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 |
835 \item topological spaces with product and disjoint union. |
835 \item topological spaces with product and disjoint union. |
836 \end{itemize} |
836 \end{itemize} |
837 For convenience, we will also assume that the objects of our auxiliary category are sets with extra structure. |
837 For convenience, we will also assume that the objects of our auxiliary category are sets with extra structure. |
838 (Otherwise, stating the axioms for identity morphisms becomes more cumbersome.) |
838 (Otherwise, stating the axioms for identity morphisms becomes more cumbersome.) |
839 |
839 |
840 Before stating the revised axioms for a disk-like $n$-category enriched in a distributive monoidal category, |
840 Before stating the revised axioms for an $n$-category enriched in a distributive monoidal category, |
841 we need a preliminary definition. |
841 we need a preliminary definition. |
842 Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the |
842 Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the |
843 category $\bbc$ of {\it $n$-balls with boundary conditions}. |
843 category $\bbc$ of {\it $n$-balls with boundary conditions}. |
844 Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \cl\cC(\bd X)$ is the ``boundary condition". |
844 Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \cl\cC(\bd X)$ is the ``boundary condition". |
845 The morphisms from $(X, c)$ to $(X', c')$, denoted $\Homeo(X,c; X', c')$, are |
845 The morphisms from $(X, c)$ to $(X', c')$, denoted $\Homeo(X,c; X', c')$, are |
846 homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$. |
846 homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$. |
847 %Let $\pi_0(\bbc)$ denote |
847 %Let $\pi_0(\bbc)$ denote |
848 |
848 |
849 \begin{axiom}[Enriched disk-like $n$-categories] |
849 \begin{axiom}[Enriched $n$-categories] |
850 \label{axiom:enriched} |
850 \label{axiom:enriched} |
851 Let $\cS$ be a distributive symmetric monoidal category. |
851 Let $\cS$ be a distributive symmetric monoidal category. |
852 A disk-like $n$-category enriched in $\cS$ satisfies the above $n$-category axioms for $k=0,\ldots,n-1$, |
852 An $n$-category enriched in $\cS$ satisfies the above $n$-category axioms for $k=0,\ldots,n-1$, |
853 and modifies the axioms for $k=n$ as follows: |
853 and modifies the axioms for $k=n$ as follows: |
854 \begin{itemize} |
854 \begin{itemize} |
855 \item Morphisms. We have a functor $\cC_n$ from $\bbc$ ($n$-balls with boundary conditions) to $\cS$. |
855 \item Morphisms. We have a functor $\cC_n$ from $\bbc$ ($n$-balls with boundary conditions) to $\cS$. |
856 %[already said this above. ack] Furthermore, $\cC_n(f)$ depends only on the path component of a homeomorphism $f: (X, c) \to (X', c')$. |
856 %[already said this above. ack] Furthermore, $\cC_n(f)$ depends only on the path component of a homeomorphism $f: (X, c) \to (X', c')$. |
857 %In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially |
857 %In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially |
873 |
873 |
874 When the enriching category $\cS$ is chain complexes or topological spaces, |
874 When the enriching category $\cS$ is chain complexes or topological spaces, |
875 or more generally an appropriate sort of $\infty$-category, |
875 or more generally an appropriate sort of $\infty$-category, |
876 we can modify the extended isotopy axiom \ref{axiom:extended-isotopies} |
876 we can modify the extended isotopy axiom \ref{axiom:extended-isotopies} |
877 to require that families of homeomorphisms act |
877 to require that families of homeomorphisms act |
878 and obtain what we shall call an $A_\infty$ disk-like $n$-category. |
878 and obtain what we shall call an $A_\infty$ $n$-category. |
879 |
879 |
880 \noop{ |
880 \noop{ |
881 We believe that abstract definitions should be guided by diverse collections |
881 We believe that abstract definitions should be guided by diverse collections |
882 of concrete examples, and a lack of diversity in our present collection of examples of $A_\infty$ $n$-categories |
882 of concrete examples, and a lack of diversity in our present collection of examples of $A_\infty$ $n$-categories |
883 makes us reluctant to commit to an all-encompassing general definition. |
883 makes us reluctant to commit to an all-encompassing general definition. |
926 In fact, compatibility implies less than this. |
926 In fact, compatibility implies less than this. |
927 For simplicity, assume that $\cJ$ is $C_*$, the singular chains functor. |
927 For simplicity, assume that $\cJ$ is $C_*$, the singular chains functor. |
928 (This is the example most relevant to this paper.) |
928 (This is the example most relevant to this paper.) |
929 Then compatibility implies that the action of $C_*(\Homeo(X,c; X', c'))$ agrees with the action |
929 Then compatibility implies that the action of $C_*(\Homeo(X,c; X', c'))$ agrees with the action |
930 of $C_0(\Homeo(X,c; X', c'))$ coming from Axiom \ref{axiom:morphisms}, so we only require associativity in degree zero. |
930 of $C_0(\Homeo(X,c; X', c'))$ coming from Axiom \ref{axiom:morphisms}, so we only require associativity in degree zero. |
931 And indeed, this is true for our main example of an $A_\infty$ disk-like $n$-category based on the blob construction. |
931 And indeed, this is true for our main example of an $A_\infty$ $n$-category based on the blob construction. |
932 Stating this sort of compatibility for general $\cS$ and $\cJ$ requires further assumptions, |
932 Stating this sort of compatibility for general $\cS$ and $\cJ$ requires further assumptions, |
933 such as the forgetful functor from $\cS$ to sets having a left adjoint, and $\cS$ having an internal Hom. |
933 such as the forgetful functor from $\cS$ to sets having a left adjoint, and $\cS$ having an internal Hom. |
934 |
934 |
935 An alternative (due to Peter Teichner) is to say that Axiom \ref{axiom:families} |
935 An alternative (due to Peter Teichner) is to say that Axiom \ref{axiom:families} |
936 supersedes the $k=n$ case of Axiom \ref{axiom:morphisms}; in dimension $n$ we just have a |
936 supersedes the $k=n$ case of Axiom \ref{axiom:morphisms}; in dimension $n$ we just have a |
948 gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom. |
948 gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom. |
949 %since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across. |
949 %since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across. |
950 For future reference we make the following definition. |
950 For future reference we make the following definition. |
951 |
951 |
952 \begin{defn} |
952 \begin{defn} |
953 A {\em strict $A_\infty$ disk-like $n$-category} is one in which the actions of Axiom \ref{axiom:families} are strictly associative. |
953 A {\em strict $A_\infty$ $n$-category} is one in which the actions of Axiom \ref{axiom:families} are strictly associative. |
954 \end{defn} |
954 \end{defn} |
955 |
955 |
956 \noop{ |
956 \noop{ |
957 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
957 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
958 into a ordinary $n$-category (enriched over graded groups). |
958 into a ordinary $n$-category (enriched over graded groups). |
964 } |
964 } |
965 |
965 |
966 |
966 |
967 \medskip |
967 \medskip |
968 |
968 |
969 We define a $j$ times monoidal disk-like $n$-category to be a disk-like $(n{+}j)$-category $\cC$ where |
969 We define a $j$ times monoidal $n$-category to be an $(n{+}j)$-category $\cC$ where |
970 $\cC(X)$ is a trivial 1-element set if $X$ is a $k$-ball with $k<j$. |
970 $\cC(X)$ is a trivial 1-element set if $X$ is a $k$-ball with $k<j$. |
971 See Example \ref{ex:bord-cat}. |
971 See Example \ref{ex:bord-cat}. |
972 |
972 |
973 \medskip |
973 \medskip |
974 |
974 |
975 The alert reader will have already noticed that our definition of an (ordinary) disk-like $n$-category |
975 The alert reader will have already noticed that our definition of an (ordinary) $n$-category |
976 is extremely similar to our definition of a system of fields. |
976 is extremely similar to our definition of a system of fields. |
977 There are two differences. |
977 There are two differences. |
978 First, for the $n$-category definition we restrict our attention to balls |
978 First, for the $n$-category definition we restrict our attention to balls |
979 (and their boundaries), while for fields we consider all manifolds. |
979 (and their boundaries), while for fields we consider all manifolds. |
980 Second, in category definition we directly impose isotopy |
980 Second, in category definition we directly impose isotopy |
981 invariance in dimension $n$, while in the fields definition we |
981 invariance in dimension $n$, while in the fields definition we |
982 instead remember a subspace of local relations which contain differences of isotopic fields. |
982 instead remember a subspace of local relations which contain differences of isotopic fields. |
983 (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
983 (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
984 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 |
984 Thus a system of fields and local relations $(\cF,U)$ determines an $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to |
985 balls and, at level $n$, quotienting out by the local relations: |
985 balls and, at level $n$, quotienting out by the local relations: |
986 \begin{align*} |
986 \begin{align*} |
987 \cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases} |
987 \cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases} |
988 \end{align*} |
988 \end{align*} |
989 This $n$-category can be thought of as the local part of the fields. |
989 This $n$-category can be thought of as the local part of the fields. |
993 \medskip |
993 \medskip |
994 |
994 |
995 In the $n$-category axioms above we have intermingled data and properties for expository reasons. |
995 In the $n$-category axioms above we have intermingled data and properties for expository reasons. |
996 Here's a summary of the definition which segregates the data from the properties. |
996 Here's a summary of the definition which segregates the data from the properties. |
997 |
997 |
998 A disk-like $n$-category consists of the following data: |
998 An $n$-category consists of the following data: |
999 \begin{itemize} |
999 \begin{itemize} |
1000 \item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms}); |
1000 \item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms}); |
1001 \item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary}); |
1001 \item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary}); |
1002 \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}); |
1002 \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}); |
1003 \item ``product'' or ``identity'' maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product}); |
1003 \item ``product'' or ``identity'' maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product}); |
1151 C_*(\Maps_c(X \to T)), |
1151 C_*(\Maps_c(X \to T)), |
1152 \] |
1152 \] |
1153 where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
1153 where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
1154 and $C_*$ denotes singular chains. |
1154 and $C_*$ denotes singular chains. |
1155 Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X \to T)$, |
1155 Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X \to T)$, |
1156 we get an $A_\infty$ disk-like $n$-category enriched over spaces. |
1156 we get an $A_\infty$ $n$-category enriched over spaces. |
1157 \end{example} |
1157 \end{example} |
1158 |
1158 |
1159 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to |
1159 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to |
1160 homotopy as the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
1160 homotopy as the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
1161 |
1161 |
1162 \begin{example}[Blob complexes of balls (with a fiber)] |
1162 \begin{example}[Blob complexes of balls (with a fiber)] |
1163 \rm |
1163 \rm |
1164 \label{ex:blob-complexes-of-balls} |
1164 \label{ex:blob-complexes-of-balls} |
1165 Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
1165 Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
1166 We will define an $A_\infty$ disk-like $k$-category $\cC$. |
1166 We will define an $A_\infty$ $k$-category $\cC$. |
1167 When $X$ is a $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$. |
1167 When $X$ is a $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$. |
1168 When $X$ is an $k$-ball, |
1168 When $X$ is an $k$-ball, |
1169 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
1169 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
1170 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
1170 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
1171 \end{example} |
1171 \end{example} |
1172 |
1172 |
1173 This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
1173 This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
1174 Notice that with $F$ a point, the above example is a construction turning an ordinary disk-like |
1174 Notice that with $F$ a point, the above example is a construction turning an ordinary |
1175 $n$-category $\cC$ into an $A_\infty$ disk-like $n$-category. |
1175 $n$-category $\cC$ into an $A_\infty$ $n$-category. |
1176 We think of this as providing a ``free resolution" |
1176 We think of this as providing a ``free resolution" |
1177 of the ordinary disk-like $n$-category. |
1177 of the ordinary $n$-category. |
1178 %\nn{say something about cofibrant replacements?} |
1178 %\nn{say something about cofibrant replacements?} |
1179 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
1179 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
1180 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
1180 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
1181 and take $\CD{B}$ to act trivially. |
1181 and take $\CD{B}$ to act trivially. |
1182 |
1182 |
1183 Beware that the ``free resolution" of the ordinary disk-like $n$-category $\pi_{\leq n}(T)$ |
1183 Beware that the ``free resolution" of the ordinary $n$-category $\pi_{\leq n}(T)$ |
1184 is not the $A_\infty$ disk-like $n$-category $\pi^\infty_{\leq n}(T)$. |
1184 is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
1185 It's easy to see that with $n=0$, the corresponding system of fields is just |
1185 It's easy to see that with $n=0$, the corresponding system of fields is just |
1186 linear combinations of connected components of $T$, and the local relations are trivial. |
1186 linear combinations of connected components of $T$, and the local relations are trivial. |
1187 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
1187 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
1188 |
1188 |
1189 \begin{example}[The bordism $n$-category of $d$-manifolds, $A_\infty$ version] |
1189 \begin{example}[The bordism $n$-category of $d$-manifolds, $A_\infty$ version] |
1223 \rm |
1223 \rm |
1224 \label{ex:e-n-alg} |
1224 \label{ex:e-n-alg} |
1225 Let $A$ be an $\cE\cB_n$-algebra. |
1225 Let $A$ be an $\cE\cB_n$-algebra. |
1226 Note that this implies a $\Diff(B^n)$ action on $A$, |
1226 Note that this implies a $\Diff(B^n)$ action on $A$, |
1227 since $\cE\cB_n$ contains a copy of $\Diff(B^n)$. |
1227 since $\cE\cB_n$ contains a copy of $\Diff(B^n)$. |
1228 We will define a strict $A_\infty$ disk-like $n$-category $\cC^A$. |
1228 We will define a strict $A_\infty$ $n$-category $\cC^A$. |
1229 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point. |
1229 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point. |
1230 In other words, the $k$-morphisms are trivial for $k<n$. |
1230 In other words, the $k$-morphisms are trivial for $k<n$. |
1231 If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction. |
1231 If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction. |
1232 (Plain colimit, not homotopy colimit.) |
1232 (Plain colimit, not homotopy colimit.) |
1233 Let $J$ be the category whose objects are embeddings of a disjoint union of copies of |
1233 Let $J$ be the category whose objects are embeddings of a disjoint union of copies of |
1235 embedded balls into a single larger embedded ball. |
1235 embedded balls into a single larger embedded ball. |
1236 To each object of $J$ we associate $A^{\times m}$ (where $m$ is the number of balls), and |
1236 To each object of $J$ we associate $A^{\times m}$ (where $m$ is the number of balls), and |
1237 to each morphism of $J$ we associate a morphism coming from the $\cE\cB_n$ action on $A$. |
1237 to each morphism of $J$ we associate a morphism coming from the $\cE\cB_n$ action on $A$. |
1238 Alternatively and more simply, we could define $\cC^A(X)$ to be |
1238 Alternatively and more simply, we could define $\cC^A(X)$ to be |
1239 $\Diff(B^n\to X)\times A$ modulo the diagonal action of $\Diff(B^n)$. |
1239 $\Diff(B^n\to X)\times A$ modulo the diagonal action of $\Diff(B^n)$. |
1240 The remaining data for the $A_\infty$ disk-like $n$-category |
1240 The remaining data for the $A_\infty$ $n$-category |
1241 --- composition and $\Diff(X\to X')$ action --- |
1241 --- composition and $\Diff(X\to X')$ action --- |
1242 also comes from the $\cE\cB_n$ action on $A$. |
1242 also comes from the $\cE\cB_n$ action on $A$. |
1243 %\nn{should we spell this out?} |
1243 %\nn{should we spell this out?} |
1244 |
1244 |
1245 Conversely, one can show that a strict $A_\infty$ disk-like $n$-category $\cC$, where the $k$-morphisms |
1245 Conversely, one can show that a disk-like strict $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
1246 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
1246 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
1247 an $\cE\cB_n$-algebra. |
1247 an $\cE\cB_n$-algebra. |
1248 %\nn{The paper is already long; is it worth giving details here?} |
1248 %\nn{The paper is already long; is it worth giving details here?} |
1249 % According to the referee, yes it is... |
1249 % According to the referee, yes it is... |
1250 Let $A = \cC(B^n)$, where $B^n$ is the standard $n$-ball. |
1250 Let $A = \cC(B^n)$, where $B^n$ is the standard $n$-ball. |
1255 \end{example} |
1255 \end{example} |
1256 |
1256 |
1257 |
1257 |
1258 \subsection{From balls to manifolds} |
1258 \subsection{From balls to manifolds} |
1259 \label{ss:ncat_fields} \label{ss:ncat-coend} |
1259 \label{ss:ncat_fields} \label{ss:ncat-coend} |
1260 In this section we show how to extend a disk-like $n$-category $\cC$ as described above |
1260 In this section we show how to extend an $n$-category $\cC$ as described above |
1261 (of either the ordinary or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
1261 (of either the ordinary or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
1262 This extension is a certain colimit, and the arrow in the notation is intended as a reminder of this. |
1262 This extension is a certain colimit, and the arrow in the notation is intended as a reminder of this. |
1263 |
1263 |
1264 In the case of ordinary disk-like $n$-categories, this construction factors into a construction of a |
1264 In the case of ordinary $n$-categories, this construction factors into a construction of a |
1265 system of fields and local relations, followed by the usual TQFT definition of a |
1265 system of fields and local relations, followed by the usual TQFT definition of a |
1266 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
1266 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
1267 For an $A_\infty$ disk-like $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
1267 For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
1268 Recall that we can take a ordinary disk-like $n$-category $\cC$ and pass to the ``free resolution", |
1268 Recall that we can take a ordinary $n$-category $\cC$ and pass to the ``free resolution", |
1269 an $A_\infty$ disk-like $n$-category $\bc_*(\cC)$, by computing the blob complex of balls |
1269 an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls |
1270 (recall Example \ref{ex:blob-complexes-of-balls} above). |
1270 (recall Example \ref{ex:blob-complexes-of-balls} above). |
1271 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
1271 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
1272 for a manifold $M$ associated to this $A_\infty$ disk-like $n$-category is actually the |
1272 for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the |
1273 same as the original blob complex for $M$ with coefficients in $\cC$. |
1273 same as the original blob complex for $M$ with coefficients in $\cC$. |
1274 |
1274 |
1275 Recall that we've already anticipated this construction Subsection \ref{ss:n-cat-def}, |
1275 Recall that we've already anticipated this construction Subsection \ref{ss:n-cat-def}, |
1276 inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
1276 inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
1277 so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
1277 so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
1278 |
1278 |
1279 \medskip |
1279 \medskip |
1280 |
1280 |
1281 We will first define the {\it decomposition poset} $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
1281 We will first define the {\it decomposition poset} $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
1282 A disk-like $n$-category $\cC$ provides a functor from this poset to the category of sets, |
1282 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
1283 and we will define $\cl{\cC}(W)$ as a suitable colimit |
1283 and we will define $\cl{\cC}(W)$ as a suitable colimit |
1284 (or homotopy colimit in the $A_\infty$ case) of this functor. |
1284 (or homotopy colimit in the $A_\infty$ case) of this functor. |
1285 We'll later give a more explicit description of this colimit. |
1285 We'll later give a more explicit description of this colimit. |
1286 In the case that the disk-like $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain |
1286 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain |
1287 complexes to $n$-balls with boundary data), |
1287 complexes to $n$-balls with boundary data), |
1288 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into |
1288 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into |
1289 subsets according to boundary data, and each of these subsets has the appropriate structure |
1289 subsets according to boundary data, and each of these subsets has the appropriate structure |
1290 (e.g. a vector space or chain complex). |
1290 (e.g. a vector space or chain complex). |
1291 |
1291 |
1400 |
1400 |
1401 Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$: |
1401 Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$: |
1402 |
1402 |
1403 \begin{defn}[System of fields functor] |
1403 \begin{defn}[System of fields functor] |
1404 \label{def:colim-fields} |
1404 \label{def:colim-fields} |
1405 If $\cC$ is a disk-like $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
1405 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
1406 That is, for each decomposition $x$ there is a map |
1406 That is, for each decomposition $x$ there is a map |
1407 $\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps |
1407 $\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps |
1408 above, and $\cl{\cC}(W)$ is universal with respect to these properties. |
1408 above, and $\cl{\cC}(W)$ is universal with respect to these properties. |
1409 \end{defn} |
1409 \end{defn} |
1410 |
1410 |
1411 \begin{defn}[System of fields functor, $A_\infty$ case] |
1411 \begin{defn}[System of fields functor, $A_\infty$ case] |
1412 When $\cC$ is an $A_\infty$ disk-like $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$ |
1412 When $\cC$ is an $A_\infty$ $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$ |
1413 is defined as above, as the colimit of $\psi_{\cC;W}$. |
1413 is defined as above, as the colimit of $\psi_{\cC;W}$. |
1414 When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
1414 When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
1415 \end{defn} |
1415 \end{defn} |
1416 |
1416 |
1417 %We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$ |
1417 %We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$ |
1583 %\nn{need to finish explaining why we have a system of fields; |
1583 %\nn{need to finish explaining why we have a system of fields; |
1584 %define $k$-cat $\cC(\cdot\times W)$} |
1584 %define $k$-cat $\cC(\cdot\times W)$} |
1585 |
1585 |
1586 \subsection{Modules} |
1586 \subsection{Modules} |
1587 |
1587 |
1588 Next we define ordinary and $A_\infty$ disk-like $n$-category modules. |
1588 Next we define ordinary and $A_\infty$ $n$-category modules. |
1589 The definition will be very similar to that of disk-like $n$-categories, |
1589 The definition will be very similar to that of $n$-categories, |
1590 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
1590 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
1591 |
1591 |
1592 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
1592 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
1593 in the context of an $m{+}1$-dimensional TQFT. |
1593 in the context of an $m{+}1$-dimensional TQFT. |
1594 Such a $W$ gives rise to a module for the disk-like $n$-category associated to $\bd W$. |
1594 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
1595 This will be explained in more detail as we present the axioms. |
1595 This will be explained in more detail as we present the axioms. |
1596 |
1596 |
1597 Throughout, we fix a disk-like $n$-category $\cC$. |
1597 Throughout, we fix an $n$-category $\cC$. |
1598 For all but one axiom, it doesn't matter whether $\cC$ is an ordinary $n$-category or an $A_\infty$ $n$-category. |
1598 For all but one axiom, it doesn't matter whether $\cC$ is an ordinary $n$-category or an $A_\infty$ $n$-category. |
1599 We state the final axiom, regarding actions of homeomorphisms, differently in the two cases. |
1599 We state the final axiom, regarding actions of homeomorphisms, differently in the two cases. |
1600 |
1600 |
1601 Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
1601 Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
1602 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
1602 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
1648 These maps, for various $M$, comprise a natural transformation of functors.} |
1648 These maps, for various $M$, comprise a natural transformation of functors.} |
1649 \end{module-axiom} |
1649 \end{module-axiom} |
1650 |
1650 |
1651 Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
1651 Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
1652 |
1652 |
1653 If the disk-like $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
1653 If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
1654 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. |
1654 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. |
1655 |
1655 |
1656 \begin{lem}[Boundary from domain and range] |
1656 \begin{lem}[Boundary from domain and range] |
1657 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$), |
1657 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$), |
1658 $M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere. |
1658 $M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere. |
1855 \] |
1855 \] |
1856 ($Y$ could be either a marked or plain ball.) |
1856 ($Y$ could be either a marked or plain ball.) |
1857 \end{enumerate} |
1857 \end{enumerate} |
1858 \end{module-axiom} |
1858 \end{module-axiom} |
1859 |
1859 |
1860 As in the disk-like $n$-category definition, once we have product morphisms we can define |
1860 As in the $n$-category definition, once we have product morphisms we can define |
1861 collar maps $\cM(M)\to \cM(M)$. |
1861 collar maps $\cM(M)\to \cM(M)$. |
1862 Note that there are two cases: |
1862 Note that there are two cases: |
1863 the collar could intersect the marking of the marked ball $M$, in which case |
1863 the collar could intersect the marking of the marked ball $M$, in which case |
1864 we use a product on a morphism of $\cM$; or the collar could be disjoint from the marking, |
1864 we use a product on a morphism of $\cM$; or the collar could be disjoint from the marking, |
1865 in which case we use a product on a morphism of $\cC$. |
1865 in which case we use a product on a morphism of $\cC$. |
1868 $a$ along a map associated to $\pi$. |
1868 $a$ along a map associated to $\pi$. |
1869 |
1869 |
1870 \medskip |
1870 \medskip |
1871 |
1871 |
1872 There are two alternatives for the next axiom, according whether we are defining |
1872 There are two alternatives for the next axiom, according whether we are defining |
1873 modules for ordinary or $A_\infty$ disk-like $n$-categories. |
1873 modules for ordinary $n$-categories or $A_\infty$ $n$-categories. |
1874 In the ordinary case we require |
1874 In the ordinary case we require |
1875 |
1875 |
1876 \begin{module-axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$] |
1876 \begin{module-axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$] |
1877 {Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
1877 {Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
1878 to the identity on $\bd M$ and is isotopic (rel boundary) to the identity. |
1878 to the identity on $\bd M$ and is isotopic (rel boundary) to the identity. |
1901 |
1901 |
1902 As with the $n$-category version of the above axiom, we should also have families of collar maps act. |
1902 As with the $n$-category version of the above axiom, we should also have families of collar maps act. |
1903 |
1903 |
1904 \medskip |
1904 \medskip |
1905 |
1905 |
1906 Note that the above axioms imply that a disk-like $n$-category module has the structure |
1906 Note that the above axioms imply that an $n$-category module has the structure |
1907 of a disk-like $n{-}1$-category. |
1907 of an $n{-}1$-category. |
1908 More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
1908 More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
1909 where $X$ is a $k$-ball and in the product $X\times J$ we pinch |
1909 where $X$ is a $k$-ball and in the product $X\times J$ we pinch |
1910 above the non-marked boundary component of $J$. |
1910 above the non-marked boundary component of $J$. |
1911 (More specifically, we collapse $X\times P$ to a single point, where |
1911 (More specifically, we collapse $X\times P$ to a single point, where |
1912 $P$ is the non-marked boundary component of $J$.) |
1912 $P$ is the non-marked boundary component of $J$.) |
1913 Then $\cE$ has the structure of a disk-like $n{-}1$-category. |
1913 Then $\cE$ has the structure of an $n{-}1$-category. |
1914 |
1914 |
1915 All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
1915 All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
1916 are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
1916 are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
1917 In this case ($k=1$ and oriented or Spin), there are two types |
1917 In this case ($k=1$ and oriented or Spin), there are two types |
1918 of marked 1-balls, call them left-marked and right-marked, |
1918 of marked 1-balls, call them left-marked and right-marked, |
1920 In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
1920 In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
1921 there is no left/right module distinction. |
1921 there is no left/right module distinction. |
1922 |
1922 |
1923 \medskip |
1923 \medskip |
1924 |
1924 |
1925 We now give some examples of modules over ordinary and $A_\infty$ disk-like $n$-categories. |
1925 We now give some examples of modules over ordinary and $A_\infty$ $n$-categories. |
1926 |
1926 |
1927 \begin{example}[Examples from TQFTs] |
1927 \begin{example}[Examples from TQFTs] |
1928 \rm |
1928 \rm |
1929 Continuing Example \ref{ex:ncats-from-tqfts}, with $\cF$ a TQFT, $W$ an $n{-}j$-manifold, |
1929 Continuing Example \ref{ex:ncats-from-tqfts}, with $\cF$ a TQFT, $W$ an $n{-}j$-manifold, |
1930 and $\cF(W)$ the disk-like $j$-category associated to $W$. |
1930 and $\cF(W)$ the $j$-category associated to $W$. |
1931 Let $Y$ be an $(n{-}j{+}1)$-manifold with $\bd Y = W$. |
1931 Let $Y$ be an $(n{-}j{+}1)$-manifold with $\bd Y = W$. |
1932 Define a $\cF(W)$ module $\cF(Y)$ as follows. |
1932 Define a $\cF(W)$ module $\cF(Y)$ as follows. |
1933 If $M = (B, N)$ is a marked $k$-ball with $k<j$ let |
1933 If $M = (B, N)$ is a marked $k$-ball with $k<j$ let |
1934 $\cF(Y)(M)\deq \cF((B\times W) \cup (N\times Y))$. |
1934 $\cF(Y)(M)\deq \cF((B\times W) \cup (N\times Y))$. |
1935 If $M = (B, N)$ is a marked $j$-ball and $c\in \cl{\cF(Y)}(\bd M)$ let |
1935 If $M = (B, N)$ is a marked $j$-ball and $c\in \cl{\cF(Y)}(\bd M)$ let |
2019 |
2019 |
2020 If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
2020 If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
2021 $\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
2021 $\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
2022 $D\times Y_i \sub \bd(D\times W)$. |
2022 $D\times Y_i \sub \bd(D\times W)$. |
2023 It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$ |
2023 It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$ |
2024 has the structure of a disk-like $n{-}k$-category. |
2024 has the structure of an $n{-}k$-category. |
2025 |
2025 |
2026 \medskip |
2026 \medskip |
2027 |
2027 |
2028 We will use a simple special case of the above |
2028 We will use a simple special case of the above |
2029 construction to define tensor products |
2029 construction to define tensor products |
2030 of modules. |
2030 of modules. |
2031 Let $\cM_1$ and $\cM_2$ be modules for a disk-like $n$-category $\cC$. |
2031 Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
2032 (If $k=1$ and our manifolds are oriented, then one should be |
2032 (If $k=1$ and our manifolds are oriented, then one should be |
2033 a left module and the other a right module.) |
2033 a left module and the other a right module.) |
2034 Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
2034 Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
2035 Define the tensor product $\cM_1 \tensor \cM_2$ to be the |
2035 Define the tensor product $\cM_1 \tensor \cM_2$ to be the |
2036 disk-like $n{-}1$-category associated as above to $J$ with its boundary labeled by $\cM_1$ and $\cM_2$. |
2036 $n{-}1$-category associated as above to $J$ with its boundary labeled by $\cM_1$ and $\cM_2$. |
2037 This of course depends (functorially) |
2037 This of course depends (functorially) |
2038 on the choice of 1-ball $J$. |
2038 on the choice of 1-ball $J$. |
2039 |
2039 |
2040 We will define a more general self tensor product (categorified coend) below. |
2040 We will define a more general self tensor product (categorified coend) below. |
2041 |
2041 |
2736 then compose the module maps. |
2736 then compose the module maps. |
2737 The proof that this composition rule is associative is similar to the proof of Lemma \ref{equator-lemma}. |
2737 The proof that this composition rule is associative is similar to the proof of Lemma \ref{equator-lemma}. |
2738 |
2738 |
2739 \medskip |
2739 \medskip |
2740 |
2740 |
2741 We end this subsection with some remarks about Morita equivalence of disk-like $n$-categories. |
2741 We end this subsection with some remarks about Morita equivalence of disklike $n$-categories. |
2742 Recall that two 1-categories $\cC$ and $\cD$ are Morita equivalent if and only if they are equivalent |
2742 Recall that two 1-categories $\cC$ and $\cD$ are Morita equivalent if and only if they are equivalent |
2743 objects in the 2-category of (linear) 1-categories, bimodules, and intertwiners. |
2743 objects in the 2-category of (linear) 1-categories, bimodules, and intertwiners. |
2744 Similarly, we define two disk-like $n$-categories to be Morita equivalent if they are equivalent objects in the |
2744 Similarly, we define two disklike $n$-categories to be Morita equivalent if they are equivalent objects in the |
2745 $n{+}1$-category of sphere modules. |
2745 $n{+}1$-category of sphere modules. |
2746 |
2746 |
2747 Because of the strong duality enjoyed by disk-like $n$-categories, the data for such an equivalence lives only in |
2747 Because of the strong duality enjoyed by disklike $n$-categories, the data for such an equivalence lives only in |
2748 dimensions 1 and $n+1$ (the middle dimensions come along for free). |
2748 dimensions 1 and $n+1$ (the middle dimensions come along for free). |
2749 The $n{+}1$-dimensional part of the data must be invertible and satisfy |
2749 The $n{+}1$-dimensional part of the data must be invertible and satisfy |
2750 identities corresponding to Morse cancellations in $n$-manifolds. |
2750 identities corresponding to Morse cancellations in $n$-manifolds. |
2751 We will treat this in detail for the $n=2$ case; the case for general $n$ is very similar. |
2751 We will treat this in detail for the $n=2$ case; the case for general $n$ is very similar. |
2752 |
2752 |
2753 Let $\cC$ and $\cD$ be (unoriented) disk-like 2-categories. |
2753 Let $\cC$ and $\cD$ be (unoriented) disklike 2-categories. |
2754 Let $\cS$ denote the 3-category of 2-category sphere modules. |
2754 Let $\cS$ denote the 3-category of 2-category sphere modules. |
2755 The 1-dimensional part of the data for a Morita equivalence between $\cC$ and $\cD$ is a 0-sphere module $\cM = {}_\cC\cM_\cD$ |
2755 The 1-dimensional part of the data for a Morita equivalence between $\cC$ and $\cD$ is a 0-sphere module $\cM = {}_\cC\cM_\cD$ |
2756 (categorified bimodule) connecting $\cC$ and $\cD$. |
2756 (categorified bimodule) connecting $\cC$ and $\cD$. |
2757 Because of the full unoriented symmetry, this can also be thought of as a |
2757 Because of the full unoriented symmetry, this can also be thought of as a |
2758 0-sphere module ${}_\cD\cM_\cC$ connecting $\cD$ and $\cC$. |
2758 0-sphere module ${}_\cD\cM_\cC$ connecting $\cD$ and $\cC$. |