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 \nn{start with (less general) tensor products; maybe change this later} |
567 \nn{start with (less general) tensor products; maybe change this later} |
568 |
568 |
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 $\cC$. |
569 Let $\cM$ and $\cM'$ be modules for an $n$-category $\cC$. |
573 (If $k=1$ and manifolds are oriented, then one should be |
570 (If $k=1$ and manifolds are oriented, then one should be |
574 a left module and the other a right module.) |
571 a left module and the other a right module.) |
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572 We will define an $n{-}1$-category $\cM\ot_\cC\cM'$, which depend (functorially) |
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573 on a choice of 1-ball (interval) $J$. |
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574 |
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575 Define a {\it doubly marked $k$-ball} to be a triple $(B, N, N')$, where $B$ is a $k$-ball |
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576 and $N$ and $N'$ are disjoint $k{-}1$-balls in $\bd B$. |
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577 |
575 Let $D = (B, N, N')$ be a doubly marked $k$-ball, $1\le k \le n$. |
578 Let $D = (B, N, N')$ be a doubly marked $k$-ball, $1\le k \le n$. |
576 We will define a set $\cM\ot_\cC\cM'(D)$. |
579 We will define a set $\cM\ot_\cC\cM'(D)$. |
577 (If $k = n$ and our $k$-categories are enriched, then |
580 (If $k = n$ and our $k$-categories are enriched, then |
578 $\cM\ot_\cC\cM'(D)$ will have additional structure; see below.) |
581 $\cM\ot_\cC\cM'(D)$ will have additional structure; see below.) |
579 $\cM\ot_\cC\cM'(D)$ will be the colimit of a functor defined on a category $\cJ(D)$, |
582 $\cM\ot_\cC\cM'(D)$ will be the colimit of a functor defined on a category $\cJ(D)$, |
594 \nn{need figures} |
597 \nn{need figures} |
595 |
598 |
596 $\cC$, $\cM$ and $\cM'$ determine |
599 $\cC$, $\cM$ and $\cM'$ determine |
597 a functor $\psi$ from $\cJ(D)$ to the category of sets |
600 a functor $\psi$ from $\cJ(D)$ to the category of sets |
598 (possibly with additional structure if $k=n$). |
601 (possibly with additional structure if $k=n$). |
599 For a decomposition $x = (X_a, M_b, M'_c)$ in $\cJ(D)$, define $\psi(x)$ to subset |
602 For a decomposition $x = (X_a, M_b, M'_c)$ in $\cJ(D)$, define $\psi(x)$ to be the subset |
600 \[ |
603 \[ |
601 \psi(x) \sub (\prod_a \cC(X_a)) \prod (\prod_b \cM(M_b)) \prod (\prod_c \cM'(M'_c)) |
604 \psi(x) \sub (\prod_a \cC(X_a)) \prod (\prod_b \cM(M_b)) \prod (\prod_c \cM'(M'_c)) |
602 \] |
605 \] |
603 such that the restrictions to the various pieces of shared boundaries amongst the |
606 such that the restrictions to the various pieces of shared boundaries amongst the |
604 $X_a$, $M_b$ and $M'_c$ all agree. |
607 $X_a$, $M_b$ and $M'_c$ all agree. |
609 Finally, define $\cM\ot_\cC\cM'(D)$ to be the colimit of $\psi$. |
612 Finally, define $\cM\ot_\cC\cM'(D)$ to be the colimit of $\psi$. |
610 In other words, for each decomposition $x$ there is a map |
613 In other words, for each decomposition $x$ there is a map |
611 $\psi(x)\to \cM\ot_\cC\cM'(D)$, these maps are compatible with the refinement maps |
614 $\psi(x)\to \cM\ot_\cC\cM'(D)$, these maps are compatible with the refinement maps |
612 above, and $\cM\ot_\cC\cM'(D)$ is universal with respect to these properties. |
615 above, and $\cM\ot_\cC\cM'(D)$ is universal with respect to these properties. |
613 |
616 |
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617 Define a {\it marked $k$-annulus} to be a manifold homeomorphic |
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618 to $S^{k-1}\times I$, with its entire boundary ``marked". |
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619 Define the boundary of a doubly marked $k$-ball $(B, N, N')$ to be the marked |
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620 $k{-}1$-annulus $\bd B \setmin(N\cup N')$. |
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621 |
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622 Using a colimit construction similar to the one above, we can define a set |
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623 $\cM\ot_\cC\cM'(A)$ for any marked $k$-annulus $A$ (for $k < n$). |
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624 |
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625 $\cM\ot_\cC\cM'$ is (among other things) a functor from the category of |
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626 doubly marked $k$-balls ($k\le n$) and homeomorphisms to the category of sets. |
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627 We have other functors, also denoted $\cM\ot_\cC\cM'$, from the category of |
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628 marked $k$-annuli ($k < n$) and homeomorphisms to the category of sets. |
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629 |
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630 For each marked $k$-ball $D$ there is a restriction map |
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631 \[ |
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632 \bd : \cM\ot_\cC\cM(D) \to \cM\ot_\cC\cM(\bd D) . |
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633 \] |
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634 These maps comprise a natural transformation of functors. |
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635 \nn{possible small problem: might need to define $\cM$ of a singly marked annulus} |
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636 |
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637 For $c \in \cM\ot_\cC\cM(\bd D)$, let |
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638 \[ |
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639 \cM\ot_\cC\cM(D; c) \deq \bd\inv(c) . |
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640 \] |
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641 |
614 Note that if $k=n$ and we fix boundary conditions $c$ on the unmarked boundary of $D$, |
642 Note that if $k=n$ and we fix boundary conditions $c$ on the unmarked boundary of $D$, |
615 then $\cM\ot_\cC\cM'(D; c)$ will be an object in the enriching category |
643 then $\cM\ot_\cC\cM'(D; c)$ will be an object in the enriching category |
616 (e.g.\ vector space or chain complex). |
644 (e.g.\ vector space or chain complex). |
617 \nn{say this more precisely?} |
645 |
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646 Let $J$ be a doubly marked 1-ball (i.e. an interval, where we think of both endpoints |
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647 as marked). |
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648 For $X$ a plain $k$-ball ($k \le n-1$) or $k$-sphere ($k \le n-2$), define |
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649 \[ |
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650 \cM\ot_\cC\cM'(X) \deq \cM\ot_\cC\cM'(X\times J) . |
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651 \] |
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652 We claim that $\cM\ot_\cC\cM'$ has the structure of an $n{-}1$-category. |
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653 We have already defined restriction maps $\bd : \cM\ot_\cC\cM'(X) \to |
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654 \cM\ot_\cC\cM'(\bd X)$. |
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655 The only data for the $n{-}1$-category that we have not defined yet are the product |
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656 morphisms. |
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657 \nn{so next define those} |
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658 |
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659 \nn{need to check whether any of the steps in verifying that we have |
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660 an $n{-}1$-category are non-trivial.} |
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661 |
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662 |
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663 |
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664 |
618 |
665 |
619 \medskip |
666 \medskip |
620 \hrule |
667 \hrule |
621 \medskip |
668 \medskip |
622 |
669 |
625 |
672 |
626 |
673 |
627 Stuff that remains to be done (either below or in an appendix or in a separate section or in |
674 Stuff that remains to be done (either below or in an appendix or in a separate section or in |
628 a separate paper): |
675 a separate paper): |
629 \begin{itemize} |
676 \begin{itemize} |
630 \item tensor products |
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631 \item traditional $n$-cat defs (e.g. *-1-cat, pivotal 2-cat) imply our def of plain $n$-cat |
677 \item traditional $n$-cat defs (e.g. *-1-cat, pivotal 2-cat) imply our def of plain $n$-cat |
632 \item conversely, our def implies other defs |
678 \item conversely, our def implies other defs |
633 \item do same for modules; maybe an appendix on relating topological |
679 \item do same for modules; maybe an appendix on relating topological |
634 vs traditional defs, $n = 1,2$, $A_\infty$ or not, cats, modules, tensor products |
680 vs traditional defs, $n = 1,2$, $A_\infty$ or not, cats, modules, tensor products |
635 \item traditional $A_\infty$ 1-cat def implies our def |
681 \item traditional $A_\infty$ 1-cat def implies our def |