18 Let $M^n = Y^k\times F^{n-k}$. |
18 Let $M^n = Y^k\times F^{n-k}$. |
19 Let $C$ be a plain $n$-category. |
19 Let $C$ be a plain $n$-category. |
20 Let $\cF$ be the $A_\infty$ $k$-category which assigns to a $k$-ball |
20 Let $\cF$ be the $A_\infty$ $k$-category which assigns to a $k$-ball |
21 $X$ the old-fashioned blob complex $\bc_*(X\times F)$. |
21 $X$ the old-fashioned blob complex $\bc_*(X\times F)$. |
22 |
22 |
23 \begin{thm} |
23 \begin{thm} \label{product_thm} |
24 The old-fashioned blob complex $\bc_*^C(Y\times F)$ is homotopy equivalent to the |
24 The old-fashioned blob complex $\bc_*^C(Y\times F)$ is homotopy equivalent to the |
25 new-fangled blob complex $\bc_*^\cF(Y)$. |
25 new-fangled blob complex $\bc_*^\cF(Y)$. |
26 \end{thm} |
26 \end{thm} |
27 |
27 |
28 \begin{proof} |
28 \begin{proof} |
29 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
29 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
30 |
30 |
31 First we define a map from $\bc_*^\cF(Y)$ to $\bc_*^C(Y\times F)$. |
31 First we define a map |
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32 \[ |
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33 \psi: \bc_*^\cF(Y) \to \bc_*^C(Y\times F) . |
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34 \] |
32 In filtration degree 0 we just glue together the various blob diagrams on $X\times F$ |
35 In filtration degree 0 we just glue together the various blob diagrams on $X\times F$ |
33 (where $X$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
36 (where $X$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
34 $Y\times F$. |
37 $Y\times F$. |
35 In filtration degrees 1 and higher we define the map to be zero. |
38 In filtration degrees 1 and higher we define the map to be zero. |
36 It is easy to check that this is a chain map. |
39 It is easy to check that this is a chain map. |
37 |
40 |
38 Next we define a map from $\phi: \bc_*^C(Y\times F) \to \bc_*^\cF(Y)$. |
41 Next we define a map |
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42 \[ |
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43 \phi: \bc_*^C(Y\times F) \to \bc_*^\cF(Y) . |
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44 \] |
39 Actually, we will define it on the homotopy equivalent subcomplex |
45 Actually, we will define it on the homotopy equivalent subcomplex |
40 $\cS_* \sub \bc_*^C(Y\times F)$ generated by blob diagrams which are small with |
46 $\cS_* \sub \bc_*^C(Y\times F)$ generated by blob diagrams which are small with |
41 respect to some open cover |
47 respect to some open cover |
42 of $Y\times F$. |
48 of $Y\times F$. |
43 \nn{need reference to small blob lemma} |
49 \nn{need reference to small blob lemma} |
54 %This defines a filtration degree 0 element of $\bc_*^\cF(Y)$ |
60 %This defines a filtration degree 0 element of $\bc_*^\cF(Y)$ |
55 |
61 |
56 We will define $\phi$ using a variant of the method of acyclic models. |
62 We will define $\phi$ using a variant of the method of acyclic models. |
57 Let $a\in \cS_m$ be a blob diagram on $Y\times F$. |
63 Let $a\in \cS_m$ be a blob diagram on $Y\times F$. |
58 For $m$ sufficiently small there exists a decomposition $K$ of $Y$ into $k$-balls such that the |
64 For $m$ sufficiently small there exists a decomposition $K$ of $Y$ into $k$-balls such that the |
59 codimension 1 cells of $K\times F$ miss the blobs of $a$, and more generally such that $a$ is splittable along $K\times F$. |
65 codimension 1 cells of $K\times F$ miss the blobs of $a$, and more generally such that $a$ is splittable along (the codimension-1 part of) $K\times F$. |
60 Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(a, \bar{K})$ |
66 Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(a, \bar{K})$ |
61 such that each $K_i$ has the aforementioned splittable property |
67 such that each $K_i$ has the aforementioned splittable property |
62 (see Subsection \ref{ss:ncat_fields}). |
68 (see Subsection \ref{ss:ncat_fields}). |
63 \nn{need to define $D(a)$ more clearly; also includes $(b_j, \bar{K})$ where |
69 \nn{need to define $D(a)$ more clearly; also includes $(b_j, \bar{K})$ where |
64 $\bd(a) = \sum b_j$.} |
70 $\bd(a) = \sum b_j$.} |
107 Choose a decomposition $M$ which has common refinements with each of |
113 Choose a decomposition $M$ which has common refinements with each of |
108 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
114 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
109 \nn{need to also require that $KLM$ antirefines to $KM$, etc.} |
115 \nn{need to also require that $KLM$ antirefines to $KM$, etc.} |
110 Then we have a filtration degree 2 chain, as shown in Figure \ref{zzz5}, which does the trick. |
116 Then we have a filtration degree 2 chain, as shown in Figure \ref{zzz5}, which does the trick. |
111 (Each small triangle in Figure \ref{zzz5} can be filled with a filtration degree 2 chain.) |
117 (Each small triangle in Figure \ref{zzz5} can be filled with a filtration degree 2 chain.) |
112 For example, .... |
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113 |
118 |
114 \begin{figure}[!ht] |
119 \begin{figure}[!ht] |
115 \begin{equation*} |
120 \begin{equation*} |
116 \mathfig{1.0}{tempkw/zz5} |
121 \mathfig{1.0}{tempkw/zz5} |
117 \end{equation*} |
122 \end{equation*} |
118 \caption{Filling in $K$-$KL$-$L$-$K'L$-$K'$-$K'L'$-$L'$-$KL'$-$K$} |
123 \caption{Filling in $K$-$KL$-$L$-$K'L$-$K'$-$K'L'$-$L'$-$KL'$-$K$} |
119 \label{zzz5} |
124 \label{zzz5} |
120 \end{figure} |
125 \end{figure} |
121 |
126 |
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127 Continuing in this way we see that $D(a)$ is acyclic. |
122 \end{proof} |
128 \end{proof} |
123 |
129 |
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130 We are now in a position to apply the method of acyclic models to get a map |
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131 $\phi:\cS_* \to \bc_*^\cF(Y)$. |
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132 This map is defined in sufficiently low degrees, sends a blob diagram $a$ to $D(a)$, |
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133 and is well-defined up to (iterated) homotopy. |
124 |
134 |
125 \nn{....} |
135 The subcomplex $\cS_* \subset \bc_*^C(Y\times F)$ depends on choice of cover of $Y\times F$. |
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136 If we refine that cover, we get a complex $\cS'_* \subset \cS_*$ |
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137 and a map $\phi':\cS'_* \to \bc_*^\cF(Y)$. |
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138 $\phi'$ is defined only on homological degrees below some bound, but this bound is higher than |
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139 the corresponding bound for $\phi$. |
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140 We must show that $\phi$ and $\phi'$ agree, up to homotopy, |
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141 on the intersection of the subcomplexes on which they are defined. |
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142 This is clear, since the acyclic subcomplexes $D(a)$ above used in the definition of |
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143 $\phi$ and $\phi'$ do not depend on the choice of cover. |
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144 |
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145 \nn{need to say (and justify) that we now have a map $\phi$ indep of choice of cover} |
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146 |
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147 We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity. |
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148 |
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149 $\psi\circ\phi$ is the identity. $\phi$ takes a blob diagram $a$ and chops it into pieces |
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150 according to some decomposition $K$ of $Y$. |
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151 $\psi$ glues those pieces back together, yielding the same $a$ we started with. |
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152 |
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153 $\phi\circ\psi$ is the identity up to homotopy by another MoAM argument... |
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154 |
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155 This concludes the proof of Theorem \ref{product_thm}. |
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156 \nn{at least I think it does; it's pretty rough at this point.} |
126 \end{proof} |
157 \end{proof} |
127 |
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128 |
158 |
129 \nn{need to say something about dim $< n$ above} |
159 \nn{need to say something about dim $< n$ above} |
130 |
160 |
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161 \medskip |
131 |
162 |
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163 \begin{cor} |
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164 The new-fangled and old-fashioned blob complexes are homotopic. |
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165 \end{cor} |
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166 \begin{proof} |
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167 Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point. |
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168 \end{proof} |
132 |
169 |
133 \medskip |
170 \medskip |
134 \hrule |
171 \hrule |
135 \medskip |
172 \medskip |
136 |
173 |