150 |
150 |
151 We have just argued that the boundary of a morphism has no preferred splitting into |
151 We have just argued that the boundary of a morphism has no preferred splitting into |
152 domain and range, but the converse meets with our approval. |
152 domain and range, but the converse meets with our approval. |
153 That is, given compatible domain and range, we should be able to combine them into |
153 That is, given compatible domain and range, we should be able to combine them into |
154 the full boundary of a morphism. |
154 the full boundary of a morphism. |
155 The following lemma follows from the colimit construction used to define $\cl{\cC}_{k-1}$ |
155 The following lemma will follow from the colimit construction used to define $\cl{\cC}_{k-1}$ |
156 on spheres. |
156 on spheres. |
157 |
157 |
158 \begin{lem}[Boundary from domain and range] |
158 \begin{lem}[Boundary from domain and range] |
159 \label{lem:domain-and-range} |
159 \label{lem:domain-and-range} |
160 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
160 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
161 $B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
161 $B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
162 Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
162 Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
163 two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$. |
163 two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$. |
164 Then we have an injective map |
164 Then we have an injective map |
165 \[ |
165 \[ |
166 \gl_E : \cC(B_1) \times_{\\cl{cC}(E)} \cC(B_2) \into \cl{\cC}(S) |
166 \gl_E : \cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2) \into \cl{\cC}(S) |
167 \] |
167 \] |
168 which is natural with respect to the actions of homeomorphisms. |
168 which is natural with respect to the actions of homeomorphisms. |
169 (When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product |
169 (When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product |
170 becomes a normal product.) |
170 becomes a normal product.) |
171 \end{lem} |
171 \end{lem} |
882 \subsection{From balls to manifolds} |
882 \subsection{From balls to manifolds} |
883 \label{ss:ncat_fields} \label{ss:ncat-coend} |
883 \label{ss:ncat_fields} \label{ss:ncat-coend} |
884 In this section we describe how to extend an $n$-category $\cC$ as described above |
884 In this section we describe how to extend an $n$-category $\cC$ as described above |
885 (of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
885 (of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
886 This extension is a certain colimit, and we've chosen the notation to remind you of this. |
886 This extension is a certain colimit, and we've chosen the notation to remind you of this. |
887 That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension |
887 Thus we show that functors $\cC_k$ satisfying the axioms above have a canonical extension |
888 from $k$-balls to arbitrary $k$-manifolds. |
888 from $k$-balls to arbitrary $k$-manifolds. |
889 Recall that we've already anticipated this construction in the previous section, |
889 Recall that we've already anticipated this construction in the previous section, |
890 inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
890 inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
891 so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
891 so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
892 In the case of plain $n$-categories, this construction factors into a construction of a |
892 In the case of plain $n$-categories, this construction factors into a construction of a |
960 (i.e. fix an element of the colimit associated to $\bd W$). |
959 (i.e. fix an element of the colimit associated to $\bd W$). |
961 |
960 |
962 Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
961 Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
963 |
962 |
964 \begin{defn}[System of fields functor] |
963 \begin{defn}[System of fields functor] |
965 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cC(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
964 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
966 That is, for each decomposition $x$ there is a map |
965 That is, for each decomposition $x$ there is a map |
967 $\psi_{\cC;W}(x)\to \cC(W)$, these maps are compatible with the refinement maps |
966 $\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps |
968 above, and $\cC(W)$ is universal with respect to these properties. |
967 above, and $\cl{\cC}(W)$ is universal with respect to these properties. |
969 \end{defn} |
968 \end{defn} |
970 |
969 |
971 \begin{defn}[System of fields functor, $A_\infty$ case] |
970 \begin{defn}[System of fields functor, $A_\infty$ case] |
972 When $\cC$ is an $A_\infty$ $n$-category, $\cC(W)$ for $W$ a $k$-manifold with $k < n$ |
971 When $\cC$ is an $A_\infty$ $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$ |
973 is defined as above, as the colimit of $\psi_{\cC;W}$. |
972 is defined as above, as the colimit of $\psi_{\cC;W}$. |
974 When $W$ is an $n$-manifold, the chain complex $\cC(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
973 When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
975 \end{defn} |
974 \end{defn} |
976 |
975 |
977 We can specify boundary data $c \in \cC(\bdy W)$, and define functors $\psi_{\cC;W,c}$ |
976 We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$ |
978 with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
977 with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
979 |
978 |
980 We now give a more concrete description of the colimit in each case. |
979 We now give a more concrete description of the colimit in each case. |
981 If $\cC$ is enriched over vector spaces, and $W$ is an $n$-manifold, |
980 If $\cC$ is enriched over vector spaces, and $W$ is an $n$-manifold, |
982 we can take the vector space $\cC(W,c)$ to be the direct sum over all permissible decompositions of $W$ |
981 we can take the vector space $\cl{\cC}(W,c)$ to be the direct sum over all permissible decompositions of $W$ |
983 \begin{equation*} |
982 \begin{equation*} |
984 \cC(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K |
983 \cl{\cC}(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K |
985 \end{equation*} |
984 \end{equation*} |
986 where $K$ is the vector space spanned by elements $a - g(a)$, with |
985 where $K$ is the vector space spanned by elements $a - g(a)$, with |
987 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
986 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
988 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
987 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
989 |
988 |
990 In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit |
989 In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit |
991 is more involved. |
990 is more involved. |
992 %\nn{should probably rewrite this to be compatible with some standard reference} |
991 %\nn{should probably rewrite this to be compatible with some standard reference} |
993 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
992 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
994 Such sequences (for all $m$) form a simplicial set in $\cell(W)$. |
993 Such sequences (for all $m$) form a simplicial set in $\cell(W)$. |
995 Define $V$ as a vector space via |
994 Define $\cl{\cC}(W)$ as a vector space via |
996 \[ |
995 \[ |
997 V = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
996 \cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
998 \] |
997 \] |
999 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. |
998 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. |
1000 (Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$, |
999 (Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$, |
1001 the complex $U[m]$ is concentrated in degree $m$.) |
1000 the complex $U[m]$ is concentrated in degree $m$.) |
1002 We endow $V$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
1001 We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
1003 summands plus another term using the differential of the simplicial set of $m$-sequences. |
1002 summands plus another term using the differential of the simplicial set of $m$-sequences. |
1004 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
1003 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
1005 summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
1004 summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
1006 \[ |
1005 \[ |
1007 \bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) , |
1006 \bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) , |
1008 \] |
1007 \] |
1009 where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
1008 where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
1010 is the usual gluing map coming from the antirefinement $x_0 \le x_1$. |
1009 is the usual gluing map coming from the antirefinement $x_0 \le x_1$. |
1019 Then we glue these together with mapping cylinders coming from gluing maps |
1018 Then we glue these together with mapping cylinders coming from gluing maps |
1020 (filtration degree 1). |
1019 (filtration degree 1). |
1021 Then we kill the extra homology we just introduced with mapping |
1020 Then we kill the extra homology we just introduced with mapping |
1022 cylinders between the mapping cylinders (filtration degree 2), and so on. |
1021 cylinders between the mapping cylinders (filtration degree 2), and so on. |
1023 |
1022 |
1024 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
1023 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1025 |
1024 |
1026 It is easy to see that |
1025 \todo{This next sentence is circular: these maps are an axiom, not a consequence of anything. -S} It is easy to see that |
1027 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
1026 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
1028 comprise a natural transformation of functors. |
1027 comprise a natural transformation of functors. |
|
1028 |
|
1029 \todo{Explicitly say somewhere: `this proves Lemma \ref{lem:domain-and-range}'} |
1029 |
1030 |
1030 \nn{need to finish explaining why we have a system of fields; |
1031 \nn{need to finish explaining why we have a system of fields; |
1031 need to say more about ``homological" fields? |
1032 need to say more about ``homological" fields? |
1032 (actions of homeomorphisms); |
1033 (actions of homeomorphisms); |
1033 define $k$-cat $\cC(\cdot\times W)$} |
1034 define $k$-cat $\cC(\cdot\times W)$} |