694 Before stating the revised axioms for an $n$-category enriched in a distributive monoidal category, |
694 Before stating the revised axioms for an $n$-category enriched in a distributive monoidal category, |
695 we need a preliminary definition. |
695 we need a preliminary definition. |
696 Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the |
696 Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the |
697 category $\bbc$ of {\it $n$-balls with boundary conditions}. |
697 category $\bbc$ of {\it $n$-balls with boundary conditions}. |
698 Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \cl\cC(\bd X)$ is the ``boundary condition". |
698 Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \cl\cC(\bd X)$ is the ``boundary condition". |
699 Its morphisms are homeomorphisms $f:X\to X$ such that $f|_{\bd X}(c) = c$. |
699 The morphisms from $(X, c)$ to $(X', c')$, denoted $\Homeo(X,c; X', c')$, are |
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700 homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$. |
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701 %Let $\pi_0(\bbc)$ denote |
700 |
702 |
701 \begin{axiom}[Enriched $n$-categories] |
703 \begin{axiom}[Enriched $n$-categories] |
702 \label{axiom:enriched} |
704 \label{axiom:enriched} |
703 Let $\cS$ be a distributive symmetric monoidal category. |
705 Let $\cS$ be a distributive symmetric monoidal category. |
704 An $n$-category enriched in $\cS$ satisfies the above $n$-category axioms for $k=0,\ldots,n-1$, |
706 An $n$-category enriched in $\cS$ satisfies the above $n$-category axioms for $k=0,\ldots,n-1$, |
705 and modifies the axioms for $k=n$ as follows: |
707 and modifies the axioms for $k=n$ as follows: |
706 \begin{itemize} |
708 \begin{itemize} |
707 \item Morphisms. We have a functor $\cC_n$ from $\bbc$ ($n$-balls with boundary conditions) to $\cS$. |
709 \item Morphisms. We have a functor $\cC_n$ from $\bbc$ ($n$-balls with boundary conditions) to $\cS$. |
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710 %[already said this above. ack] Furthermore, $\cC_n(f)$ depends only on the path component of a homeomorphism $f: (X, c) \to (X', c')$. |
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711 %In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially |
708 \item Composition. Let $B = B_1\cup_Y B_2$ as in Axiom \ref{axiom:composition}. |
712 \item Composition. Let $B = B_1\cup_Y B_2$ as in Axiom \ref{axiom:composition}. |
709 Let $Y_i = \bd B_i \setmin Y$. |
713 Let $Y_i = \bd B_i \setmin Y$. |
710 Note that $\bd B = Y_1\cup Y_2$. |
714 Note that $\bd B = Y_1\cup Y_2$. |
711 Let $c_i \in \cC(Y_i)$ with $\bd c_1 = \bd c_2 = d \in \cl\cC(E)$. |
715 Let $c_i \in \cC(Y_i)$ with $\bd c_1 = \bd c_2 = d \in \cl\cC(E)$. |
712 Then we have a map |
716 Then we have a map |
725 or more generally an appropriate sort of $\infty$-category, |
729 or more generally an appropriate sort of $\infty$-category, |
726 we can modify the extended isotopy axiom \ref{axiom:extended-isotopies} |
730 we can modify the extended isotopy axiom \ref{axiom:extended-isotopies} |
727 to require that families of homeomorphisms act |
731 to require that families of homeomorphisms act |
728 and obtain an $A_\infty$ $n$-category. |
732 and obtain an $A_\infty$ $n$-category. |
729 |
733 |
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734 \noop{ |
730 We believe that abstract definitions should be guided by diverse collections |
735 We believe that abstract definitions should be guided by diverse collections |
731 of concrete examples, and a lack of diversity in our present collection of examples of $A_\infty$ $n$-categories |
736 of concrete examples, and a lack of diversity in our present collection of examples of $A_\infty$ $n$-categories |
732 makes us reluctant to commit to an all-encompassing general definition. |
737 makes us reluctant to commit to an all-encompassing general definition. |
733 Instead, we will give a relatively narrow definition which covers the examples we consider in this paper. |
738 Instead, we will give a relatively narrow definition which covers the examples we consider in this paper. |
734 After stating it, we will briefly discuss ways in which it can be made more general. |
739 After stating it, we will briefly discuss ways in which it can be made more general. |
735 |
740 } |
736 Assume that our $n$-morphisms are enriched over chain complexes. |
741 |
737 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
742 Recall the category $\bbc$ of balls with boundary conditions. |
738 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
743 Note that the morphisms $\Homeo(X,c; X', c')$ from $(X, c)$ to $(X', c')$ form a topological space. |
739 |
744 Let $\cS$ be an appropriate $\infty$-category (e.g.\ chain complexes) |
740 \nn{need to loosen for bbc reasons} |
745 and let $\cJ$ be an $\infty$-functor from topological spaces to $\cS$ |
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746 (e.g.\ the singular chain functor $C_*$). |
741 |
747 |
742 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
748 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
743 \label{axiom:families} |
749 \label{axiom:families} |
744 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
750 For each pair of $n$-balls $X$ and $X'$ and each pair $c\in \cl{\cC}(\bd X)$ and $c'\in \cl{\cC}(\bd X')$ we have an $\cS$-morphism |
745 \[ |
751 \[ |
746 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
752 \cJ(\Homeo(X,c; X', c')) \ot \cC(X; c) \to \cC(X'; c') . |
747 \] |
753 \] |
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754 Similarly, we have an $\cS$-morphism |
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755 \[ |
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756 \cJ(\Coll(X,c)) \ot \cC(X; c) \to \cC(X; c), |
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757 \] |
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758 where $\Coll(X,c)$ denotes the space of collar maps. |
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759 (See below for further discussion.) |
748 These action maps are required to be associative up to coherent homotopy, |
760 These action maps are required to be associative up to coherent homotopy, |
749 and also compatible with composition (gluing) in the sense that |
761 and also compatible with composition (gluing) in the sense that |
750 a diagram like the one in Theorem \ref{thm:CH} commutes. |
762 a diagram like the one in Theorem \ref{thm:CH} commutes. |
751 %\nn{repeat diagram here?} |
763 % say something about compatibility with product morphisms? |
752 %\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
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753 On $C_0(\Homeo_\bd(X))\ot \cC(X; c)$ the action should coincide |
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754 with the one coming from Axiom \ref{axiom:morphisms}. |
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755 \end{axiom} |
764 \end{axiom} |
756 |
765 |
757 We should strengthen the above $A_\infty$ axiom to apply to families of collar maps. |
766 We now describe the topology on $\Coll(X; c)$. |
758 To do this we need to explain how collar maps form a topological space. |
767 We retain notation from the above definition of collar map. |
759 Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
768 Each collaring homeomorphism $X \cup (Y\times J) \to X$ determines a map from points $p$ of $\bd X$ to |
760 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
769 (possibly zero-width) embedded intervals in $X$ terminating at $p$. |
761 Having chains on the space of collar maps act gives rise to coherence maps involving |
770 If $p \in Y$ this interval is the image of $\{p\}\times J$. |
762 weak identities. |
771 If $p \notin Y$ then $p$ is assigned the zero-width interval $\{p\}$. |
763 We will not pursue this in detail here. |
772 Such collections of intervals have a natural topology, and $\Coll(X; c)$ inherits its topology from this. |
764 |
773 Note in particular that parts of the collar are allowed to shrink continuously to zero width. |
765 One potential variant of the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. |
774 (This is the real content; if nothing shrinks to zero width then the action of families of collar |
766 (In fact, the alternative construction of the blob complex described in \S \ref{ss:alt-def} |
775 maps follows from the action of families of homeomorphisms and compatibility with gluing.) |
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776 |
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777 The $k=n$ case of Axiom \ref{axiom:morphisms} posits a {\it strictly} associative action of {\it sets} |
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778 $\Homeo(X,c; X', c') \times \cC(X; c) \to \cC(X'; c')$, and at first it might seem that this would force the above |
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779 action of $\cJ(\Homeo(X,c; X', c'))$ to be strictly associative as well (assuming the two actions are compatible). |
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780 In fact, compatibility implies less than this. |
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781 For simplicity, assume that $\cJ$ is $C_*$, the singular chains functor. |
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782 (This is the example most relevant to this paper.) |
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783 Then compatibility implies that the action of $C_*(\Homeo(X,c; X', c'))$ agrees with the action |
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784 of $C_0(\Homeo(X,c; X', c'))$ coming from Axiom \ref{axiom:morphisms}, so we only require associativity in degree zero. |
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785 And indeed, this is true for our main example of an $A_\infty$ $n$-category based on the blob construction. |
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786 Stating this sort of compatibility for general $\cS$ and $\cJ$ requires further assumptions, |
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787 such as the forgetful functor from $\cS$ to sets having a left adjoint, and $\cS$ having an internal Hom. |
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788 An alternative (due to Peter Teichner) is to say that Axiom \ref{axiom:families} |
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789 supersedes the $k=n$ case of Axiom \ref{axiom:morphisms}; in dimension $n$ we just have a |
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790 functor $\bbc \to \cS$ of $A_\infty$ 1-categories. |
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791 (This assumes some prior notion of $A_\infty$ 1-category.) |
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792 We are not currently aware of any examples which require this sort of greater generality, so we think it best |
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793 to refrain from settling on a preferred version of the axiom until |
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794 we have a greater variety of examples to guide the choice. |
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795 |
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796 Another variant of the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. |
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797 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; |
798 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.) |
799 since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across. |
769 |
800 |
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801 \noop{ |
770 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
802 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
771 into a ordinary $n$-category (enriched over graded groups). |
803 into a ordinary $n$-category (enriched over graded groups). |
772 In a different direction, if we enrich over topological spaces instead of chain complexes, |
804 In a different direction, if we enrich over topological spaces instead of chain complexes, |
773 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
805 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
774 instead of $C_*(\Homeo_\bd(X))$. |
806 instead of $C_*(\Homeo_\bd(X))$. |
775 Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex |
807 Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex |
776 type $A_\infty$ $n$-category. |
808 type $A_\infty$ $n$-category. |
777 |
809 } |
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 |
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786 |
810 |
787 |
811 |
788 \medskip |
812 \medskip |
789 |
813 |
790 We define a $j$ times monoidal $n$-category to be an $(n{+}j)$-category $\cC$ where |
814 We define a $j$ times monoidal $n$-category to be an $(n{+}j)$-category $\cC$ where |