717 This map is natural with respect to the action of homeomorphisms and with respect to restrictions. |
717 This map is natural with respect to the action of homeomorphisms and with respect to restrictions. |
718 %\item Product morphisms. \nn{Hmm... not sure what to say here. maybe we need sets with structure after all.} |
718 %\item Product morphisms. \nn{Hmm... not sure what to say here. maybe we need sets with structure after all.} |
719 \end{itemize} |
719 \end{itemize} |
720 \end{axiom} |
720 \end{axiom} |
721 |
721 |
722 |
722 \medskip |
723 |
723 |
724 |
724 When the enriching category $\cS$ is chain complexes or topological spaces, |
725 \nn{a-inf starts by enriching over inf-cat; maybe simple version first, then peter's version (attrib to peter)} |
725 or more generally an appropriate sort of $\infty$-category, |
726 |
726 we can modify the extended isotopy axiom \ref{axiom:extended-isotopies} |
727 \nn{blarg} |
727 to require that families of homeomorphisms act |
728 |
728 and obtain an $A_\infty$ $n$-category. |
729 \nn{$k=n$ injectivity for a-inf (necessary?)} |
729 |
730 or if $k=n$ and we are in the $A_\infty$ case, |
730 We believe that abstract definitions should be guided by diverse collections |
731 |
731 of concrete examples, and a lack of diversity in our present collection of examples of $A_\infty$ $n$-categories |
732 |
732 makes us reluctant to commit to an all-encompassing general definition. |
733 \nn{resume revising here} |
733 Instead, we will give a relatively narrow definition which covers the examples we consider in this paper. |
734 |
734 After stating it, we will briefly discuss ways in which it can be made more general. |
735 |
735 |
736 \smallskip |
736 Assume that our $n$-morphisms are enriched over chain complexes. |
737 |
|
738 For $A_\infty$ $n$-categories, we replace |
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739 isotopy invariance with the requirement that families of homeomorphisms act. |
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740 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
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741 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
737 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
742 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
738 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
743 |
739 |
744 |
740 \nn{need to loosen for bbc reasons} |
745 |
741 |
746 %\addtocounter{axiom}{-1} |
|
747 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
742 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
748 \label{axiom:families} |
743 \label{axiom:families} |
749 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
744 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
750 \[ |
745 \[ |
751 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
746 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
752 \] |
747 \] |
753 These action maps are required to be associative up to homotopy, |
748 These action maps are required to be associative up to coherent homotopy, |
754 %\nn{iterated homotopy?} |
|
755 and also compatible with composition (gluing) in the sense that |
749 and also compatible with composition (gluing) in the sense that |
756 a diagram like the one in Theorem \ref{thm:CH} commutes. |
750 a diagram like the one in Theorem \ref{thm:CH} commutes. |
757 %\nn{repeat diagram here?} |
751 %\nn{repeat diagram here?} |
758 %\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
752 %\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
759 On $C_0(\Homeo_\bd(X))\ot \cC(X; c)$ the action should coincide |
753 On $C_0(\Homeo_\bd(X))\ot \cC(X; c)$ the action should coincide |
766 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
760 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
767 Having chains on the space of collar maps act gives rise to coherence maps involving |
761 Having chains on the space of collar maps act gives rise to coherence maps involving |
768 weak identities. |
762 weak identities. |
769 We will not pursue this in detail here. |
763 We will not pursue this in detail here. |
770 |
764 |
771 A potential variant on the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. (In fact, the alternative construction of the blob complex described in \S \ref{ss:alt-def} gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom; since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across.) |
765 One potential variant of the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. |
|
766 (In fact, the alternative construction of the blob complex described in \S \ref{ss:alt-def} |
|
767 gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom; |
|
768 since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across.) |
772 |
769 |
773 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
770 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
774 into a ordinary $n$-category (enriched over graded groups). |
771 into a ordinary $n$-category (enriched over graded groups). |
775 In a different direction, if we enrich over topological spaces instead of chain complexes, |
772 In a different direction, if we enrich over topological spaces instead of chain complexes, |
776 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
773 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
777 instead of $C_*(\Homeo_\bd(X))$. |
774 instead of $C_*(\Homeo_\bd(X))$. |
778 Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex |
775 Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex |
779 type $A_\infty$ $n$-category. |
776 type $A_\infty$ $n$-category. |
|
777 |
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778 One possibility for generalizing the above axiom to encompass a wider variety of examples goes as follows. |
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779 (Credit for these ideas goes to Peter Teichner, but of course the blame for any flaws goes to us.) |
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780 Let $\cS$ be an $A_\infty$ 1-category. |
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781 (We assume some prior notion of $A_\infty$ 1-category.) |
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782 Note that the category $\cB\cB\cC$ of balls with boundary conditions, defined above, is enriched in the category |
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783 of topological spaces, and hence can also be regarded as an $A_\infty$ 1-category. |
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784 \nn{...} |
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785 |
|
786 |
780 |
787 |
781 \medskip |
788 \medskip |
782 |
789 |
783 We define a $j$ times monoidal $n$-category to be an $(n{+}j)$-category $\cC$ where |
790 We define a $j$ times monoidal $n$-category to be an $(n{+}j)$-category $\cC$ where |
784 $\cC(X)$ is a trivial 1-element set if $X$ is a $k$-ball with $k<j$. |
791 $\cC(X)$ is a trivial 1-element set if $X$ is a $k$-ball with $k<j$. |