825 \label{zzz3} |
825 \label{zzz3} |
826 \end{figure} |
826 \end{figure} |
827 |
827 |
828 First, we can compose two module morphisms to get another module morphism. |
828 First, we can compose two module morphisms to get another module morphism. |
829 |
829 |
830 \xxpar{Module composition:} |
830 \mmpar{Module axiom 6}{Module composition} |
831 {Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls ($0\le k\le n$) |
831 {Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls ($0\le k\le n$) |
832 and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
832 and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
833 Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
833 Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
834 Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
834 Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
835 We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$. |
835 We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$. |
847 |
847 |
848 Second, we can compose an $n$-category morphism with a module morphism to get another |
848 Second, we can compose an $n$-category morphism with a module morphism to get another |
849 module morphism. |
849 module morphism. |
850 We'll call this the action map to distinguish it from the other kind of composition. |
850 We'll call this the action map to distinguish it from the other kind of composition. |
851 |
851 |
852 \xxpar{$n$-category action:} |
852 \mmpar{Module axiom 7}{$n$-category action} |
853 {Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
853 {Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
854 $X$ is a plain $k$-ball, |
854 $X$ is a plain $k$-ball, |
855 and $Y = X\cap M'$ is a $k{-}1$-ball. |
855 and $Y = X\cap M'$ is a $k{-}1$-ball. |
856 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
856 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
857 We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$. |
857 We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$. |
863 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
863 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
864 to the intersection of the boundaries of $X$ and $M'$. |
864 to the intersection of the boundaries of $X$ and $M'$. |
865 If $k < n$ we require that $\gl_Y$ is injective. |
865 If $k < n$ we require that $\gl_Y$ is injective. |
866 (For $k=n$, see below.)} |
866 (For $k=n$, see below.)} |
867 |
867 |
868 \xxpar{Module strict associativity:} |
868 \mmpar{Module axiom 8}{Strict associativity} |
869 {The composition and action maps above are strictly associative.} |
869 {The composition and action maps above are strictly associative.} |
870 |
870 |
871 Note that the above associativity axiom applies to mixtures of module composition, |
871 Note that the above associativity axiom applies to mixtures of module composition, |
872 action maps and $n$-category composition. |
872 action maps and $n$-category composition. |
873 See Figure \ref{zzz1b}. |
873 See Figure \ref{zzz1b}. |
901 and these various multifold composition maps satisfy an |
901 and these various multifold composition maps satisfy an |
902 operad-type strict associativity condition.} |
902 operad-type strict associativity condition.} |
903 |
903 |
904 (The above operad-like structure is analogous to the swiss cheese operad |
904 (The above operad-like structure is analogous to the swiss cheese operad |
905 \cite{MR1718089}.) |
905 \cite{MR1718089}.) |
906 \nn{need to double-check that this is true.} |
906 %\nn{need to double-check that this is true.} |
907 |
907 |
908 \xxpar{Module product (identity) morphisms:} |
908 \mmpar{Module axiom 9}{Product/identity morphisms} |
909 {Let $M$ be a marked $k$-ball and $D$ be a plain $m$-ball, with $k+m \le n$. |
909 {Let $M$ be a marked $k$-ball and $D$ be a plain $m$-ball, with $k+m \le n$. |
910 Then we have a map $\cM(M)\to \cM(M\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cM(M)$. |
910 Then we have a map $\cM(M)\to \cM(M\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cM(M)$. |
911 If $f:M\to M'$ and $\tilde{f}:M\times D \to M'\times D'$ are maps such that the diagram |
911 If $f:M\to M'$ and $\tilde{f}:M\times D \to M'\times D'$ are maps such that the diagram |
912 \[ \xymatrix{ |
912 \[ \xymatrix{ |
913 M\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & M'\times D' \ar[d]^{\pi} \\ |
913 M\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & M'\times D' \ar[d]^{\pi} \\ |
915 } \] |
915 } \] |
916 commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
916 commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
917 |
917 |
918 \nn{Need to add compatibility with various things, as in the n-cat version of this axiom above.} |
918 \nn{Need to add compatibility with various things, as in the n-cat version of this axiom above.} |
919 |
919 |
920 \nn{** marker --- resume revising here **} |
920 \nn{postpone finalizing the above axiom until the n-cat version is finalized} |
921 |
921 |
922 There are two alternatives for the next axiom, according whether we are defining |
922 There are two alternatives for the next axiom, according whether we are defining |
923 modules for plain $n$-categories or $A_\infty$ $n$-categories. |
923 modules for plain $n$-categories or $A_\infty$ $n$-categories. |
924 In the plain case we require |
924 In the plain case we require |
925 |
925 |
926 \xxpar{Extended isotopy invariance in dimension $n$:} |
926 \mmpar{Module axiom 10a}{Extended isotopy invariance in dimension $n$} |
927 {Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
927 {Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
928 to the identity on $\bd M$ and is extended isotopic (rel boundary) to the identity. |
928 to the identity on $\bd M$ and is extended isotopic (rel boundary) to the identity. |
929 Then $f$ acts trivially on $\cM(M)$.} |
929 Then $f$ acts trivially on $\cM(M)$.} |
930 |
930 |
931 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
931 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
934 In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
934 In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
935 on $\bd B \setmin N$. |
935 on $\bd B \setmin N$. |
936 |
936 |
937 For $A_\infty$ modules we require |
937 For $A_\infty$ modules we require |
938 |
938 |
939 \xxpar{Families of homeomorphisms act.} |
939 \mmpar{Module axiom 10b}{Families of homeomorphisms act} |
940 {For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
940 {For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
941 \[ |
941 \[ |
942 C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
942 C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
943 \] |
943 \] |
944 Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |
944 Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |
954 Note that the above axioms imply that an $n$-category module has the structure |
954 Note that the above axioms imply that an $n$-category module has the structure |
955 of an $n{-}1$-category. |
955 of an $n{-}1$-category. |
956 More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
956 More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
957 where $X$ is a $k$-ball or $k{-}1$-sphere and in the product $X\times J$ we pinch |
957 where $X$ is a $k$-ball or $k{-}1$-sphere and in the product $X\times J$ we pinch |
958 above the non-marked boundary component of $J$. |
958 above the non-marked boundary component of $J$. |
959 \nn{give figure for this, or say more?} |
959 (More specifically, we collapse $X\times P$ to a single point, where |
|
960 $P$ is the non-marked boundary component of $J$.) |
|
961 \nn{give figure for this?} |
960 Then $\cE$ has the structure of an $n{-}1$-category. |
962 Then $\cE$ has the structure of an $n{-}1$-category. |
961 |
963 |
962 All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
964 All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
963 are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
965 are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
964 In this case ($k=1$ and oriented or Spin), there are two types |
966 In this case ($k=1$ and oriented or Spin), there are two types |