked annulus} |
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824 |
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825 For $c \in \cM\ot_\cC\cM(\bd D)$, let |
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826 \[ |
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827 \cM\ot_\cC\cM(D; c) \deq \bd\inv(c) . |
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828 \] |
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829 |
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830 Note that if $k=n$ and we fix boundary conditions $c$ on the unmarked boundary of $D$, |
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831 then $\cM\ot_\cC\cM'(D; c)$ will be an object in the enriching category |
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832 (e.g.\ vector space or chain complex). |
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833 |
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834 Let $J$ be a doubly marked 1-ball (i.e. an interval, where we think of both endpoints |
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835 as marked). |
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836 For $X$ a plain $k$-ball ($k \le n-1$) or $k$-sphere ($k \le n-2$), define |
|
837 \[ |
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838 \cM\ot_\cC\cM'(X) \deq \cM\ot_\cC\cM'(X\times J) . |
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839 \] |
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840 We claim that $\cM\ot_\cC\cM'$ has the structure of an $n{-}1$-category. |
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841 We have already defined restriction maps $\bd : \cM\ot_\cC\cM'(X) \to |
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842 \cM\ot_\cC\cM'(\bd X)$. |
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843 The only data for the $n{-}1$-category that we have not defined yet are the product |
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844 morphisms. |
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845 \nn{so next define those} |
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846 |
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847 \nn{need to check whether any of the steps in verifying that we have |
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848 an $n{-}1$-category are non-trivial.} |
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849 |
805 |
850 |
806 |
851 |
807 |
852 |
808 |
853 |
809 |