659 In addition, collar maps act trivially on $\cC(X)$. |
659 In addition, collar maps act trivially on $\cC(X)$. |
660 \end{axiom} |
660 \end{axiom} |
661 |
661 |
662 \medskip |
662 \medskip |
663 |
663 |
664 |
664 This completes the definition of an $n$-category. |
665 |
665 Next we define enriched $n$-categories. |
666 |
666 |
667 \nn{begin temp relocation} |
667 \medskip |
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668 |
668 |
669 |
669 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
670 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
670 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
671 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
671 all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category |
672 all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some appropriate auxiliary category |
672 with sufficient limits and colimits |
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673 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
673 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
674 %\nn{actually, need both disj-union/sum and product/tensor-product; what's the name for this sort of cat?} |
674 and all the structure maps of the $n$-category are compatible with the auxiliary |
675 and all the structure maps of the $n$-category should be compatible with the auxiliary |
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676 category structure. |
675 category structure. |
677 Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then |
676 Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then |
678 $\cC(Y; c)$ is just a plain set. |
677 $\cC(Y; c)$ is just a plain set. |
679 |
678 |
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679 We will aim for a little bit more generality than we need and not assume that the objects |
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680 of our auxiliary category are sets with extra structure. |
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681 First we must specify requirements for the auxiliary category. |
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682 It should have a {\it distributive monoidal structure} in the sense of |
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683 \nn{Stolz and Teichner, Traces in monoidal categories, 1010.4527}. |
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684 This means that there is a monoidal structure $\otimes$ and also coproduct $\oplus$, |
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685 and these two structures interact in the appropriate way. |
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686 Examples include |
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687 \begin{itemize} |
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688 \item vector spaces (or $R$-modules or chain complexes) with tensor product and direct sum; and |
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689 \item topological spaces with product and disjoint union. |
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690 \end{itemize} |
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691 Before stating the revised axioms for an $n$-category enriched in a distributive monoidal category, |
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692 we need a preliminary definition. |
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693 Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the |
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694 category $\bbc$ of {\it $n$-balls with boundary conditions}. |
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695 Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \cl\cC(\bd X)$ is the ``boundary condition". |
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696 Its morphisms are homeomorphisms $f:X\to X$ such that $f|_{\bd X}(c) = c$. |
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697 |
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698 \begin{axiom}[Enriched $n$-categories] |
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699 \label{axiom:enriched} |
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700 Let $\cS$ be a distributive symmetric monoidal category. |
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701 An $n$-category enriched in $\cS$ satisfies the above $n$-category axioms for $k=0,\ldots,n-1$, |
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702 and modifies the axioms for $k=n$ as follows: |
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703 \begin{itemize} |
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704 \item Morphisms. We have a functor $\cC_n$ from $\bbc$ ($n$-balls with boundary conditions) to $\cS$. |
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705 \item Composition. Let $B = B_1\cup_Y B_2$ as in Axiom \ref{axiom:composition}. |
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706 Let $Y_i = \bd B_i \setmin Y$. |
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707 Note that $\bd B = Y_1\cup Y_2$. |
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708 Let $c_i \in \cC(Y_i)$ with $\bd c_1 = \bd c_2 = d \in \cl\cC(E)$. |
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709 Then we have a map |
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710 \[ |
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711 \gl_Y : \bigoplus_c \cC(B_1; c_1 \bullet c) \otimes \cC(B_2; c_2\bullet c) \to \cC(B; c_1\bullet c_2), |
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712 \] |
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713 where the sum is over $c\in\cC(Y)$ such that $\bd c = d$. |
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714 This map is natural with respect to the action of homeomorphisms and with respect to restrictions. |
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715 \item Product morphisms. \nn{Hmm... not sure what to say here. maybe we need sets with structure after all.} |
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716 \end{itemize} |
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717 \end{axiom} |
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718 |
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719 |
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720 |
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721 |
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722 \nn{a-inf starts by enriching over inf-cat; maybe simple version first, then peter's version (attrib to peter)} |
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723 |
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724 \nn{blarg} |
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725 |
680 \nn{$k=n$ injectivity for a-inf (necessary?)} |
726 \nn{$k=n$ injectivity for a-inf (necessary?)} |
681 or if $k=n$ and we are in the $A_\infty$ case, |
727 or if $k=n$ and we are in the $A_\infty$ case, |
682 |
728 |
683 |
729 |
684 \nn{end temp relocation} |
730 \nn{resume revising here} |
685 |
731 |
686 |
732 |
687 \smallskip |
733 \smallskip |
688 |
734 |
689 For $A_\infty$ $n$-categories, we replace |
735 For $A_\infty$ $n$-categories, we replace |
735 $\cC(X)$ is a trivial 1-element set if $X$ is a $k$-ball with $k<j$. |
781 $\cC(X)$ is a trivial 1-element set if $X$ is a $k$-ball with $k<j$. |
736 See Example \ref{ex:bord-cat}. |
782 See Example \ref{ex:bord-cat}. |
737 |
783 |
738 \medskip |
784 \medskip |
739 |
785 |
740 The alert reader will have already noticed that our definition of a (ordinary) $n$-category |
786 The alert reader will have already noticed that our definition of an (ordinary) $n$-category |
741 is extremely similar to our definition of a system of fields. |
787 is extremely similar to our definition of a system of fields. |
742 There are two differences. |
788 There are two differences. |
743 First, for the $n$-category definition we restrict our attention to balls |
789 First, for the $n$-category definition we restrict our attention to balls |
744 (and their boundaries), while for fields we consider all manifolds. |
790 (and their boundaries), while for fields we consider all manifolds. |
745 Second, in category definition we directly impose isotopy |
791 Second, in category definition we directly impose isotopy |