819 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
819 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
820 \end{example} |
820 \end{example} |
821 |
821 |
822 This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
822 This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
823 Notice that with $F$ a point, the above example is a construction turning a topological |
823 Notice that with $F$ a point, the above example is a construction turning a topological |
824 $n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. |
824 $n$-category $\cC$ into an $A_\infty$ $n$-category. |
825 \nn{do we use this notation elsewhere (anymore)?} |
|
826 We think of this as providing a ``free resolution" |
825 We think of this as providing a ``free resolution" |
827 of the topological $n$-category. |
826 of the topological $n$-category. |
828 \nn{say something about cofibrant replacements?} |
827 \nn{say something about cofibrant replacements?} |
829 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: |
830 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 $0$, |
1023 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
1022 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
1024 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
1023 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
1025 |
1024 |
1026 In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit |
1025 In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit |
1027 is more involved. |
1026 is more involved. |
1028 \nn{should change to less strange terminology: ``filtration" to ``simplex" |
|
1029 (search for all occurrences of ``filtration")} |
|
1030 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
1027 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
1031 Such sequences (for all $m$) form a simplicial set in $\cell(W)$. |
1028 Such sequences (for all $m$) form a simplicial set in $\cell(W)$. |
1032 Define $\cl{\cC}(W)$ as a vector space via |
1029 Define $\cl{\cC}(W)$ as a vector space via |
1033 \[ |
1030 \[ |
1034 \cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
1031 \cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
1035 \] |
1032 \] |
1036 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. |
1033 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. |
1037 (Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$, |
|
1038 the complex $U[m]$ is concentrated in degree $m$.) |
|
1039 \nn{if there is a std convention, should we use it? or are we deliberately bucking tradition?} |
|
1040 We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
1034 We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
1041 summands plus another term using the differential of the simplicial set of $m$-sequences. |
1035 summands plus another term using the differential of the simplicial set of $m$-sequences. |
1042 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
1036 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
1043 summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
1037 summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
1044 \[ |
1038 \[ |
1049 %\nn{need to say this better} |
1043 %\nn{need to say this better} |
1050 %\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
1044 %\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
1051 %combine only two balls at a time; for $n=1$ this version will lead to usual definition |
1045 %combine only two balls at a time; for $n=1$ this version will lead to usual definition |
1052 %of $A_\infty$ category} |
1046 %of $A_\infty$ category} |
1053 |
1047 |
1054 We will call $m$ the filtration degree of the complex. |
1048 We will call $m$ the simplex degree of the complex. |
1055 \nn{is there a more standard term for this?} |
|
1056 We can think of this construction as starting with a disjoint copy of a complex for each |
1049 We can think of this construction as starting with a disjoint copy of a complex for each |
1057 permissible decomposition (filtration degree 0). |
1050 permissible decomposition (simplex degree 0). |
1058 Then we glue these together with mapping cylinders coming from gluing maps |
1051 Then we glue these together with mapping cylinders coming from gluing maps |
1059 (filtration degree 1). |
1052 (simplex degree 1). |
1060 Then we kill the extra homology we just introduced with mapping |
1053 Then we kill the extra homology we just introduced with mapping |
1061 cylinders between the mapping cylinders (filtration degree 2), and so on. |
1054 cylinders between the mapping cylinders (simplex degree 2), and so on. |
1062 |
1055 |
1063 $\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1056 $\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1064 |
1057 |
1065 It is easy to see that |
1058 It is easy to see that |
1066 there are well-defined maps $\cl{\cC}(W)\to\cl{\cC}(\bd W)$, and that these maps |
1059 there are well-defined maps $\cl{\cC}(W)\to\cl{\cC}(\bd W)$, and that these maps |
2218 |
2211 |
2219 The second movie move replaces two successive pushes in the same direction, |
2212 The second movie move replaces two successive pushes in the same direction, |
2220 across $B_1$ and $B_2$, say, with a single push across $B_1\cup B_2$. |
2213 across $B_1$ and $B_2$, say, with a single push across $B_1\cup B_2$. |
2221 (See Figure \ref{jun23d}.) |
2214 (See Figure \ref{jun23d}.) |
2222 \begin{figure}[t] |
2215 \begin{figure}[t] |
2223 \begin{equation*} |
2216 \begin{tikzpicture} |
2224 \mathfig{.9}{tempkw/jun23d} |
2217 \node(L) { |
2225 \end{equation*} |
2218 \scalebox{0.5}{ |
|
2219 \begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=1.5cm] |
|
2220 \draw[red] (0.75,0) -- +(2,0); |
|
2221 \draw[red] (0,0) node(R) {} |
|
2222 -- (0.75,0) node[below] {} |
|
2223 --(1.5,0) node[circle,fill=black,inner sep=2pt] {}; |
|
2224 \draw[fill] (150:1.5) circle (2pt) node[above=4pt] {}; |
|
2225 \draw (1.5,0) arc (0:149:1.5); |
|
2226 \draw[red] |
|
2227 (R) node[circle,fill=black,inner sep=2pt] {} |
|
2228 arc (-45:-135:3) node[circle,fill=black,inner sep=2pt] {}; |
|
2229 \draw[red] (-5.5,0) -- (-4.2,0); |
|
2230 \draw (R) arc (45:75:3); |
|
2231 \draw (150:1.5) arc (74:135:3); |
|
2232 \node at (-2,0) {\scalebox{2.0}{$B_1$}}; |
|
2233 \node at (0.2,0.8) {\scalebox{2.0}{$B_2$}}; |
|
2234 \node at (-4,1.2) {\scalebox{2.0}{$A$}}; |
|
2235 \node at (-4,-1.2) {\scalebox{2.0}{$C$}}; |
|
2236 \node[red] at (2.53,0.35) {\scalebox{2.0}{$E$}}; |
|
2237 \end{tikzpicture} |
|
2238 } |
|
2239 }; |
|
2240 \node(M) at (5,4) { |
|
2241 \scalebox{0.5}{ |
|
2242 \begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=1.5cm] |
|
2243 \draw[red] (0.75,0) -- +(2,0); |
|
2244 \draw[red] (0,0) node(R) {} |
|
2245 -- (0.75,0) node[below] {} |
|
2246 --(1.5,0) node[circle,fill=black,inner sep=2pt] {}; |
|
2247 \draw[fill] (150:1.5) circle (2pt) node[above=4pt] {}; |
|
2248 \draw(1.5,0) arc (0:149:1.5); |
|
2249 \draw |
|
2250 (R) node[circle,fill=black,inner sep=2pt] {} |
|
2251 arc (-45:-135:3) node[circle,fill=black,inner sep=2pt] {}; |
|
2252 \draw[red] (-5.5,0) -- (-4.2,0); |
|
2253 \draw[red] (R) arc (45:75:3); |
|
2254 \draw[red] (150:1.5) arc (74:135:3); |
|
2255 \node at (-2,0) {\scalebox{2.0}{$B_1$}}; |
|
2256 \node at (0.2,0.8) {\scalebox{2.0}{$B_2$}}; |
|
2257 \node at (-4,1.2) {\scalebox{2.0}{$A$}}; |
|
2258 \node at (-4,-1.2) {\scalebox{2.0}{$C$}}; |
|
2259 \node[red] at (2.53,0.35) {\scalebox{2.0}{$E$}}; |
|
2260 \end{tikzpicture} |
|
2261 } |
|
2262 }; |
|
2263 \node(R) at (10,0) { |
|
2264 \scalebox{0.5}{ |
|
2265 \begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=1.5cm] |
|
2266 \draw[red] (0.75,0) -- +(2,0); |
|
2267 \draw (0,0) node(R) {} |
|
2268 -- (0.75,0) node[below] {} |
|
2269 --(1.5,0) node[circle,fill=black,inner sep=2pt] {}; |
|
2270 \draw[fill] (150:1.5) circle (2pt) node[above=4pt] {}; |
|
2271 \draw[red] (1.5,0) arc (0:149:1.5); |
|
2272 \draw |
|
2273 (R) node[circle,fill=black,inner sep=2pt] {} |
|
2274 arc (-45:-135:3) node[circle,fill=black,inner sep=2pt] {}; |
|
2275 \draw[red] (-5.5,0) -- (-4.2,0); |
|
2276 \draw (R) arc (45:75:3); |
|
2277 \draw[red] (150:1.5) arc (74:135:3); |
|
2278 \node at (-2,0) {\scalebox{2.0}{$B_1$}}; |
|
2279 \node at (0.2,0.8) {\scalebox{2.0}{$B_2$}}; |
|
2280 \node at (-4,1.2) {\scalebox{2.0}{$A$}}; |
|
2281 \node at (-4,-1.2) {\scalebox{2.0}{$C$}}; |
|
2282 \node[red] at (2.53,0.35) {\scalebox{2.0}{$E$}}; |
|
2283 \end{tikzpicture} |
|
2284 } |
|
2285 }; |
|
2286 \draw[->] (L) to[out=90,in=225] node[sloped, above] {push $B_1$} (M); |
|
2287 \draw[->] (M) to[out=-45,in=90] node[sloped, above] {push $B_2$} (R); |
|
2288 \draw[->] (L) to[out=-35,in=-145] node[sloped, below] {push $B_1 \cup B_2$} (R); |
|
2289 \end{tikzpicture} |
2226 \caption{A movie move} |
2290 \caption{A movie move} |
2227 \label{jun23d} |
2291 \label{jun23d} |
2228 \end{figure} |
2292 \end{figure} |
2229 Invariance under this movie move follows from the compatibility of the inner |
2293 Invariance under this movie move follows from the compatibility of the inner |
2230 product for $B_1\cup B_2$ with the inner products for $B_1$ and $B_2$. |
2294 product for $B_1\cup B_2$ with the inner products for $B_1$ and $B_2$. |