1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
2 |
3 \def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip} |
3 \def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip} |
4 \def\mmpar#1#2#3{\smallskip\noindent{\bf #1} (#2). {\it #3} \smallskip} |
4 \def\mmpar#1#2#3{\smallskip\noindent{\bf #1} (#2). {\it #3} \smallskip} |
5 |
5 |
6 \section{Disk-like \texorpdfstring{$n$}{n}-categories and their modules} |
6 \section{\texorpdfstring{$n$}{n}-categories and their modules} |
7 \label{sec:ncats} |
7 \label{sec:ncats} |
8 |
8 |
9 \subsection{Definition of disk-like \texorpdfstring{$n$}{n}-categories} |
9 \subsection{Definition of \texorpdfstring{$n$}{n}-categories} |
10 \label{ss:n-cat-def} |
10 \label{ss:n-cat-def} |
11 |
11 |
12 Before proceeding, we need more appropriate definitions of $n$-categories, |
12 Before proceeding, we need more appropriate definitions of $n$-categories, |
13 $A_\infty$ $n$-categories, as well as modules for these, and tensor products of these modules. |
13 $A_\infty$ $n$-categories, as well as modules for these, and tensor products of these modules. |
14 (As is the case throughout this paper, by ``$n$-category" we mean some notion of |
14 (As is the case throughout this paper, by ``$n$-category" we mean some notion of |
30 %\nn{Say something explicit about Lurie's work here? |
30 %\nn{Say something explicit about Lurie's work here? |
31 %It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen} |
31 %It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen} |
32 |
32 |
33 \medskip |
33 \medskip |
34 |
34 |
35 The axioms for a disk-like $n$-category are spread throughout this section. |
35 The axioms for an $n$-category are spread throughout this section. |
36 Collecting these together, a disk-like $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, |
36 Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, |
37 \ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:extended-isotopies} and \ref{axiom:vcones}. |
37 \ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:extended-isotopies} and \ref{axiom:vcones}. |
38 For an enriched disk-like $n$-category we add Axiom \ref{axiom:enriched}. |
38 For an enriched $n$-category we add Axiom \ref{axiom:enriched}. |
39 For an $A_\infty$ disk-like $n$-category, we replace |
39 For an $A_\infty$ $n$-category, we replace |
40 Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. |
40 Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. |
41 |
41 |
42 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms |
42 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms |
43 for $k{-}1$-morphisms. |
43 for $k{-}1$-morphisms. |
44 Readers who prefer things to be presented in a strictly logical order should read this |
44 Readers who prefer things to be presented in a strictly logical order should read this |
86 They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
86 They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
87 They could be topological or PL or smooth. |
87 They could be topological or PL or smooth. |
88 %\nn{need to check whether this makes much difference} |
88 %\nn{need to check whether this makes much difference} |
89 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
89 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
90 to be fussier about corners and boundaries.) |
90 to be fussier about corners and boundaries.) |
91 For each flavor of manifold there is a corresponding flavor of disk-like $n$-category. |
91 For each flavor of manifold there is a corresponding flavor of $n$-category. |
92 For simplicity, we will concentrate on the case of PL unoriented manifolds. |
92 For simplicity, we will concentrate on the case of PL unoriented manifolds. |
93 |
93 |
94 An ambitious reader may want to keep in mind two other classes of balls. |
94 An ambitious reader may want to keep in mind two other classes of balls. |
95 The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). |
95 The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). |
96 This will be used below (see the end of \S \ref{ss:product-formula}) to describe the blob complex of a fiber bundle with |
96 This will be used below (see the end of \S \ref{ss:product-formula}) to describe the blob complex of a fiber bundle with |
842 \item topological spaces with product and disjoint union. |
842 \item topological spaces with product and disjoint union. |
843 \end{itemize} |
843 \end{itemize} |
844 For convenience, we will also assume that the objects of our auxiliary category are sets with extra structure. |
844 For convenience, we will also assume that the objects of our auxiliary category are sets with extra structure. |
845 (Otherwise, stating the axioms for identity morphisms becomes more cumbersome.) |
845 (Otherwise, stating the axioms for identity morphisms becomes more cumbersome.) |
846 |
846 |
847 Before stating the revised axioms for a disk-like $n$-category enriched in a distributive monoidal category, |
847 Before stating the revised axioms for an $n$-category enriched in a distributive monoidal category, |
848 we need a preliminary definition. |
848 we need a preliminary definition. |
849 Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the |
849 Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the |
850 category $\bbc$ of {\it $n$-balls with boundary conditions}. |
850 category $\bbc$ of {\it $n$-balls with boundary conditions}. |
851 Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \cl\cC(\bd X)$ is the ``boundary condition". |
851 Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \cl\cC(\bd X)$ is the ``boundary condition". |
852 The morphisms from $(X, c)$ to $(X', c')$, denoted $\Homeo(X,c; X', c')$, are |
852 The morphisms from $(X, c)$ to $(X', c')$, denoted $\Homeo(X,c; X', c')$, are |
853 homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$. |
853 homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$. |
854 %Let $\pi_0(\bbc)$ denote |
854 %Let $\pi_0(\bbc)$ denote |
855 |
855 |
856 \begin{axiom}[Enriched disk-like $n$-categories] |
856 \begin{axiom}[Enriched $n$-categories] |
857 \label{axiom:enriched} |
857 \label{axiom:enriched} |
858 Let $\cS$ be a distributive symmetric monoidal category. |
858 Let $\cS$ be a distributive symmetric monoidal category. |
859 A disk-like $n$-category enriched in $\cS$ satisfies the above $n$-category axioms for $k=0,\ldots,n-1$, |
859 An $n$-category enriched in $\cS$ satisfies the above $n$-category axioms for $k=0,\ldots,n-1$, |
860 and modifies the axioms for $k=n$ as follows: |
860 and modifies the axioms for $k=n$ as follows: |
861 \begin{itemize} |
861 \begin{itemize} |
862 \item Morphisms. We have a functor $\cC_n$ from $\bbc$ ($n$-balls with boundary conditions) to $\cS$. |
862 \item Morphisms. We have a functor $\cC_n$ from $\bbc$ ($n$-balls with boundary conditions) to $\cS$. |
863 %[already said this above. ack] Furthermore, $\cC_n(f)$ depends only on the path component of a homeomorphism $f: (X, c) \to (X', c')$. |
863 %[already said this above. ack] Furthermore, $\cC_n(f)$ depends only on the path component of a homeomorphism $f: (X, c) \to (X', c')$. |
864 %In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially |
864 %In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially |
880 |
880 |
881 When the enriching category $\cS$ is chain complexes or topological spaces, |
881 When the enriching category $\cS$ is chain complexes or topological spaces, |
882 or more generally an appropriate sort of $\infty$-category, |
882 or more generally an appropriate sort of $\infty$-category, |
883 we can modify the extended isotopy axiom \ref{axiom:extended-isotopies} |
883 we can modify the extended isotopy axiom \ref{axiom:extended-isotopies} |
884 to require that families of homeomorphisms act |
884 to require that families of homeomorphisms act |
885 and obtain what we shall call an $A_\infty$ disk-like $n$-category. |
885 and obtain what we shall call an $A_\infty$ $n$-category. |
886 |
886 |
887 \noop{ |
887 \noop{ |
888 We believe that abstract definitions should be guided by diverse collections |
888 We believe that abstract definitions should be guided by diverse collections |
889 of concrete examples, and a lack of diversity in our present collection of examples of $A_\infty$ $n$-categories |
889 of concrete examples, and a lack of diversity in our present collection of examples of $A_\infty$ $n$-categories |
890 makes us reluctant to commit to an all-encompassing general definition. |
890 makes us reluctant to commit to an all-encompassing general definition. |
933 In fact, compatibility implies less than this. |
933 In fact, compatibility implies less than this. |
934 For simplicity, assume that $\cJ$ is $C_*$, the singular chains functor. |
934 For simplicity, assume that $\cJ$ is $C_*$, the singular chains functor. |
935 (This is the example most relevant to this paper.) |
935 (This is the example most relevant to this paper.) |
936 Then compatibility implies that the action of $C_*(\Homeo(X,c; X', c'))$ agrees with the action |
936 Then compatibility implies that the action of $C_*(\Homeo(X,c; X', c'))$ agrees with the action |
937 of $C_0(\Homeo(X,c; X', c'))$ coming from Axiom \ref{axiom:morphisms}, so we only require associativity in degree zero. |
937 of $C_0(\Homeo(X,c; X', c'))$ coming from Axiom \ref{axiom:morphisms}, so we only require associativity in degree zero. |
938 And indeed, this is true for our main example of an $A_\infty$ disk-like $n$-category based on the blob construction. |
938 And indeed, this is true for our main example of an $A_\infty$ $n$-category based on the blob construction. |
939 Stating this sort of compatibility for general $\cS$ and $\cJ$ requires further assumptions, |
939 Stating this sort of compatibility for general $\cS$ and $\cJ$ requires further assumptions, |
940 such as the forgetful functor from $\cS$ to sets having a left adjoint, and $\cS$ having an internal Hom. |
940 such as the forgetful functor from $\cS$ to sets having a left adjoint, and $\cS$ having an internal Hom. |
941 |
941 |
942 An alternative (due to Peter Teichner) is to say that Axiom \ref{axiom:families} |
942 An alternative (due to Peter Teichner) is to say that Axiom \ref{axiom:families} |
943 supersedes the $k=n$ case of Axiom \ref{axiom:morphisms}; in dimension $n$ we just have a |
943 supersedes the $k=n$ case of Axiom \ref{axiom:morphisms}; in dimension $n$ we just have a |
955 gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom. |
955 gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom. |
956 %since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across. |
956 %since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across. |
957 For future reference we make the following definition. |
957 For future reference we make the following definition. |
958 |
958 |
959 \begin{defn} |
959 \begin{defn} |
960 A {\em strict $A_\infty$ disk-like $n$-category} is one in which the actions of Axiom \ref{axiom:families} are strictly associative. |
960 A {\em strict $A_\infty$ $n$-category} is one in which the actions of Axiom \ref{axiom:families} are strictly associative. |
961 \end{defn} |
961 \end{defn} |
962 |
962 |
963 \noop{ |
963 \noop{ |
964 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
964 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
965 into a ordinary $n$-category (enriched over graded groups). |
965 into a ordinary $n$-category (enriched over graded groups). |
971 } |
971 } |
972 |
972 |
973 |
973 |
974 \medskip |
974 \medskip |
975 |
975 |
976 We define a $j$ times monoidal disk-like $n$-category to be a disk-like $(n{+}j)$-category $\cC$ where |
976 We define a $j$ times monoidal $n$-category to be an $(n{+}j)$-category $\cC$ where |
977 $\cC(X)$ is a trivial 1-element set if $X$ is a $k$-ball with $k<j$. |
977 $\cC(X)$ is a trivial 1-element set if $X$ is a $k$-ball with $k<j$. |
978 See Example \ref{ex:bord-cat}. |
978 See Example \ref{ex:bord-cat}. |
979 |
979 |
980 \medskip |
980 \medskip |
981 |
981 |
982 The alert reader will have already noticed that our definition of an (ordinary) disk-like $n$-category |
982 The alert reader will have already noticed that our definition of an (ordinary) $n$-category |
983 is extremely similar to our definition of a system of fields. |
983 is extremely similar to our definition of a system of fields. |
984 There are two differences. |
984 There are two differences. |
985 First, for the $n$-category definition we restrict our attention to balls |
985 First, for the $n$-category definition we restrict our attention to balls |
986 (and their boundaries), while for fields we consider all manifolds. |
986 (and their boundaries), while for fields we consider all manifolds. |
987 Second, in the category definition we directly impose isotopy |
987 Second, in the category definition we directly impose isotopy |
988 invariance in dimension $n$, while in the fields definition we |
988 invariance in dimension $n$, while in the fields definition we |
989 instead remember a subspace of local relations which contain differences of isotopic fields. |
989 instead remember a subspace of local relations which contain differences of isotopic fields. |
990 (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
990 (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
991 Thus a system of fields and local relations $(\cF,U)$ determines a disk-like $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to |
991 Thus a system of fields and local relations $(\cF,U)$ determines an $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to |
992 balls and, at level $n$, quotienting out by the local relations: |
992 balls and, at level $n$, quotienting out by the local relations: |
993 \begin{align*} |
993 \begin{align*} |
994 \cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases} |
994 \cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases} |
995 \end{align*} |
995 \end{align*} |
996 This $n$-category can be thought of as the local part of the fields. |
996 This $n$-category can be thought of as the local part of the fields. |
1002 In the $n$-category axioms above we have intermingled data and properties for expository reasons. |
1002 In the $n$-category axioms above we have intermingled data and properties for expository reasons. |
1003 Here's a summary of the definition which segregates the data from the properties. |
1003 Here's a summary of the definition which segregates the data from the properties. |
1004 We also remind the reader of the inductive nature of the definition: All the data for $k{-}1$-morphisms must be in place |
1004 We also remind the reader of the inductive nature of the definition: All the data for $k{-}1$-morphisms must be in place |
1005 before we can describe the data for $k$-morphisms. |
1005 before we can describe the data for $k$-morphisms. |
1006 |
1006 |
1007 A disk-like $n$-category consists of the following data: |
1007 An $n$-category consists of the following data: |
1008 \begin{itemize} |
1008 \begin{itemize} |
1009 \item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms}); |
1009 \item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms}); |
1010 \item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary}); |
1010 \item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary}); |
1011 \item ``composition'' or ``gluing'' maps $\gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B_1\cup_Y B_2)\trans E$ (Axiom \ref{axiom:composition}); |
1011 \item ``composition'' or ``gluing'' maps $\gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B_1\cup_Y B_2)\trans E$ (Axiom \ref{axiom:composition}); |
1012 \item ``product'' or ``identity'' maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product}); |
1012 \item ``product'' or ``identity'' maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product}); |
1160 C_*(\Maps_c(X \to T)), |
1160 C_*(\Maps_c(X \to T)), |
1161 \] |
1161 \] |
1162 where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
1162 where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
1163 and $C_*$ denotes singular chains. |
1163 and $C_*$ denotes singular chains. |
1164 Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X \to T)$, |
1164 Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X \to T)$, |
1165 we get an $A_\infty$ disk-like $n$-category enriched over spaces. |
1165 we get an $A_\infty$ $n$-category enriched over spaces. |
1166 \end{example} |
1166 \end{example} |
1167 |
1167 |
1168 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to |
1168 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to |
1169 homotopy as the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
1169 homotopy as the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
1170 |
1170 |
1171 \begin{example}[Blob complexes of balls (with a fiber)] |
1171 \begin{example}[Blob complexes of balls (with a fiber)] |
1172 \rm |
1172 \rm |
1173 \label{ex:blob-complexes-of-balls} |
1173 \label{ex:blob-complexes-of-balls} |
1174 Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
1174 Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
1175 We will define an $A_\infty$ disk-like $k$-category $\cC$. |
1175 We will define an $A_\infty$ $k$-category $\cC$. |
1176 When $X$ is an $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$. |
1176 When $X$ is an $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$. |
1177 When $X$ is a $k$-ball, |
1177 When $X$ is a $k$-ball, |
1178 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
1178 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
1179 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
1179 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
1180 \end{example} |
1180 \end{example} |
1181 |
1181 |
1182 This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
1182 This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
1183 Notice that with $F$ a point, the above example is a construction turning an ordinary disk-like |
1183 Notice that with $F$ a point, the above example is a construction turning an ordinary |
1184 $n$-category $\cC$ into an $A_\infty$ disk-like $n$-category. |
1184 $n$-category $\cC$ into an $A_\infty$ $n$-category. |
1185 We think of this as providing a ``free resolution" |
1185 We think of this as providing a ``free resolution" |
1186 of the ordinary disk-like $n$-category. |
1186 of the ordinary $n$-category. |
1187 %\nn{say something about cofibrant replacements?} |
1187 %\nn{say something about cofibrant replacements?} |
1188 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
1188 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
1189 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
1189 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
1190 and take $\CD{B}$ to act trivially. |
1190 and take $\CD{B}$ to act trivially. |
1191 |
1191 |
1192 Beware that the ``free resolution" of the ordinary disk-like $n$-category $\pi_{\leq n}(T)$ |
1192 Beware that the ``free resolution" of the ordinary $n$-category $\pi_{\leq n}(T)$ |
1193 is not the $A_\infty$ disk-like $n$-category $\pi^\infty_{\leq n}(T)$. |
1193 is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
1194 It's easy to see that with $n=0$, the corresponding system of fields is just |
1194 It's easy to see that with $n=0$, the corresponding system of fields is just |
1195 linear combinations of connected components of $T$, and the local relations are trivial. |
1195 linear combinations of connected components of $T$, and the local relations are trivial. |
1196 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
1196 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
1197 |
1197 |
1198 \begin{example}[The bordism $n$-category of $d$-manifolds, $A_\infty$ version] |
1198 \begin{example}[The bordism $n$-category of $d$-manifolds, $A_\infty$ version] |
1232 \rm |
1232 \rm |
1233 \label{ex:e-n-alg} |
1233 \label{ex:e-n-alg} |
1234 Let $A$ be an $\cE\cB_n$-algebra. |
1234 Let $A$ be an $\cE\cB_n$-algebra. |
1235 Note that this implies a $\Diff(B^n)$ action on $A$, |
1235 Note that this implies a $\Diff(B^n)$ action on $A$, |
1236 since $\cE\cB_n$ contains a copy of $\Diff(B^n)$. |
1236 since $\cE\cB_n$ contains a copy of $\Diff(B^n)$. |
1237 We will define a strict $A_\infty$ disk-like $n$-category $\cC^A$. |
1237 We will define a strict $A_\infty$ $n$-category $\cC^A$. |
1238 (We enrich in topological spaces, though this could easily be adapted to, say, chain complexes.) |
1238 (We enrich in topological spaces, though this could easily be adapted to, say, chain complexes.) |
1239 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point. |
1239 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point. |
1240 In other words, the $k$-morphisms are trivial for $k<n$. |
1240 In other words, the $k$-morphisms are trivial for $k<n$. |
1241 If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction. |
1241 If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction. |
1242 (Plain colimit, not homotopy colimit.) |
1242 (Plain colimit, not homotopy colimit.) |
1245 embedded balls into a single larger embedded ball. |
1245 embedded balls into a single larger embedded ball. |
1246 To each object of $J$ we associate $A^{\times m}$ (where $m$ is the number of balls), and |
1246 To each object of $J$ we associate $A^{\times m}$ (where $m$ is the number of balls), and |
1247 to each morphism of $J$ we associate a morphism coming from the $\cE\cB_n$ action on $A$. |
1247 to each morphism of $J$ we associate a morphism coming from the $\cE\cB_n$ action on $A$. |
1248 Alternatively and more simply, we could define $\cC^A(X)$ to be |
1248 Alternatively and more simply, we could define $\cC^A(X)$ to be |
1249 $\Diff(B^n\to X)\times A$ modulo the diagonal action of $\Diff(B^n)$. |
1249 $\Diff(B^n\to X)\times A$ modulo the diagonal action of $\Diff(B^n)$. |
1250 The remaining data for the $A_\infty$ disk-like $n$-category |
1250 The remaining data for the $A_\infty$ $n$-category |
1251 --- composition and $\Diff(X\to X')$ action --- |
1251 --- composition and $\Diff(X\to X')$ action --- |
1252 also comes from the $\cE\cB_n$ action on $A$. |
1252 also comes from the $\cE\cB_n$ action on $A$. |
1253 %\nn{should we spell this out?} |
1253 %\nn{should we spell this out?} |
1254 |
1254 |
1255 Conversely, one can show that a strict $A_\infty$ disk-like $n$-category $\cC$, where the $k$-morphisms |
1255 Conversely, one can show that a disk-like strict $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
1256 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
1256 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
1257 an $\cE\cB_n$-algebra. |
1257 an $\cE\cB_n$-algebra. |
1258 %\nn{The paper is already long; is it worth giving details here?} |
1258 %\nn{The paper is already long; is it worth giving details here?} |
1259 % According to the referee, yes it is... |
1259 % According to the referee, yes it is... |
1260 Let $A = \cC(B^n)$, where $B^n$ is the standard $n$-ball. |
1260 Let $A = \cC(B^n)$, where $B^n$ is the standard $n$-ball. |
1275 \end{example} |
1275 \end{example} |
1276 |
1276 |
1277 |
1277 |
1278 \subsection{From balls to manifolds} |
1278 \subsection{From balls to manifolds} |
1279 \label{ss:ncat_fields} \label{ss:ncat-coend} |
1279 \label{ss:ncat_fields} \label{ss:ncat-coend} |
1280 In this section we show how to extend a disk-like $n$-category $\cC$ as described above |
1280 In this section we show how to extend an $n$-category $\cC$ as described above |
1281 (of either the ordinary or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
1281 (of either the ordinary or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
1282 This extension is a certain colimit, and the arrow in the notation is intended as a reminder of this. |
1282 This extension is a certain colimit, and the arrow in the notation is intended as a reminder of this. |
1283 |
1283 |
1284 In the case of ordinary disk-like $n$-categories, this construction factors into a construction of a |
1284 In the case of ordinary $n$-categories, this construction factors into a construction of a |
1285 system of fields and local relations, followed by the usual TQFT definition of a |
1285 system of fields and local relations, followed by the usual TQFT definition of a |
1286 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
1286 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
1287 For an $A_\infty$ disk-like $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
1287 For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
1288 Recall that we can take an ordinary disk-like $n$-category $\cC$ and pass to the ``free resolution", |
1288 Recall that we can take an ordinary $n$-category $\cC$ and pass to the ``free resolution", |
1289 an $A_\infty$ disk-like $n$-category $\bc_*(\cC)$, by computing the blob complex of balls |
1289 an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls |
1290 (recall Example \ref{ex:blob-complexes-of-balls} above). |
1290 (recall Example \ref{ex:blob-complexes-of-balls} above). |
1291 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
1291 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
1292 for a manifold $M$ associated to this $A_\infty$ disk-like $n$-category is actually the |
1292 for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the |
1293 same as the original blob complex for $M$ with coefficients in $\cC$. |
1293 same as the original blob complex for $M$ with coefficients in $\cC$. |
1294 |
1294 |
1295 Recall that we've already anticipated this construction Subsection \ref{ss:n-cat-def}, |
1295 Recall that we've already anticipated this construction Subsection \ref{ss:n-cat-def}, |
1296 inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
1296 inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
1297 so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
1297 so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
1298 |
1298 |
1299 \medskip |
1299 \medskip |
1300 |
1300 |
1301 We will first define the {\it decomposition poset} $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
1301 We will first define the {\it decomposition poset} $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
1302 A disk-like $n$-category $\cC$ provides a functor from this poset to the category of sets, |
1302 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
1303 and we will define $\cl{\cC}(W)$ as a suitable colimit |
1303 and we will define $\cl{\cC}(W)$ as a suitable colimit |
1304 (or homotopy colimit in the $A_\infty$ case) of this functor. |
1304 (or homotopy colimit in the $A_\infty$ case) of this functor. |
1305 We'll later give a more explicit description of this colimit. |
1305 We'll later give a more explicit description of this colimit. |
1306 In the case that the disk-like $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain |
1306 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain |
1307 complexes to $n$-balls with boundary data), |
1307 complexes to $n$-balls with boundary data), |
1308 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into |
1308 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into |
1309 subsets according to boundary data, and each of these subsets has the appropriate structure |
1309 subsets according to boundary data, and each of these subsets has the appropriate structure |
1310 (e.g. a vector space or chain complex). |
1310 (e.g. a vector space or chain complex). |
1311 |
1311 |
1420 |
1420 |
1421 Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$: |
1421 Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$: |
1422 |
1422 |
1423 \begin{defn}[System of fields functor] |
1423 \begin{defn}[System of fields functor] |
1424 \label{def:colim-fields} |
1424 \label{def:colim-fields} |
1425 If $\cC$ is a disk-like $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
1425 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
1426 That is, for each decomposition $x$ there is a map |
1426 That is, for each decomposition $x$ there is a map |
1427 $\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps |
1427 $\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps |
1428 above, and $\cl{\cC}(W)$ is universal with respect to these properties. |
1428 above, and $\cl{\cC}(W)$ is universal with respect to these properties. |
1429 \end{defn} |
1429 \end{defn} |
1430 |
1430 |
1431 \begin{defn}[System of fields functor, $A_\infty$ case] |
1431 \begin{defn}[System of fields functor, $A_\infty$ case] |
1432 When $\cC$ is an $A_\infty$ disk-like $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$ |
1432 When $\cC$ is an $A_\infty$ $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$ |
1433 is defined as above, as the colimit of $\psi_{\cC;W}$. |
1433 is defined as above, as the colimit of $\psi_{\cC;W}$. |
1434 When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
1434 When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
1435 \end{defn} |
1435 \end{defn} |
1436 |
1436 |
1437 %We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$ |
1437 %We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$ |
1603 %\nn{need to finish explaining why we have a system of fields; |
1603 %\nn{need to finish explaining why we have a system of fields; |
1604 %define $k$-cat $\cC(\cdot\times W)$} |
1604 %define $k$-cat $\cC(\cdot\times W)$} |
1605 |
1605 |
1606 \subsection{Modules} |
1606 \subsection{Modules} |
1607 \label{sec:modules} |
1607 \label{sec:modules} |
1608 Next we define ordinary and $A_\infty$ disk-like $n$-category modules. |
1608 Next we define ordinary and $A_\infty$ $n$-category modules. |
1609 The definition will be very similar to that of disk-like $n$-categories, |
1609 The definition will be very similar to that of $n$-categories, |
1610 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
1610 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
1611 |
1611 |
1612 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
1612 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
1613 in the context of an $m{+}1$-dimensional TQFT. |
1613 in the context of an $m{+}1$-dimensional TQFT. |
1614 Such a $W$ gives rise to a module for the disk-like $n$-category associated to $\bd W$. |
1614 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
1615 This will be explained in more detail as we present the axioms. |
1615 This will be explained in more detail as we present the axioms. |
1616 |
1616 |
1617 Throughout, we fix a disk-like $n$-category $\cC$. |
1617 Throughout, we fix an $n$-category $\cC$. |
1618 For all but one axiom, it doesn't matter whether $\cC$ is an ordinary $n$-category or an $A_\infty$ $n$-category. |
1618 For all but one axiom, it doesn't matter whether $\cC$ is an ordinary $n$-category or an $A_\infty$ $n$-category. |
1619 We state the final axiom, regarding actions of homeomorphisms, differently in the two cases. |
1619 We state the final axiom, regarding actions of homeomorphisms, differently in the two cases. |
1620 |
1620 |
1621 Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
1621 Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
1622 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
1622 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
1668 These maps, for various $M$, comprise a natural transformation of functors.} |
1668 These maps, for various $M$, comprise a natural transformation of functors.} |
1669 \end{module-axiom} |
1669 \end{module-axiom} |
1670 |
1670 |
1671 Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
1671 Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
1672 |
1672 |
1673 If the disk-like $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
1673 If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
1674 then for each marked $n$-ball $M=(B,N)$ and $c\in \cC(\bd B \setminus N)$, the set $\cM(M; c)$ should be an object in that category. |
1674 then for each marked $n$-ball $M=(B,N)$ and $c\in \cC(\bd B \setminus N)$, the set $\cM(M; c)$ should be an object in that category. |
1675 |
1675 |
1676 \begin{lem}[Boundary from domain and range] |
1676 \begin{lem}[Boundary from domain and range] |
1677 \label{lem:module-boundary} |
1677 \label{lem:module-boundary} |
1678 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$), |
1678 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$), |
1901 \] |
1901 \] |
1902 ($Y$ could be either a marked or plain ball.) |
1902 ($Y$ could be either a marked or plain ball.) |
1903 \end{enumerate} |
1903 \end{enumerate} |
1904 \end{module-axiom} |
1904 \end{module-axiom} |
1905 |
1905 |
1906 As in the disk-like $n$-category definition, once we have product morphisms we can define |
1906 As in the $n$-category definition, once we have product morphisms we can define |
1907 collar maps $\cM(M)\to \cM(M)$. |
1907 collar maps $\cM(M)\to \cM(M)$. |
1908 Note that there are two cases: |
1908 Note that there are two cases: |
1909 the collar could intersect the marking of the marked ball $M$, in which case |
1909 the collar could intersect the marking of the marked ball $M$, in which case |
1910 we use a product on a morphism of $\cM$; or the collar could be disjoint from the marking, |
1910 we use a product on a morphism of $\cM$; or the collar could be disjoint from the marking, |
1911 in which case we use a product on a morphism of $\cC$. |
1911 in which case we use a product on a morphism of $\cC$. |
1914 $a$ along a map associated to $\pi$. |
1914 $a$ along a map associated to $\pi$. |
1915 |
1915 |
1916 \medskip |
1916 \medskip |
1917 |
1917 |
1918 There are two alternatives for the next axiom, according whether we are defining |
1918 There are two alternatives for the next axiom, according whether we are defining |
1919 modules for ordinary or $A_\infty$ disk-like $n$-categories. |
1919 modules for ordinary $n$-categories or $A_\infty$ $n$-categories. |
1920 In the ordinary case we require |
1920 In the ordinary case we require |
1921 |
1921 |
1922 \begin{module-axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$] |
1922 \begin{module-axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$] |
1923 {Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
1923 {Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
1924 to the identity on $\bd M$ and is isotopic (rel boundary) to the identity. |
1924 to the identity on $\bd M$ and is isotopic (rel boundary) to the identity. |
1947 |
1947 |
1948 As with the $n$-category version of the above axiom, we should also have families of collar maps act. |
1948 As with the $n$-category version of the above axiom, we should also have families of collar maps act. |
1949 |
1949 |
1950 \medskip |
1950 \medskip |
1951 |
1951 |
1952 Note that the above axioms imply that a disk-like $n$-category module has the structure |
1952 Note that the above axioms imply that an $n$-category module has the structure |
1953 of a disk-like $n{-}1$-category. |
1953 of an $n{-}1$-category. |
1954 More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
1954 More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
1955 where $X$ is a $k$-ball and in the product $X\times J$ we pinch |
1955 where $X$ is a $k$-ball and in the product $X\times J$ we pinch |
1956 above the non-marked boundary component of $J$. |
1956 above the non-marked boundary component of $J$. |
1957 (More specifically, we collapse $X\times P$ to a single point, where |
1957 (More specifically, we collapse $X\times P$ to a single point, where |
1958 $P$ is the non-marked boundary component of $J$.) |
1958 $P$ is the non-marked boundary component of $J$.) |
1959 Then $\cE$ has the structure of a disk-like $n{-}1$-category. |
1959 Then $\cE$ has the structure of an $n{-}1$-category. |
1960 |
1960 |
1961 All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
1961 All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
1962 are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
1962 are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
1963 In this case ($k=1$ and oriented or Spin), there are two types |
1963 In this case ($k=1$ and oriented or Spin), there are two types |
1964 of marked 1-balls, call them left-marked and right-marked, |
1964 of marked 1-balls, call them left-marked and right-marked, |
1966 In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
1966 In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
1967 there is no left/right module distinction. |
1967 there is no left/right module distinction. |
1968 |
1968 |
1969 \medskip |
1969 \medskip |
1970 |
1970 |
1971 We now give some examples of modules over ordinary and $A_\infty$ disk-like $n$-categories. |
1971 We now give some examples of modules over ordinary and $A_\infty$ $n$-categories. |
1972 |
1972 |
1973 \begin{example}[Examples from TQFTs] |
1973 \begin{example}[Examples from TQFTs] |
1974 \rm |
1974 \rm |
1975 Continuing Example \ref{ex:ncats-from-tqfts}, with $\cF$ a TQFT, $W$ an $n{-}j$-manifold, |
1975 Continuing Example \ref{ex:ncats-from-tqfts}, with $\cF$ a TQFT, $W$ an $n{-}j$-manifold, |
1976 and $\cF(W)$ the disk-like $j$-category associated to $W$. |
1976 and $\cF(W)$ the $j$-category associated to $W$. |
1977 Let $Y$ be an $(n{-}j{+}1)$-manifold with $\bd Y = W$. |
1977 Let $Y$ be an $(n{-}j{+}1)$-manifold with $\bd Y = W$. |
1978 Define a $\cF(W)$ module $\cF(Y)$ as follows. |
1978 Define a $\cF(W)$ module $\cF(Y)$ as follows. |
1979 If $M = (B, N)$ is a marked $k$-ball with $k<j$ let |
1979 If $M = (B, N)$ is a marked $k$-ball with $k<j$ let |
1980 $\cF(Y)(M)\deq \cF((B\times W) \cup (N\times Y))$. |
1980 $\cF(Y)(M)\deq \cF((B\times W) \cup (N\times Y))$. |
1981 If $M = (B, N)$ is a marked $j$-ball and $c\in \cl{\cF(Y)}(\bd M)$ let |
1981 If $M = (B, N)$ is a marked $j$-ball and $c\in \cl{\cF(Y)}(\bd M)$ let |
2065 |
2065 |
2066 If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
2066 If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
2067 $\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
2067 $\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
2068 $D\times Y_i \sub \bd(D\times W)$. |
2068 $D\times Y_i \sub \bd(D\times W)$. |
2069 It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$ |
2069 It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$ |
2070 has the structure of a disk-like $n{-}k$-category. |
2070 has the structure of an $n{-}k$-category. |
2071 |
2071 |
2072 \medskip |
2072 \medskip |
2073 |
2073 |
2074 We will use a simple special case of the above |
2074 We will use a simple special case of the above |
2075 construction to define tensor products |
2075 construction to define tensor products |
2076 of modules. |
2076 of modules. |
2077 Let $\cM_1$ and $\cM_2$ be modules for a disk-like $n$-category $\cC$. |
2077 Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
2078 (If $k=1$ and our manifolds are oriented, then one should be |
2078 (If $k=1$ and our manifolds are oriented, then one should be |
2079 a left module and the other a right module.) |
2079 a left module and the other a right module.) |
2080 Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
2080 Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
2081 Define the tensor product $\cM_1 \tensor \cM_2$ to be the |
2081 Define the tensor product $\cM_1 \tensor \cM_2$ to be the |
2082 disk-like $n{-}1$-category associated as above to $J$ with its boundary labeled by $\cM_1$ and $\cM_2$. |
2082 $n{-}1$-category associated as above to $J$ with its boundary labeled by $\cM_1$ and $\cM_2$. |
2083 This of course depends (functorially) |
2083 This of course depends (functorially) |
2084 on the choice of 1-ball $J$. |
2084 on the choice of 1-ball $J$. |
2085 |
2085 |
2086 We will define a more general self tensor product (categorified coend) below. |
2086 We will define a more general self tensor product (categorified coend) below. |
2087 |
2087 |