562 \medskip |
562 \medskip |
563 |
563 |
564 Next we consider tensor products (or, more generally, self tensor products |
564 Next we consider tensor products (or, more generally, self tensor products |
565 or coends). |
565 or coends). |
566 |
566 |
567 |
567 \nn{start with (less general) tensor products; maybe change this later} |
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568 |
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569 Define a {\it doubly marked $k$-ball} to be a triple $(B, N, N')$, where $B$ is a $k$-ball |
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570 and $N$ and $N'$ are disjoint $k{-}1$-balls in $\bd B$. |
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571 |
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572 Let $\cM$ and $\cM'$ be modules for an $n$-category $\cC$. |
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573 (If $k=1$ and manifolds are oriented, then one should be |
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574 a left module and the other a right module.) |
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575 Let $D = (B, N, N')$ be a doubly marked $k$-ball, $1\le k \le n$. |
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576 We will define a set $\cM\ot_\cC\cM'(D)$. |
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577 (If $k = n$ and our $k$-categories are enriched, then |
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578 $\cM\ot_\cC\cM'(D)$ will have additional structure; see below.) |
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579 $\cM\ot_\cC\cM'(D)$ will be the colimit of a functor defined on a category $\cJ(D)$, |
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580 which we define next. |
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581 |
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582 Define a permissible decomposition of $D$ to be a decomposition |
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583 \[ |
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584 D = (\cup_a X_a) \cup (\cup_b M_b) \cup (\cup_c M'_c) , |
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585 \] |
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586 Where each $X_a$ is a plain $k$-ball (disjoint from the markings $N$ and $N'$ of $D$), |
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587 each $M_b$ is a marked $k$-ball intersecting $N$, and |
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588 each $M'_b$ is a marked $k$-ball intersecting $N'$. |
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589 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
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590 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
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591 This defines a partial ordering $\cJ(D)$, which we will think of as a category. |
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592 (The objects of $\cJ(D)$ are permissible decompositions of $D$, and there is a unique |
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593 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
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594 \nn{need figures} |
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595 |
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596 $\cC$, $\cM$ and $\cM'$ determine |
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597 a functor $\psi$ from $\cJ(D)$ to the category of sets |
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598 (possibly with additional structure if $k=n$). |
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599 For a decomposition $x = (X_a, M_b, M'_c)$ in $\cJ(D)$, define $\psi(x)$ to subset |
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600 \[ |
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601 \psi(x) \sub (\prod_a \cC(X_a)) \prod (\prod_b \cM(M_b)) \prod (\prod_c \cM'(M'_c)) |
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602 \] |
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603 such that the restrictions to the various pieces of shared boundaries amongst the |
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604 $X_a$, $M_b$ and $M'_c$ all agree. |
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605 (Think fibered product.) |
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606 If $x$ is a refinement of $y$, define a map $\psi(x)\to\psi(y)$ |
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607 via the gluing (composition or action) maps from $\cC$, $\cM$ and $\cM'$. |
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608 |
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609 Finally, define $\cM\ot_\cC\cM'(D)$ to be the colimit of $\psi$. |
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610 In other words, for each decomposition $x$ there is a map |
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611 $\psi(x)\to \cM\ot_\cC\cM'(D)$, these maps are compatible with the refinement maps |
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612 above, and $\cM\ot_\cC\cM'(D)$ is universal with respect to these properties. |
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613 |
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614 Note that if $k=n$ and we fix boundary conditions $c$ on the unmarked boundary of $D$, |
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615 then $\cM\ot_\cC\cM'(D; c)$ will be an object in the enriching category |
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616 (e.g.\ vector space or chain complex). |
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617 \nn{say this more precisely?} |
568 |
618 |
569 \medskip |
619 \medskip |
570 \hrule |
620 \hrule |
571 \medskip |
621 \medskip |
572 |
622 |