827 %\nn{say something about cofibrant replacements?} |
827 %\nn{say something about cofibrant replacements?} |
828 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
828 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
829 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
829 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
830 and take $\CD{B}$ to act trivially. |
830 and take $\CD{B}$ to act trivially. |
831 |
831 |
832 Be careful that the ``free resolution" of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
832 Beware that the ``free resolution" of the topological $n$-category $\pi_{\leq n}(T)$ |
|
833 is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
833 It's easy to see that with $n=0$, the corresponding system of fields is just |
834 It's easy to see that with $n=0$, the corresponding system of fields is just |
834 linear combinations of connected components of $T$, and the local relations are trivial. |
835 linear combinations of connected components of $T$, and the local relations are trivial. |
835 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
836 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
836 |
837 |
837 \begin{example}[The bordism $n$-category, $A_\infty$ version] |
838 \begin{example}[The bordism $n$-category, $A_\infty$ version] |
901 \end{example} |
902 \end{example} |
902 |
903 |
903 |
904 |
904 \subsection{From balls to manifolds} |
905 \subsection{From balls to manifolds} |
905 \label{ss:ncat_fields} \label{ss:ncat-coend} |
906 \label{ss:ncat_fields} \label{ss:ncat-coend} |
906 In this section we describe how to extend an $n$-category $\cC$ as described above |
907 In this section we show how to extend an $n$-category $\cC$ as described above |
907 (of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
908 (of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
908 This extension is a certain colimit, and we've chosen the notation to remind you of this. |
909 This extension is a certain colimit, and the arrow in the notation is intended as a reminder of this. |
909 Thus we show that functors $\cC_k$ satisfying the axioms above have a canonical extension |
910 |
910 from $k$-balls to arbitrary $k$-manifolds. |
|
911 Recall that we've already anticipated this construction in the previous section, |
|
912 inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
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913 so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
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914 In the case of plain $n$-categories, this construction factors into a construction of a |
911 In the case of plain $n$-categories, this construction factors into a construction of a |
915 system of fields and local relations, followed by the usual TQFT definition of a |
912 system of fields and local relations, followed by the usual TQFT definition of a |
916 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
913 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
917 For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
914 For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
918 Recall that we can take a plain $n$-category $\cC$ and pass to the ``free resolution", |
915 Recall that we can take a plain $n$-category $\cC$ and pass to the ``free resolution", |
919 an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls |
916 an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls |
920 (recall Example \ref{ex:blob-complexes-of-balls} above). |
917 (recall Example \ref{ex:blob-complexes-of-balls} above). |
921 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
918 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
922 for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the |
919 for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the |
923 same as the original blob complex for $M$ with coefficients in $\cC$. |
920 same as the original blob complex for $M$ with coefficients in $\cC$. |
|
921 |
|
922 Recall that we've already anticipated this construction in the previous section, |
|
923 inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
|
924 so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
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925 |
|
926 \medskip |
924 |
927 |
925 We will first define the ``decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
928 We will first define the ``decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
926 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
929 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
927 and we will define $\cl{\cC}(W)$ as a suitable colimit |
930 and we will define $\cl{\cC}(W)$ as a suitable colimit |
928 (or homotopy colimit in the $A_\infty$ case) of this functor. |
931 (or homotopy colimit in the $A_\infty$ case) of this functor. |
1083 One can show that the usual hocolimit and the local hocolimit are homotopy equivalent using an |
1086 One can show that the usual hocolimit and the local hocolimit are homotopy equivalent using an |
1084 Eilenberg-Zilber type subdivision argument. |
1087 Eilenberg-Zilber type subdivision argument. |
1085 |
1088 |
1086 \medskip |
1089 \medskip |
1087 |
1090 |
1088 $\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1091 $\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
|
1092 Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1089 |
1093 |
1090 It is easy to see that |
1094 It is easy to see that |
1091 there are well-defined maps $\cl{\cC}(W)\to\cl{\cC}(\bd W)$, and that these maps |
1095 there are well-defined maps $\cl{\cC}(W)\to\cl{\cC}(\bd W)$, and that these maps |
1092 comprise a natural transformation of functors. |
1096 comprise a natural transformation of functors. |
1093 |
1097 |
1139 But $a'_0$ and $a'_k$ are both elements of $\psi(x'_0)$ (because $x'_k = x'_0$). |
1143 But $a'_0$ and $a'_k$ are both elements of $\psi(x'_0)$ (because $x'_k = x'_0$). |
1140 So by the injectivity clause of the composition axiom, we must have that $a'_0 = a'_k$. |
1144 So by the injectivity clause of the composition axiom, we must have that $a'_0 = a'_k$. |
1141 But this implies that $a = a_0 = a_k = \hat{a}$, contrary to our assumption that $a\ne \hat{a}$. |
1145 But this implies that $a = a_0 = a_k = \hat{a}$, contrary to our assumption that $a\ne \hat{a}$. |
1142 \end{proof} |
1146 \end{proof} |
1143 |
1147 |
1144 \nn{need to finish explaining why we have a system of fields; |
1148 %\nn{need to finish explaining why we have a system of fields; |
1145 define $k$-cat $\cC(\cdot\times W)$} |
1149 %define $k$-cat $\cC(\cdot\times W)$} |
1146 |
1150 |
1147 \subsection{Modules} |
1151 \subsection{Modules} |
1148 |
1152 |
1149 Next we define plain and $A_\infty$ $n$-category modules. |
1153 Next we define plain and $A_\infty$ $n$-category modules. |
1150 The definition will be very similar to that of $n$-categories, |
1154 The definition will be very similar to that of $n$-categories, |
2225 f: \cS(X; c; E) \to \cS(X'; f(c); f(E)) . |
2229 f: \cS(X; c; E) \to \cS(X'; f(c); f(E)) . |
2226 \] |
2230 \] |
2227 It is easy to show that this is independent of the choice of $E$. |
2231 It is easy to show that this is independent of the choice of $E$. |
2228 Note also that this map depends only on the restriction of $f$ to $\bd X$. |
2232 Note also that this map depends only on the restriction of $f$ to $\bd X$. |
2229 In particular, if $F: X\to X$ is the identity on $\bd X$ then $f$ acts trivially, as required by |
2233 In particular, if $F: X\to X$ is the identity on $\bd X$ then $f$ acts trivially, as required by |
2230 Axiom \ref{axiom:extended-isotopies} of \S\ref{ss:n-cat-def}. |
2234 Axiom \ref{axiom:extended-isotopies}. |
2231 |
2235 |
2232 We define product $n{+}1$-morphisms to be identity maps of modules. |
2236 We define product $n{+}1$-morphisms to be identity maps of modules. |
2233 |
2237 |
2234 To define (binary) composition of $n{+}1$-morphisms, choose the obvious common equator |
2238 To define (binary) composition of $n{+}1$-morphisms, choose the obvious common equator |
2235 then compose the module maps. |
2239 then compose the module maps. |