661 \subsection{Examples of $n$-categories} |
661 \subsection{Examples of $n$-categories} |
662 \label{ss:ncat-examples} |
662 \label{ss:ncat-examples} |
663 |
663 |
664 |
664 |
665 We now describe several classes of examples of $n$-categories satisfying our axioms. |
665 We now describe several classes of examples of $n$-categories satisfying our axioms. |
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666 We typically specify only the morphisms; the rest of the data for the category |
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667 (restriction maps, gluing, product morphisms, action of homeomorphisms) is usually obvious. |
666 |
668 |
667 \begin{example}[Maps to a space] |
669 \begin{example}[Maps to a space] |
668 \rm |
670 \rm |
669 \label{ex:maps-to-a-space}% |
671 \label{ex:maps-to-a-space}% |
670 Fix a ``target space" $T$, any topological space. |
672 Let $T$be a topological space. |
671 We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
673 We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
672 For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
674 For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
673 all continuous maps from $X$ to $T$. |
675 all continuous maps from $X$ to $T$. |
674 For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
676 For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
675 homotopies fixed on $\bd X$. |
677 homotopies fixed on $\bd X$. |
676 (Note that homotopy invariance implies isotopy invariance.) |
678 (Note that homotopy invariance implies isotopy invariance.) |
677 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
679 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
678 be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
680 be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
679 |
681 \end{example} |
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682 |
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683 \noop{ |
680 Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above. |
684 Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above. |
681 Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example. |
685 Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example. |
682 \end{example} |
686 \nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from |
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687 an n-cat} |
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688 } |
683 |
689 |
684 \begin{example}[Maps to a space, with a fiber] |
690 \begin{example}[Maps to a space, with a fiber] |
685 \rm |
691 \rm |
686 \label{ex:maps-to-a-space-with-a-fiber}% |
692 \label{ex:maps-to-a-space-with-a-fiber}% |
687 We can modify the example above, by fixing a |
693 We can modify the example above, by fixing a |
699 For $X$ of dimension less than $n$ define $\pi^{\alpha, \times F}_{\leq n}(T)(X)$ as before, ignoring $\alpha$. |
705 For $X$ of dimension less than $n$ define $\pi^{\alpha, \times F}_{\leq n}(T)(X)$ as before, ignoring $\alpha$. |
700 For $X$ an $n$-ball and $c\in \Maps(\bdy X \times F \to T)$ define $\pi^{\alpha, \times F}_{\leq n}(T)(X; c)$ to be |
706 For $X$ an $n$-ball and $c\in \Maps(\bdy X \times F \to T)$ define $\pi^{\alpha, \times F}_{\leq n}(T)(X; c)$ to be |
701 the $R$-module of finite linear combinations of continuous maps from $X\times F$ to $T$, |
707 the $R$-module of finite linear combinations of continuous maps from $X\times F$ to $T$, |
702 modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
708 modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
703 $h: X\times F\times I \to T$, then $a = \alpha(h)b$. |
709 $h: X\times F\times I \to T$, then $a = \alpha(h)b$. |
704 \nn{need to say something about fundamental classes, or choose $\alpha$ carefully} |
710 (In order for this to be well-defined we must choose $\alpha$ to be zero on degenerate simplices. |
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711 Alternatively, we could equip the balls with fundamental classes.) |
705 \end{example} |
712 \end{example} |
706 |
713 |
707 The next example is only intended to be illustrative, as we don't specify which definition of a ``traditional $n$-category" we intend. |
714 The next example is only intended to be illustrative, as we don't specify which definition of a ``traditional $n$-category" we intend. |
708 Further, most of these definitions don't even have an agreed-upon notion of ``strong duality", which we assume here. |
715 Further, most of these definitions don't even have an agreed-upon notion of ``strong duality", which we assume here. |
709 \begin{example}[Traditional $n$-categories] |
716 \begin{example}[Traditional $n$-categories] |
721 More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
728 More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
722 Define $\cC(X)$, for $\dim(X) < n$, |
729 Define $\cC(X)$, for $\dim(X) < n$, |
723 to be the set of all $C$-labeled embedded cell complexes of $X\times F$. |
730 to be the set of all $C$-labeled embedded cell complexes of $X\times F$. |
724 Define $\cC(X; c)$, for $X$ an $n$-ball, |
731 Define $\cC(X; c)$, for $X$ an $n$-ball, |
725 to be the dual Hilbert space $A(X\times F; c)$. |
732 to be the dual Hilbert space $A(X\times F; c)$. |
726 \nn{refer elsewhere for details?} |
733 (See Subsection \ref{sec:constructing-a-tqft}.) |
727 |
734 \end{example} |
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735 |
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736 \noop{ |
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737 \nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from |
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738 an n-cat} |
728 Recall we described a system of fields and local relations based on a ``traditional $n$-category" |
739 Recall we described a system of fields and local relations based on a ``traditional $n$-category" |
729 $C$ in Example \ref{ex:traditional-n-categories(fields)} above. |
740 $C$ in Example \ref{ex:traditional-n-categories(fields)} above. |
730 \nn{KW: We already refer to \S \ref{sec:fields} above} |
741 \nn{KW: We already refer to \S \ref{sec:fields} above} |
731 Constructing a system of fields from $\cC$ recovers that example. |
742 Constructing a system of fields from $\cC$ recovers that example. |
732 \todo{Except that it doesn't: pasting diagrams v.s. string diagrams.} |
743 \todo{Except that it doesn't: pasting diagrams v.s. string diagrams.} |
733 \nn{KW: but the above example is all about string diagrams. the only difference is at the top level, |
744 \nn{KW: but the above example is all about string diagrams. the only difference is at the top level, |
734 where the quotient is built in. |
745 where the quotient is built in. |
735 but (string diagrams)/(relations) is isomorphic to |
746 but (string diagrams)/(relations) is isomorphic to |
736 (pasting diagrams composed of smaller string diagrams)/(relations)} |
747 (pasting diagrams composed of smaller string diagrams)/(relations)} |
737 \end{example} |
748 } |
738 |
749 |
739 Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
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740 |
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741 \nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example} |
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742 |
750 |
743 \newcommand{\Bord}{\operatorname{Bord}} |
751 \newcommand{\Bord}{\operatorname{Bord}} |
744 \begin{example}[The bordism $n$-category, plain version] |
752 \begin{example}[The bordism $n$-category, plain version] |
745 \label{ex:bord-cat} |
753 \label{ex:bord-cat} |
746 \rm |
754 \rm |
764 %\end{example} |
772 %\end{example} |
765 |
773 |
766 |
774 |
767 %We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex. |
775 %We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex. |
768 |
776 |
769 \begin{example}[Chains of maps to a space] |
777 \begin{example}[Chains (or space) of maps to a space] |
770 \rm |
778 \rm |
771 \label{ex:chains-of-maps-to-a-space} |
779 \label{ex:chains-of-maps-to-a-space} |
772 We can modify Example \ref{ex:maps-to-a-space} above to define the fundamental $A_\infty$ $n$-category $\pi^\infty_{\le n}(T)$ of a topological space $T$. |
780 We can modify Example \ref{ex:maps-to-a-space} above to define the fundamental $A_\infty$ $n$-category $\pi^\infty_{\le n}(T)$ of a topological space $T$. |
773 For a $k$-ball $X$, with $k < n$, the set $\pi^\infty_{\leq n}(T)(X)$ is just $\Maps(X \to T)$. |
781 For a $k$-ball $X$, with $k < n$, the set $\pi^\infty_{\leq n}(T)(X)$ is just $\Maps(X \to T)$. |
774 Define $\pi^\infty_{\leq n}(T)(X; c)$ for an $n$-ball $X$ and $c \in \pi^\infty_{\leq n}(T)(\bdy X)$ to be the chain complex |
782 Define $\pi^\infty_{\leq n}(T)(X; c)$ for an $n$-ball $X$ and $c \in \pi^\infty_{\leq n}(T)(\bdy X)$ to be the chain complex |
775 $$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
783 \[ |
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784 C_*(\Maps_c(X\times F \to T)), |
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785 \] |
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786 where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
776 and $C_*$ denotes singular chains. |
787 and $C_*$ denotes singular chains. |
777 \nn{maybe should also mention version where we enrich over spaces rather than chain complexes} |
788 Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X\times F \to T)$, |
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789 we get an $A_\infty$ $n$-category enriched over spaces. |
778 \end{example} |
790 \end{example} |
779 |
791 |
780 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to |
792 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to |
781 homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
793 homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
782 |
794 |
783 \begin{example}[Blob complexes of balls (with a fiber)] |
795 \begin{example}[Blob complexes of balls (with a fiber)] |
784 \rm |
796 \rm |
785 \label{ex:blob-complexes-of-balls} |
797 \label{ex:blob-complexes-of-balls} |
786 Fix an $n-k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
798 Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
787 We will define an $A_\infty$ $k$-category $\cC$. |
799 We will define an $A_\infty$ $k$-category $\cC$. |
788 When $X$ is a $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$. |
800 When $X$ is a $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$. |
789 When $X$ is an $k$-ball, |
801 When $X$ is an $k$-ball, |
790 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
802 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
791 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
803 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
793 |
805 |
794 This example will be essential for Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
806 This example will be essential for Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
795 Notice that with $F$ a point, the above example is a construction turning a topological |
807 Notice that with $F$ a point, the above example is a construction turning a topological |
796 $n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. |
808 $n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. |
797 We think of this as providing a ``free resolution" |
809 We think of this as providing a ``free resolution" |
798 \nn{``cofibrant replacement"?} |
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799 of the topological $n$-category. |
810 of the topological $n$-category. |
800 \todo{Say more here!} |
811 \nn{say something about cofibrant replacements?} |
801 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
812 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
802 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
813 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
803 and take $\CD{B}$ to act trivially. |
814 and take $\CD{B}$ to act trivially. |
804 |
815 |
805 Be careful that the ``free resolution" of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
816 Be careful that the ``free resolution" of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |