18 \subsection{The small blob complex} |
18 \subsection{The small blob complex} |
19 |
19 |
20 \input{text/smallblobs} |
20 \input{text/smallblobs} |
21 |
21 |
22 \subsection{A product formula} |
22 \subsection{A product formula} |
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23 \label{ss:product-formula} |
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24 |
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25 \noop{ |
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26 Let $Y$ be a $k$-manifold, $F$ be an $n{-}k$-manifold, and |
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27 \[ |
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28 E = Y\times F . |
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29 \] |
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30 Let $\cC$ be an $n$-category. |
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31 Let $\cF$ be the $k$-category of Example \ref{ex:blob-complexes-of-balls}, |
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32 \[ |
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33 \cF(X) = \cC(X\times F) |
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34 \] |
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35 for $X$ an $m$-ball with $m\le k$. |
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36 } |
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37 |
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38 \nn{need to settle on notation; proof and statement are inconsistent} |
23 |
39 |
24 \begin{thm} \label{product_thm} |
40 \begin{thm} \label{product_thm} |
25 Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $C^{\times F}$ defined by |
41 Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $C^{\times F}$ defined by |
26 \begin{equation*} |
42 \begin{equation*} |
27 C^{\times F}(B) = \cB_*(B \times F, C). |
43 C^{\times F}(B) = \cB_*(B \times F, C). |
28 \end{equation*} |
44 \end{equation*} |
29 Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned' blob complex for $Y \times F$ with coefficients in $C$ and the `new-fangled' (i.e. homotopy colimit) blob complex for $Y$ with coefficients in $C^{\times F}$: |
45 Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned' blob complex for $Y \times F$ with coefficients in $C$ and the `new-fangled' (i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $C^{\times F}$: |
30 \begin{align*} |
46 \begin{align*} |
31 \cB_*(Y \times F, C) & \htpy \cB_*(Y, C^{\times F}) |
47 \cB_*(Y \times F, C) & \htpy \cB_*(Y, C^{\times F}) |
32 \end{align*} |
48 \end{align*} |
33 \end{thm} |
49 \end{thm} |
34 |
50 |
35 \nn{To do: remark on the case of a nontrivial fiber bundle. |
51 |
36 I can think of two approaches. |
52 \begin{proof}%[Proof of Theorem \ref{product_thm}] |
37 In the first (slick but maybe a little too tautological), we generalize the |
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38 notion of an $n$-category to an $n$-category {\it over a space $B$}. |
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39 (Should be able to find precedent for this in a paper of PT. |
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40 This idea came up in a conversation with him, so maybe should site him.) |
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41 In this generalization, we replace the categories of balls with the categories |
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42 of balls equipped with maps to $B$. |
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43 A fiber bundle $F\to E\to B$ gives an example of such an $n$-category: |
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44 assign to $p:D\to B$ the blob complex $\bc_*(p^*(E))$. |
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45 We can do the colimit thing over $B$ with coefficients in a n-cat-over-B. |
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46 The proof below works essentially unchanged in this case to show that the colimit is the blob complex of the total space $E$. |
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47 } |
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48 |
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49 \nn{The second approach: Choose a decomposition $B = \cup X_i$ |
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50 such that the restriction of $E$ to $X_i$ is a product $F\times X_i$. |
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51 Choose the product structure as well. |
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52 To each codim-1 face $D_i\cap D_j$ we have a bimodule ($S^0$-module). |
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53 And more generally to each codim-$j$ face we have an $S^{j-1}$-module. |
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54 Decorate the decomposition with these modules and do the colimit. |
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55 } |
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56 |
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57 \nn{There is a version of this last construction for arbitrary maps $E \to B$ |
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58 (not necessarily a fibration).} |
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59 |
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60 |
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61 \begin{proof}[Proof of Theorem \ref{product_thm}] |
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62 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
53 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
63 |
54 |
64 First we define a map |
55 First we define a map |
65 \[ |
56 \[ |
66 \psi: \bc_*^\cF(Y) \to \bc_*^C(Y\times F) . |
57 \psi: \bc_*^\cF(Y) \to \bc_*^C(Y\times F) . |
76 \phi: \bc_*^C(Y\times F) \to \bc_*^\cF(Y) . |
67 \phi: \bc_*^C(Y\times F) \to \bc_*^\cF(Y) . |
77 \] |
68 \] |
78 Actually, we will define it on the homotopy equivalent subcomplex |
69 Actually, we will define it on the homotopy equivalent subcomplex |
79 $\cS_* \sub \bc_*^C(Y\times F)$ generated by blob diagrams which are small with |
70 $\cS_* \sub \bc_*^C(Y\times F)$ generated by blob diagrams which are small with |
80 respect to some open cover |
71 respect to some open cover |
81 of $Y\times F$. |
72 of $Y\times F$ |
82 \nn{need reference to small blob lemma} |
73 (Proposition \ref{thm:small-blobs}). |
83 We will have to show eventually that this is independent (up to homotopy) of the choice of cover. |
74 We will have to show eventually that this is independent (up to homotopy) of the choice of cover. |
84 Also, for a fixed choice of cover we will only be able to define the map for blob degree less than |
75 Also, for a fixed choice of cover we will only be able to define the map for blob degree less than |
85 some bound, but this bound goes to infinity as the cover become finer. |
76 some bound, but this bound goes to infinity as the cover become finer. |
86 |
77 |
87 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
78 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
94 |
85 |
95 We will define $\phi$ using a variant of the method of acyclic models. |
86 We will define $\phi$ using a variant of the method of acyclic models. |
96 Let $a\in \cS_m$ be a blob diagram on $Y\times F$. |
87 Let $a\in \cS_m$ be a blob diagram on $Y\times F$. |
97 For $m$ sufficiently small there exists a decomposition $K$ of $Y$ into $k$-balls such that the |
88 For $m$ sufficiently small there exists a decomposition $K$ of $Y$ into $k$-balls such that the |
98 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) $K\times F$. |
89 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) $K\times F$. |
99 Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(a, \bar{K})$ |
90 Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(a, \ol{K})$ |
100 such that each $K_i$ has the aforementioned splittable property |
91 such that each $K_i$ has the aforementioned splittable property. |
101 (see Subsection \ref{ss:ncat_fields}). |
92 (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; |
102 \nn{need to define $D(a)$ more clearly; also includes $(b_j, \bar{K})$ where |
93 see Subsection \ref{ss:ncat_fields}.) |
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94 \nn{need to define $D(a)$ more clearly; also includes $(b_j, \ol{K})$ where |
103 $\bd(a) = \sum b_j$.} |
95 $\bd(a) = \sum b_j$.} |
104 (By $(a, \bar{K})$ we really mean $(a^\sharp, \bar{K})$, where $a^\sharp$ is |
96 (By $(a, \ol{K})$ we really mean $(a^\sharp, \ol{K})$, where $a^\sharp$ is |
105 $a$ split according to $K_0\times F$. |
97 $a$ split according to $K_0\times F$. |
106 To simplify notation we will just write plain $a$ instead of $a^\sharp$.) |
98 To simplify notation we will just write plain $a$ instead of $a^\sharp$.) |
107 Roughly speaking, $D(a)$ consists of filtration degree 0 stuff which glues up to give |
99 Roughly speaking, $D(a)$ consists of filtration degree 0 stuff which glues up to give |
108 $a$, filtration degree 1 stuff which makes all of the filtration degree 0 stuff homologous, |
100 $a$, filtration degree 1 stuff which makes all of the filtration degree 0 stuff homologous, |
109 filtration degree 2 stuff which kills the homology created by the |
101 filtration degree 2 stuff which kills the homology created by the |
218 |
210 |
219 \nn{need to say (and justify) that we now have a map $\phi$ indep of choice of cover} |
211 \nn{need to say (and justify) that we now have a map $\phi$ indep of choice of cover} |
220 |
212 |
221 We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity. |
213 We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity. |
222 |
214 |
223 $\psi\circ\phi$ is the identity. $\phi$ takes a blob diagram $a$ and chops it into pieces |
215 $\psi\circ\phi$ is the identity on the nose. |
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216 $\phi$ takes a blob diagram $a$ and chops it into pieces |
224 according to some decomposition $K$ of $Y$. |
217 according to some decomposition $K$ of $Y$. |
225 $\psi$ glues those pieces back together, yielding the same $a$ we started with. |
218 $\psi$ glues those pieces back together, yielding the same $a$ we started with. |
226 |
219 |
227 $\phi\circ\psi$ is the identity up to homotopy by another MoAM argument... |
220 $\phi\circ\psi$ is the identity up to homotopy by another MoAM argument... |
228 |
221 |
241 \begin{proof} |
234 \begin{proof} |
242 Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point. |
235 Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point. |
243 \end{proof} |
236 \end{proof} |
244 |
237 |
245 \medskip |
238 \medskip |
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239 |
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240 \nn{To do: remark on the case of a nontrivial fiber bundle. |
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241 I can think of two approaches. |
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242 In the first (slick but maybe a little too tautological), we generalize the |
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243 notion of an $n$-category to an $n$-category {\it over a space $B$}. |
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244 (Should be able to find precedent for this in a paper of PT. |
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245 This idea came up in a conversation with him, so maybe should site him.) |
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246 In this generalization, we replace the categories of balls with the categories |
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247 of balls equipped with maps to $B$. |
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248 A fiber bundle $F\to E\to B$ gives an example of such an $n$-category: |
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249 assign to $p:D\to B$ the blob complex $\bc_*(p^*(E))$. |
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250 We can do the colimit thing over $B$ with coefficients in a n-cat-over-B. |
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251 The proof below works essentially unchanged in this case to show that the colimit is the blob complex of the total space $E$. |
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252 } |
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253 |
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254 \nn{The second approach: Choose a decomposition $B = \cup X_i$ |
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255 such that the restriction of $E$ to $X_i$ is a product $F\times X_i$. |
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256 Choose the product structure as well. |
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257 To each codim-1 face $D_i\cap D_j$ we have a bimodule ($S^0$-module). |
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258 And more generally to each codim-$j$ face we have an $S^{j-1}$-module. |
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259 Decorate the decomposition with these modules and do the colimit. |
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260 } |
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261 |
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262 \nn{There is a version of this last construction for arbitrary maps $E \to B$ |
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263 (not necessarily a fibration).} |
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264 |
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265 |
246 |
266 |
247 \subsection{A gluing theorem} |
267 \subsection{A gluing theorem} |
248 \label{sec:gluing} |
268 \label{sec:gluing} |
249 |
269 |
250 Next we prove a gluing theorem. |
270 Next we prove a gluing theorem. |