1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
2 |
3 \section{Commutative algebras as $n$-categories} |
3 \section{Commutative algebras as $n$-categories} |
4 \label{sec:comm_alg} |
4 \label{sec:comm_alg} |
5 |
5 |
6 \nn{should consider leaving this out; for now, make it an appendix.} |
|
7 |
|
8 \nn{also, this section needs a little updating to be compatible with the rest of the paper.} |
|
9 |
|
10 If $C$ is a commutative algebra it |
6 If $C$ is a commutative algebra it |
11 can also be thought of as an $n$-category whose $j$-morphisms are trivial for |
7 can also be thought of as an $n$-category whose $j$-morphisms are trivial for |
12 $j<n$ and whose $n$-morphisms are $C$. |
8 $j<n$ and whose $n$-morphisms are $C$. |
13 The goal of this \nn{subsection?} is to compute |
9 The goal of this appendix is to compute |
14 $\bc_*(M^n, C)$ for various commutative algebras $C$. |
10 $\bc_*(M^n, C)$ for various commutative algebras $C$. |
15 |
11 |
16 Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative |
12 Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative |
17 algebra is homotopy equivalent to the higher Hochschild complex for $M^n$ with |
13 algebra is homotopy equivalent to the higher Hochschild complex for $M^n$ with |
18 coefficients in $C$ (see \cite{MR0339132, MR1755114, MR2383113}). |
14 coefficients in $C$ (see \cite{MR0339132, MR1755114, MR2383113}). |
33 \begin{prop} \label{sympowerprop} |
29 \begin{prop} \label{sympowerprop} |
34 $\bc_*(M, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$. |
30 $\bc_*(M, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$. |
35 \end{prop} |
31 \end{prop} |
36 |
32 |
37 \begin{proof} |
33 \begin{proof} |
38 To define the chain maps between the two complexes we will use the following lemma: |
34 %To define the chain maps between the two complexes we will use the following lemma: |
39 |
35 % |
40 \begin{lemma} |
36 %\begin{lemma} |
41 Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with |
37 %Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with |
42 a basis (e.g.\ blob diagrams or singular simplices). |
38 %a basis (e.g.\ blob diagrams or singular simplices). |
43 For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$ |
39 %For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$ |
44 such that $R(c')_* \sub R(c)_*$ whenever $c'$ is a basis element which is part of $\bd c$. |
40 %such that $R(c')_* \sub R(c)_*$ whenever $c'$ is a basis element which is part of $\bd c$. |
45 Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that |
41 %Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that |
46 $f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty). |
42 %$f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty). |
47 \end{lemma} |
43 %\end{lemma} |
48 |
44 % |
49 \begin{proof} |
45 %\begin{proof} |
50 \nn{easy, but should probably write the details eventually} |
46 %\nn{easy, but should probably write the details eventually} |
51 \nn{this is just the standard ``method of acyclic models" set up, so we should just give a reference for that} |
47 %\nn{this is just the standard ``method of acyclic models" set up, so we should just give a reference for that} |
52 \end{proof} |
48 %\end{proof} |
53 |
49 We will use acyclic models \nn{need ref}. |
54 Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$ |
50 Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$ |
55 satisfying the conditions of the above lemma. |
51 satisfying the conditions of \nn{need ref}. |
56 If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a |
52 If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a |
57 finite unordered collection of points of $M$ with multiplicities, which is |
53 finite unordered collection of points of $M$ with multiplicities, which is |
58 a point in $\Sigma^\infty(M)$. |
54 a point in $\Sigma^\infty(M)$. |
59 Define $R(b)_*$ to be the singular chain complex of this point. |
55 Define $R(b)_*$ to be the singular chain complex of this point. |
60 If $(B, u, r)$ is an $i$-blob diagram, let $D\sub M$ be its support (the union of the blobs). |
56 If $(B, u, r)$ is an $i$-blob diagram, let $D\sub M$ be its support (the union of the blobs). |
64 $u$ picks out a component $X \sub \Sigma^\infty(D)$. |
60 $u$ picks out a component $X \sub \Sigma^\infty(D)$. |
65 The field $r$ on $M\setminus D$ can be thought of as a point in $\Sigma^\infty(M\setminus D)$, |
61 The field $r$ on $M\setminus D$ can be thought of as a point in $\Sigma^\infty(M\setminus D)$, |
66 and using this point we can embed $X$ in $\Sigma^\infty(M)$. |
62 and using this point we can embed $X$ in $\Sigma^\infty(M)$. |
67 Define $R(B, u, r)_*$ to be the singular chain complex of $X$, thought of as a |
63 Define $R(B, u, r)_*$ to be the singular chain complex of $X$, thought of as a |
68 subspace of $\Sigma^\infty(M)$. |
64 subspace of $\Sigma^\infty(M)$. |
69 It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from the above lemma. |
65 It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from |
|
66 \nn{need ref, or state condition}. |
70 Thus we have defined (up to homotopy) a map from |
67 Thus we have defined (up to homotopy) a map from |
71 $\bc_*(M^n, k[t])$ to $C_*(\Sigma^\infty(M))$. |
68 $\bc_*(M^n, k[t])$ to $C_*(\Sigma^\infty(M))$. |
72 |
69 |
73 Next we define, for each simplex $c$ of $C_*(\Sigma^\infty(M))$, a contractible subspace |
70 Next we define, for each simplex $c$ of $C_*(\Sigma^\infty(M))$, a contractible subspace |
74 $R(c)_* \sub \bc_*(M^n, k[t])$. |
71 $R(c)_* \sub \bc_*(M^n, k[t])$. |
82 Let $T\sub M$ be the ``track" of $c$ in $M$. |
79 Let $T\sub M$ be the ``track" of $c$ in $M$. |
83 \nn{do we need to define this precisely?} |
80 \nn{do we need to define this precisely?} |
84 Choose a neighborhood $D$ of $T$ which is a disjoint union of balls of small diameter. |
81 Choose a neighborhood $D$ of $T$ which is a disjoint union of balls of small diameter. |
85 \nn{need to say more precisely how small} |
82 \nn{need to say more precisely how small} |
86 Define $R(c)_*$ to be $\bc_*(D, k[t]) \sub \bc_*(M^n, k[t])$. |
83 Define $R(c)_*$ to be $\bc_*(D, k[t]) \sub \bc_*(M^n, k[t])$. |
87 This is contractible by \ref{bcontract}. |
84 This is contractible by Proposition \ref{bcontract}. |
88 We can arrange that the boundary/inclusion condition is satisfied if we start with |
85 We can arrange that the boundary/inclusion condition is satisfied if we start with |
89 low-dimensional simplices and work our way up. |
86 low-dimensional simplices and work our way up. |
90 \nn{need to be more precise} |
87 \nn{need to be more precise} |
91 |
88 |
92 \nn{still to do: show indep of choice of metric; show compositions are homotopic to the identity |
89 \nn{still to do: show indep of choice of metric; show compositions are homotopic to the identity |