5 Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the anticlimactically tautological definition of the blob |
5 Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the anticlimactically tautological definition of the blob |
6 complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}. |
6 complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}. |
7 |
7 |
8 We will show below |
8 We will show below |
9 in Corollary \ref{cor:new-old} |
9 in Corollary \ref{cor:new-old} |
10 that when $\cC$ is obtained from a topological $n$-category $\cD$ as the blob complex of a point, this agrees (up to homotopy) with our original definition of the blob complex |
10 that when $\cC$ is obtained from a system of fields $\cD$ |
11 for $\cD$. |
11 as the blob complex of an $n$-ball (see Example \ref{ex:blob-complexes-of-balls}), |
12 |
12 $\cl{\cC}(M)$ is homotopy equivalent to |
13 An important technical tool in the proofs of this section is provided by the idea of `small blobs'. |
13 our original definition of the blob complex $\bc_*^\cD(M)$. |
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14 |
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15 \medskip |
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16 |
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17 An important technical tool in the proofs of this section is provided by the idea of ``small blobs". |
14 Fix $\cU$, an open cover of $M$. |
18 Fix $\cU$, an open cover of $M$. |
15 Define the `small blob complex' $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$, and moreover each field labeling a region cut out by the blobs is splittable into fields on smaller regions, each of which is contained in some open set. |
19 Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$, and moreover each field labeling a region cut out by the blobs is splittable into fields on smaller regions, each of which is contained in some open set of $\cU$. |
16 |
20 |
17 \begin{thm}[Small blobs] \label{thm:small-blobs} |
21 \begin{thm}[Small blobs] \label{thm:small-blobs} |
18 The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
22 The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
19 \end{thm} |
23 \end{thm} |
20 The proof appears in \S \ref{appendix:small-blobs}. |
24 The proof appears in \S \ref{appendix:small-blobs}. |
21 |
25 |
22 \subsection{A product formula} |
26 \subsection{A product formula} |
23 \label{ss:product-formula} |
27 \label{ss:product-formula} |
24 |
28 |
25 \noop{ |
29 |
26 Let $Y$ be a $k$-manifold, $F$ be an $n{-}k$-manifold, and |
30 Given a system of fields $\cE$ and a $n{-}k$-manifold $F$, recall from |
27 \[ |
31 Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $\cC_F$ |
28 E = Y\times F . |
32 defined by $\cC_F(X) = \cE(X\times F)$ if $\dim(X) < k$ and |
29 \] |
33 $\cC_F(X) = \bc_*^\cE(X\times F)$ if $\dim(X) = k$. |
30 Let $\cC$ be an $n$-category. |
34 |
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} |
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39 |
35 |
40 \begin{thm} \label{thm:product} |
36 \begin{thm} \label{thm:product} |
41 Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from |
37 Let $Y$ be a $k$-manifold. |
42 Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $\bc_*(F; C)$ defined by |
38 Then there is a homotopy equivalence between ``old-fashioned" (blob diagrams) |
43 \begin{equation*} |
39 and ``new-fangled" (hocolimit) blob complexes |
44 \bc_*(F; C)(B) = \cB_*(F \times B; C). |
40 \[ |
45 \end{equation*} |
41 \cB_*(Y \times F) \htpy \cl{\cC_F}(Y) . |
46 Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the ``old-fashioned'' |
42 \]\end{thm} |
47 blob complex for $Y \times F$ with coefficients in $C$ and the ``new-fangled" |
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48 (i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $\bc_*(F; C)$: |
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49 \begin{align*} |
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50 \cB_*(Y \times F; C) & \htpy \cl{\bc_*(F; C)}(Y) |
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51 \end{align*} |
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52 \end{thm} |
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53 |
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54 |
43 |
55 \begin{proof} |
44 \begin{proof} |
56 We will use the concrete description of the colimit from \S\ref{ss:ncat_fields}. |
45 We will use the concrete description of the homotopy colimit from \S\ref{ss:ncat_fields}. |
57 |
46 |
58 First we define a map |
47 First we define a map |
59 \[ |
48 \[ |
60 \psi: \cl{\bc_*(F; C)}(Y) \to \bc_*(Y\times F;C) . |
49 \psi: \cl{\cC_F}(Y) \to \bc_*(Y\times F;C) . |
61 \] |
50 \] |
62 In filtration degree 0 we just glue together the various blob diagrams on $X_i\times F$ |
51 In filtration degree 0 we just glue together the various blob diagrams on $X_i\times F$ |
63 (where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
52 (where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
64 $Y\times F$. |
53 $Y\times F$. |
65 In filtration degrees 1 and higher we define the map to be zero. |
54 In filtration degrees 1 and higher we define the map to be zero. |
66 It is easy to check that this is a chain map. |
55 It is easy to check that this is a chain map. |
67 |
56 |
68 In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;C)$ |
57 In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;C)$ |
69 and a map |
58 and a map |
70 \[ |
59 \[ |
71 \phi: G_* \to \cl{\bc_*(F; C)}(Y) . |
60 \phi: G_* \to \cl{\cC_F}(Y) . |
72 \] |
61 \] |
73 |
62 |
74 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
63 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
75 decomposition of $Y\times F$ into the pieces $X_i\times F$. |
64 decomposition of $Y\times F$ into the pieces $X_i\times F$. |
76 |
65 |
79 It follows from Proposition \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ is homotopic to a subcomplex of $G_*$. |
68 It follows from Proposition \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ is homotopic to a subcomplex of $G_*$. |
80 (If the blobs of $a$ are small with respect to a sufficiently fine cover then their |
69 (If the blobs of $a$ are small with respect to a sufficiently fine cover then their |
81 projections to $Y$ are contained in some disjoint union of balls.) |
70 projections to $Y$ are contained in some disjoint union of balls.) |
82 Note that the image of $\psi$ is equal to $G_*$. |
71 Note that the image of $\psi$ is equal to $G_*$. |
83 |
72 |
84 We will define $\phi: G_* \to \cl{\bc_*(F; C)}(Y)$ using the method of acyclic models. |
73 We will define $\phi: G_* \to \cl{\cC_F}(Y)$ using the method of acyclic models. |
85 Let $a$ be a generator of $G_*$. |
74 Let $a$ be a generator of $G_*$. |
86 Let $D(a)$ denote the subcomplex of $\cl{\bc_*(F; C)}(Y)$ generated by all $(b, \ol{K})$ |
75 Let $D(a)$ denote the subcomplex of $\cl{\cC_F}(Y)$ generated by all $(b, \ol{K})$ |
87 such that $a$ splits along $K_0\times F$ and $b$ is a generator appearing |
76 such that $a$ splits along $K_0\times F$ and $b$ is a generator appearing |
88 in an iterated boundary of $a$ (this includes $a$ itself). |
77 in an iterated boundary of $a$ (this includes $a$ itself). |
89 (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; |
78 (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; |
90 see \S\ref{ss:ncat_fields}.) |
79 see \S\ref{ss:ncat_fields}.) |
91 By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is |
80 By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is |