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10 In this section we analyze the blob complex in dimension $n=1$. |
10 In this section we analyze the blob complex in dimension $n=1$. |
11 We find that $\bc_*(S^1, \cC)$ is homotopy equivalent to the |
11 We find that $\bc_*(S^1, \cC)$ is homotopy equivalent to the |
12 Hochschild complex of the 1-category $\cC$. |
12 Hochschild complex of the 1-category $\cC$. |
13 (Recall from \S \ref{sec:example:traditional-n-categories(fields)} that a |
13 (Recall from \S \ref{sec:example:traditional-n-categories(fields)} that a |
14 $1$-category gives rise to a $1$-dimensional system of fields; as usual, |
14 $1$-category gives rise to a $1$-dimensional system of fields; as usual, |
15 talking about the blob complex with coefficients in a $n$-category means |
15 talking about the blob complex with coefficients in an $n$-category means |
16 first passing to the corresponding $n$ dimensional system of fields.) |
16 first passing to the corresponding $n$ dimensional system of fields.) |
17 Thus the blob complex is a natural generalization of something already |
17 Thus the blob complex is a natural generalization of something already |
18 known to be interesting in higher homological degrees. |
18 known to be interesting in higher homological degrees. |
19 |
19 |
20 It is also worth noting that the original idea for the blob complex came from trying |
20 It is also worth noting that the original idea for the blob complex came from trying |