88 At first it might seem that we need another axiom for this, but in fact once we have |
88 At first it might seem that we need another axiom for this, but in fact once we have |
89 all the axioms in the subsection for $0$ through $k-1$ we can use a colimit |
89 all the axioms in the subsection for $0$ through $k-1$ we can use a colimit |
90 construction, as described in Subsection \ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ |
90 construction, as described in Subsection \ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ |
91 to spheres (and any other manifolds): |
91 to spheres (and any other manifolds): |
92 |
92 |
93 \begin{prop} |
93 \begin{lem} |
94 \label{axiom:spheres} |
94 \label{lem:spheres} |
95 For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from |
95 For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from |
96 the category of $k{-}1$-spheres and |
96 the category of $k{-}1$-spheres and |
97 homeomorphisms to the category of sets and bijections. |
97 homeomorphisms to the category of sets and bijections. |
98 \end{prop} |
98 \end{lem} |
99 |
99 |
100 We postpone the proof \todo{} of this result until after we've actually given all the axioms. Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, along with the data described in the other Axioms at lower levels. |
100 We postpone the proof \todo{} of this result until after we've actually given all the axioms. Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, along with the data described in the other Axioms at lower levels. |
101 |
101 |
102 %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
102 %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
103 |
103 |
140 |
140 |
141 We have just argued that the boundary of a morphism has no preferred splitting into |
141 We have just argued that the boundary of a morphism has no preferred splitting into |
142 domain and range, but the converse meets with our approval. |
142 domain and range, but the converse meets with our approval. |
143 That is, given compatible domain and range, we should be able to combine them into |
143 That is, given compatible domain and range, we should be able to combine them into |
144 the full boundary of a morphism. |
144 the full boundary of a morphism. |
145 The following proposition follows from the colimit construction used to define $\cl{\cC}_{k-1}$ |
145 The following lemma follows from the colimit construction used to define $\cl{\cC}_{k-1}$ |
146 on spheres. |
146 on spheres. |
147 |
147 |
148 \begin{prop}[Boundary from domain and range] |
148 \begin{lem}[Boundary from domain and range] |
|
149 \label{lem:domain-and-range} |
149 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
150 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
150 $B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
151 $B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
151 Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
152 Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
152 two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$. |
153 two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$. |
153 Then we have an injective map |
154 Then we have an injective map |
785 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
786 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
786 We call $B$ the ball and $N$ the marking. |
787 We call $B$ the ball and $N$ the marking. |
787 A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
788 A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
788 restricts to a homeomorphism of markings. |
789 restricts to a homeomorphism of markings. |
789 |
790 |
790 \mmpar{Module axiom 1}{Module morphisms} |
791 \begin{module-axiom}[Module morphisms] |
791 {For each $0 \le k \le n$, we have a functor $\cM_k$ from |
792 {For each $0 \le k \le n$, we have a functor $\cM_k$ from |
792 the category of marked $k$-balls and |
793 the category of marked $k$-balls and |
793 homeomorphisms to the category of sets and bijections.} |
794 homeomorphisms to the category of sets and bijections.} |
|
795 \end{module-axiom} |
794 |
796 |
795 (As with $n$-categories, we will usually omit the subscript $k$.) |
797 (As with $n$-categories, we will usually omit the subscript $k$.) |
796 |
798 |
797 For example, let $\cD$ be the $m{+}1$-dimensional TQFT which assigns to a $k$-manifold $N$ the set |
799 For example, let $\cD$ be the $m{+}1$-dimensional TQFT which assigns to a $k$-manifold $N$ the set |
798 of maps from $N$ to $T$, modulo homotopy (and possibly linearized) if $k=m$. |
800 of maps from $N$ to $T$, modulo homotopy (and possibly linearized) if $k=m$. |
808 \caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
810 \caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
809 |
811 |
810 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
812 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
811 Call such a thing a {marked $k{-}1$-hemisphere}. |
813 Call such a thing a {marked $k{-}1$-hemisphere}. |
812 |
814 |
813 \mmpar{Module axiom 2}{Module boundaries (hemispheres)} |
815 \begin{lem} |
|
816 \label{lem:hemispheres} |
814 {For each $0 \le k \le n-1$, we have a functor $\cM_k$ from |
817 {For each $0 \le k \le n-1$, we have a functor $\cM_k$ from |
815 the category of marked $k$-hemispheres and |
818 the category of marked $k$-hemispheres and |
816 homeomorphisms to the category of sets and bijections.} |
819 homeomorphisms to the category of sets and bijections.} |
|
820 \end{lem} |
|
821 The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. We use the same type of colimit construction. |
817 |
822 |
818 In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$. |
823 In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$. |
819 |
824 |
820 \mmpar{Module axiom 3}{Module boundaries (maps)} |
825 \begin{module-axiom}[Module boundaries (maps)] |
821 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. |
826 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. |
822 These maps, for various $M$, comprise a natural transformation of functors.} |
827 These maps, for various $M$, comprise a natural transformation of functors.} |
|
828 \end{module-axiom} |
823 |
829 |
824 Given $c\in\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
830 Given $c\in\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
825 |
831 |
826 If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
832 If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
827 then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
833 then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
828 and $c\in \cC(\bd M)$. |
834 and $c\in \cC(\bd M)$. |
829 |
835 |
830 \mmpar{Module axiom 4}{Boundary from domain and range} |
836 \begin{lem}[Boundary from domain and range] |
831 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k$-hemisphere ($0\le k\le n-1$), |
837 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k$-hemisphere ($0\le k\le n-1$), |
832 $M_i$ is a marked $k$-ball, and $E = M_1\cap M_2$ is a marked $k{-}1$-hemisphere. |
838 $M_i$ is a marked $k$-ball, and $E = M_1\cap M_2$ is a marked $k{-}1$-hemisphere. |
833 Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
839 Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
834 two maps $\bd: \cM(M_i)\to \cM(E)$. |
840 two maps $\bd: \cM(M_i)\to \cM(E)$. |
835 Then (axiom) we have an injective map |
841 Then (axiom) we have an injective map |
836 \[ |
842 \[ |
837 \gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \hookrightarrow \cM(H) |
843 \gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \hookrightarrow \cM(H) |
838 \] |
844 \] |
839 which is natural with respect to the actions of homeomorphisms.} |
845 which is natural with respect to the actions of homeomorphisms.} |
|
846 \end{lem} |
|
847 Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}. |
840 |
848 |
841 Let $\cM(H)_E$ denote the image of $\gl_E$. |
849 Let $\cM(H)_E$ denote the image of $\gl_E$. |
842 We will refer to elements of $\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". |
850 We will refer to elements of $\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". |
843 |
851 |
844 |
852 |
845 \mmpar{Module axiom 5}{Module to category restrictions} |
853 \begin{module-axiom}[Module to category restrictions] |
846 {For each marked $k$-hemisphere $H$ there is a restriction map |
854 {For each marked $k$-hemisphere $H$ there is a restriction map |
847 $\cM(H)\to \cC(H)$. |
855 $\cM(H)\to \cC(H)$. |
848 ($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
856 ($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
849 These maps comprise a natural transformation of functors.} |
857 These maps comprise a natural transformation of functors.} |
|
858 \end{module-axiom} |
850 |
859 |
851 Note that combining the various boundary and restriction maps above |
860 Note that combining the various boundary and restriction maps above |
852 (for both modules and $n$-categories) |
861 (for both modules and $n$-categories) |
853 we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
862 we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
854 a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
863 a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
871 \label{zzz3} |
880 \label{zzz3} |
872 \end{figure} |
881 \end{figure} |
873 |
882 |
874 First, we can compose two module morphisms to get another module morphism. |
883 First, we can compose two module morphisms to get another module morphism. |
875 |
884 |
876 \mmpar{Module axiom 6}{Module composition} |
885 \begin{module-axiom}[Module composition] |
877 {Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls (with $0\le k\le n$) |
886 {Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls (with $0\le k\le n$) |
878 and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
887 and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
879 Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
888 Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
880 Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
889 Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
881 We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$. |
890 We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$. |
886 \] |
895 \] |
887 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
896 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
888 to the intersection of the boundaries of $M$ and $M_i$. |
897 to the intersection of the boundaries of $M$ and $M_i$. |
889 If $k < n$ we require that $\gl_Y$ is injective. |
898 If $k < n$ we require that $\gl_Y$ is injective. |
890 (For $k=n$, see below.)} |
899 (For $k=n$, see below.)} |
891 |
900 \end{module-axiom} |
892 |
901 |
893 |
902 |
894 Second, we can compose an $n$-category morphism with a module morphism to get another |
903 Second, we can compose an $n$-category morphism with a module morphism to get another |
895 module morphism. |
904 module morphism. |
896 We'll call this the action map to distinguish it from the other kind of composition. |
905 We'll call this the action map to distinguish it from the other kind of composition. |
897 |
906 |
898 \mmpar{Module axiom 7}{$n$-category action} |
907 \begin{module-axiom}[$n$-category action] |
899 {Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
908 {Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
900 $X$ is a plain $k$-ball, |
909 $X$ is a plain $k$-ball, |
901 and $Y = X\cap M'$ is a $k{-}1$-ball. |
910 and $Y = X\cap M'$ is a $k{-}1$-ball. |
902 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
911 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
903 We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$. |
912 We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$. |
908 \] |
917 \] |
909 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
918 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
910 to the intersection of the boundaries of $X$ and $M'$. |
919 to the intersection of the boundaries of $X$ and $M'$. |
911 If $k < n$ we require that $\gl_Y$ is injective. |
920 If $k < n$ we require that $\gl_Y$ is injective. |
912 (For $k=n$, see below.)} |
921 (For $k=n$, see below.)} |
913 |
922 \end{module-axiom} |
914 \mmpar{Module axiom 8}{Strict associativity} |
923 |
|
924 \begin{module-axiom}[Strict associativity] |
915 {The composition and action maps above are strictly associative.} |
925 {The composition and action maps above are strictly associative.} |
|
926 \end{module-axiom} |
916 |
927 |
917 Note that the above associativity axiom applies to mixtures of module composition, |
928 Note that the above associativity axiom applies to mixtures of module composition, |
918 action maps and $n$-category composition. |
929 action maps and $n$-category composition. |
919 See Figure \ref{zzz1b}. |
930 See Figure \ref{zzz1b}. |
920 |
931 |
949 |
960 |
950 (The above operad-like structure is analogous to the swiss cheese operad |
961 (The above operad-like structure is analogous to the swiss cheese operad |
951 \cite{MR1718089}.) |
962 \cite{MR1718089}.) |
952 %\nn{need to double-check that this is true.} |
963 %\nn{need to double-check that this is true.} |
953 |
964 |
954 \mmpar{Module axiom 9}{Product/identity morphisms} |
965 \begin{module-axiom}[Product/identity morphisms] |
955 {Let $M$ be a marked $k$-ball and $D$ be a plain $m$-ball, with $k+m \le n$. |
966 {Let $M$ be a marked $k$-ball and $D$ be a plain $m$-ball, with $k+m \le n$. |
956 Then we have a map $\cM(M)\to \cM(M\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cM(M)$. |
967 Then we have a map $\cM(M)\to \cM(M\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cM(M)$. |
957 If $f:M\to M'$ and $\tilde{f}:M\times D \to M'\times D'$ are maps such that the diagram |
968 If $f:M\to M'$ and $\tilde{f}:M\times D \to M'\times D'$ are maps such that the diagram |
958 \[ \xymatrix{ |
969 \[ \xymatrix{ |
959 M\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & M'\times D' \ar[d]^{\pi} \\ |
970 M\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & M'\times D' \ar[d]^{\pi} \\ |
960 M \ar[r]^{f} & M' |
971 M \ar[r]^{f} & M' |
961 } \] |
972 } \] |
962 commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
973 commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
|
974 \end{module-axiom} |
963 |
975 |
964 \nn{Need to add compatibility with various things, as in the n-cat version of this axiom above.} |
976 \nn{Need to add compatibility with various things, as in the n-cat version of this axiom above.} |
965 |
977 |
966 \nn{postpone finalizing the above axiom until the n-cat version is finalized} |
978 \nn{postpone finalizing the above axiom until the n-cat version is finalized} |
967 |
979 |
968 There are two alternatives for the next axiom, according whether we are defining |
980 There are two alternatives for the next axiom, according whether we are defining |
969 modules for plain $n$-categories or $A_\infty$ $n$-categories. |
981 modules for plain $n$-categories or $A_\infty$ $n$-categories. |
970 In the plain case we require |
982 In the plain case we require |
971 |
983 |
972 \mmpar{Module axiom 10a}{Extended isotopy invariance in dimension $n$} |
984 \begin{module-axiom}[\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$] |
973 {Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
985 {Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
974 to the identity on $\bd M$ and is extended isotopic (rel boundary) to the identity. |
986 to the identity on $\bd M$ and is extended isotopic (rel boundary) to the identity. |
975 Then $f$ acts trivially on $\cM(M)$.} |
987 Then $f$ acts trivially on $\cM(M)$.} |
|
988 \end{module-axiom} |
976 |
989 |
977 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
990 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
978 |
991 |
979 We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
992 We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
980 In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
993 In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
981 on $\bd B \setmin N$. |
994 on $\bd B \setmin N$. |
982 |
995 |
983 For $A_\infty$ modules we require |
996 For $A_\infty$ modules we require |
984 |
997 |
985 \mmpar{Module axiom 10b}{Families of homeomorphisms act} |
998 \addtocounter{module-axiom}{-1} |
|
999 \begin{module-axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act] |
986 {For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
1000 {For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
987 \[ |
1001 \[ |
988 C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
1002 C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
989 \] |
1003 \] |
990 Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |
1004 Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |