4 \label{sec:comm_alg} |
4 \label{sec:comm_alg} |
5 |
5 |
6 \nn{this should probably not be a section by itself. i'm just trying to write down the outline |
6 \nn{this should probably not be a section by itself. i'm just trying to write down the outline |
7 while it's still fresh in my mind.} |
7 while it's still fresh in my mind.} |
8 |
8 |
9 \nn{I strongly suspect that [blob complex |
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10 for $M^n$ based on comm alg $C$ thought of as an $n$-category] |
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11 is homotopy equivalent to [higher Hochschild complex for $M^n$ with coefficients in $C$]. |
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12 (Thomas Tradler's idea.) |
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13 Should prove (or at least conjecture) that here.} |
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14 |
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15 \nn{also, this section needs a little updating to be compatible with the rest of the paper.} |
9 \nn{also, this section needs a little updating to be compatible with the rest of the paper.} |
16 |
10 |
17 If $C$ is a commutative algebra it |
11 If $C$ is a commutative algebra it |
18 can also be thought of as an $n$-category whose $j$-morphisms are trivial for |
12 can also be thought of as an $n$-category whose $j$-morphisms are trivial for |
19 $j<n$ and whose $n$-morphisms are $C$. |
13 $j<n$ and whose $n$-morphisms are $C$. |
20 The goal of this \nn{subsection?} is to compute |
14 The goal of this \nn{subsection?} is to compute |
21 $\bc_*(M^n, C)$ for various commutative algebras $C$. |
15 $\bc_*(M^n, C)$ for various commutative algebras $C$. |
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16 |
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17 Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative algebra is homotopy equivalent to the higher Hochschild complex for $M^n$ with coefficients in $C$ (see \cite{MR0339132, MR1755114, MR2383113}). This possibility was suggested to us by Thomas Tradler. |
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18 |
22 |
19 |
23 \medskip |
20 \medskip |
24 |
21 |
25 Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$. |
22 Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$. |
26 |
23 |