3 \section{Hochschild homology when $n=1$} |
3 \section{Hochschild homology when $n=1$} |
4 \label{sec:hochschild} |
4 \label{sec:hochschild} |
5 |
5 |
6 So far we have provided no evidence that blob homology is interesting in degrees |
6 So far we have provided no evidence that blob homology is interesting in degrees |
7 greater than zero. |
7 greater than zero. |
8 In this section we analyze the blob complex in dimension $n=1$ |
8 In this section we analyze the blob complex in dimension $n=1$. |
9 and find that for $S^1$ the blob complex is homotopy equivalent to the |
9 We find that $\bc_*(S^1, \cC)$ is homotopy equivalent to the |
10 Hochschild complex of the category (algebroid) that we started with. |
10 Hochschild complex of the 1-category $\cC$. |
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11 \nn{cat vs fields --- need to make sure this is clear} |
11 Thus the blob complex is a natural generalization of something already |
12 Thus the blob complex is a natural generalization of something already |
12 known to be interesting in higher homological degrees. |
13 known to be interesting in higher homological degrees. |
13 |
14 |
14 It is also worth noting that the original idea for the blob complex came from trying |
15 It is also worth noting that the original idea for the blob complex came from trying |
15 to find a more ``local" description of the Hochschild complex. |
16 to find a more ``local" description of the Hochschild complex. |
16 |
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17 \nn{need to be consistent about quasi-isomorphic versus homotopy equivalent |
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18 in this section. |
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19 since the various complexes are free, q.i. implies h.e.} |
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20 |
17 |
21 Let $C$ be a *-1-category. |
18 Let $C$ be a *-1-category. |
22 Then specializing the definitions from above to the case $n=1$ we have: |
19 Then specializing the definitions from above to the case $n=1$ we have: |
23 \begin{itemize} |
20 \begin{itemize} |
24 \item $\cC(pt) = \ob(C)$ . |
21 \item $\cC(pt) = \ob(C)$ . |