1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
2 |
3 \section{The blob complex for $A_\infty$ $n$-categories} |
3 \section{The blob complex for $A_\infty$ $n$-categories} |
4 \label{sec:ainfblob} |
4 \label{sec:ainfblob} |
5 |
5 Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the anticlimactically tautological definition of the blob |
6 Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we define the blob |
6 complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of Section \ref{ss:ncat_fields}. |
7 complex $\bc_*(M)$ to the be the homotopy colimit $\cC(M)$ of Section \ref{sec:ncats}. |
7 |
8 \nn{say something about this being anticlimatically tautological?} |
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9 We will show below |
8 We will show below |
10 in Corollary \ref{cor:new-old} |
9 in Corollary \ref{cor:new-old} |
11 that this agrees (up to homotopy) with our original definition of the blob complex |
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 |
12 in the case of plain $n$-categories. |
11 for $\cD$. |
13 When we need to distinguish between the new and old definitions, we will refer to the |
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14 new-fangled and old-fashioned blob complex. |
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15 |
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16 \medskip |
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17 |
12 |
18 An important technical tool in the proofs of this section is provided by the idea of `small blobs'. |
13 An important technical tool in the proofs of this section is provided by the idea of `small blobs'. |
19 Fix $\cU$, an open cover of $M$. |
14 Fix $\cU$, an open cover of $M$. |
20 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. |
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. |
21 |
16 |
42 |
37 |
43 \nn{need to settle on notation; proof and statement are inconsistent} |
38 \nn{need to settle on notation; proof and statement are inconsistent} |
44 |
39 |
45 \begin{thm} \label{thm:product} |
40 \begin{thm} \label{thm:product} |
46 Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from |
41 Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from |
47 Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $C^{\times F}$ defined by |
42 Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $\bc_*(F; C)$ defined by |
48 \begin{equation*} |
43 \begin{equation*} |
49 C^{\times F}(B) = \cB_*(B \times F, C). |
44 \bc_*(F; C) = \cB_*(B \times F, C). |
50 \end{equation*} |
45 \end{equation*} |
51 Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned' |
46 Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned' |
52 blob complex for $Y \times F$ with coefficients in $C$ and the `new-fangled' |
47 blob complex for $Y \times F$ with coefficients in $C$ and the `new-fangled' |
53 (i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $C^{\times F}$: |
48 (i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $\bc_*(F; C)$: |
54 \begin{align*} |
49 \begin{align*} |
55 \cB_*(Y \times F, C) & \htpy \cB_*(Y, C^{\times F}) |
50 \cB_*(Y \times F; C) & \htpy \cl{\bc_*(F; C)}(Y) |
56 \end{align*} |
51 \end{align*} |
57 \end{thm} |
52 \end{thm} |
58 |
53 |
59 |
54 |
60 \begin{proof} |
55 \begin{proof} |
61 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
56 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
62 |
57 |
63 First we define a map |
58 First we define a map |
64 \[ |
59 \[ |
65 \psi: \bc_*^\cF(Y) \to \bc_*^C(Y\times F) . |
60 \psi: \cl{\bc_*(F; C)}(Y) \to \bc_*(Y\times F;C) . |
66 \] |
61 \] |
67 In filtration degree 0 we just glue together the various blob diagrams on $X_i\times F$ |
62 In filtration degree 0 we just glue together the various blob diagrams on $X_i\times F$ |
68 (where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
63 (where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
69 $Y\times F$. |
64 $Y\times F$. |
70 In filtration degrees 1 and higher we define the map to be zero. |
65 In filtration degrees 1 and higher we define the map to be zero. |
71 It is easy to check that this is a chain map. |
66 It is easy to check that this is a chain map. |
72 |
67 |
73 In the other direction, we will define a subcomplex $G_*\sub \bc_*^C(Y\times F)$ |
68 In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;C)$ |
74 and a map |
69 and a map |
75 \[ |
70 \[ |
76 \phi: G_* \to \bc_*^\cF(Y) . |
71 \phi: G_* \to \cl{\bc_*(F; C)}(Y) . |
77 \] |
72 \] |
78 |
73 |
79 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
74 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
80 decomposition of $Y\times F$ into the pieces $X_i\times F$. |
75 decomposition of $Y\times F$ into the pieces $X_i\times F$. |
81 |
76 |
82 Let $G_*\sub \bc_*^C(Y\times F)$ be the subcomplex generated by blob diagrams $a$ such that there |
77 Let $G_*\sub \bc_*(Y\times F;C)$ be the subcomplex generated by blob diagrams $a$ such that there |
83 exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$. |
78 exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$. |
84 It follows from Proposition \ref{thm:small-blobs} that $\bc_*^C(Y\times F)$ is homotopic to a subcomplex of $G_*$. |
79 It follows from Proposition \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ is homotopic to a subcomplex of $G_*$. |
85 (If the blobs of $a$ are small with respect to a sufficiently fine cover then their |
80 (If the blobs of $a$ are small with respect to a sufficiently fine cover then their |
86 projections to $Y$ are contained in some disjoint union of balls.) |
81 projections to $Y$ are contained in some disjoint union of balls.) |
87 Note that the image of $\psi$ is equal to $G_*$. |
82 Note that the image of $\psi$ is equal to $G_*$. |
88 |
83 |
89 We will define $\phi: G_* \to \bc_*^\cF(Y)$ using the method of acyclic models. |
84 We will define $\phi: G_* \to \cl{\bc_*(F; C)}(Y)$ using the method of acyclic models. |
90 Let $a$ be a generator of $G_*$. |
85 Let $a$ be a generator of $G_*$. |
91 Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(b, \ol{K})$ |
86 Let $D(a)$ denote the subcomplex of $\cl{\bc_*(F; C)}(Y)$ generated by all $(b, \ol{K})$ |
92 such that $a$ splits along $K_0\times F$ and $b$ is a generator appearing |
87 such that $a$ splits along $K_0\times F$ and $b$ is a generator appearing |
93 in an iterated boundary of $a$ (this includes $a$ itself). |
88 in an iterated boundary of $a$ (this includes $a$ itself). |
94 (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; |
89 (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; |
95 see Subsection \ref{ss:ncat_fields}.) |
90 see Subsection \ref{ss:ncat_fields}.) |
96 By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is |
91 By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is |
192 |
187 |
193 Continuing in this way we see that $D(a)$ is acyclic. |
188 Continuing in this way we see that $D(a)$ is acyclic. |
194 \end{proof} |
189 \end{proof} |
195 |
190 |
196 We are now in a position to apply the method of acyclic models to get a map |
191 We are now in a position to apply the method of acyclic models to get a map |
197 $\phi:G_* \to \bc_*^\cF(Y)$. |
192 $\phi:G_* \to \cl{\bc_*(F; C)}(Y)$. |
198 We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is in filtration degree zero |
193 We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is in filtration degree zero |
199 and $r$ has filtration degree greater than zero. |
194 and $r$ has filtration degree greater than zero. |
200 |
195 |
201 We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity. |
196 We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity. |
202 |
197 |
203 $\psi\circ\phi$ is the identity on the nose: |
198 First, $\psi\circ\phi$ is the identity on the nose: |
204 \[ |
199 \[ |
205 \psi(\phi(a)) = \psi((a,K)) + \psi(r) = a + 0. |
200 \psi(\phi(a)) = \psi((a,K)) + \psi(r) = a + 0. |
206 \] |
201 \] |
207 Roughly speaking, $(a, K)$ is just $a$ chopped up into little pieces, and |
202 Roughly speaking, $(a, K)$ is just $a$ chopped up into little pieces, and |
208 $\psi$ glues those pieces back together, yielding $a$. |
203 $\psi$ glues those pieces back together, yielding $a$. |
209 We have $\psi(r) = 0$ since $\psi$ is zero in positive filtration degrees. |
204 We have $\psi(r) = 0$ since $\psi$ is zero in positive filtration degrees. |
210 |
205 |
211 $\phi\circ\psi$ is the identity up to homotopy by another MoAM argument. |
206 Second, $\phi\circ\psi$ is the identity up to homotopy by another argument based on the method of acyclic models. |
212 To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above. |
207 To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above. |
213 Both the identity map and $\phi\circ\psi$ are compatible with this |
208 Both the identity map and $\phi\circ\psi$ are compatible with this |
214 collection of acyclic subcomplexes, so by the usual MoAM argument these two maps |
209 collection of acyclic subcomplexes, so by the usual method of acyclic models argument these two maps |
215 are homotopic. |
210 are homotopic. |
216 |
211 |
217 This concludes the proof of Theorem \ref{thm:product}. |
212 This concludes the proof of Theorem \ref{thm:product}. |
218 \end{proof} |
213 \end{proof} |
219 |
214 |
220 \nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category} |
215 \nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category} |
221 |
216 |
222 \medskip |
217 \medskip |
223 |
218 |
224 \todo{rephrase this} |
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225 \begin{cor} |
219 \begin{cor} |
226 \label{cor:new-old} |
220 \label{cor:new-old} |
227 The new-fangled and old-fashioned blob complexes are homotopic. |
221 The blob complex of a manifold $M$ with coefficients in a topological $n$-category $\cC$ is homotopic to the homotopy colimit invariant of $M$ defined using the $A_\infty$ $n$-category obtained by applying the blob complex to a point: |
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222 $$\bc_*(M; \cC) \htpy \cl{\bc_*(pt; \cC)}(M).$$ |
228 \end{cor} |
223 \end{cor} |
229 \begin{proof} |
224 \begin{proof} |
230 Apply Theorem \ref{thm:product} with the fiber $F$ equal to a point. |
225 Apply Theorem \ref{thm:product} with the fiber $F$ equal to a point. |
231 \end{proof} |
226 \end{proof} |
232 |
227 |