6 \subsection{Outline} |
6 \subsection{Outline} |
7 |
7 |
8 So far we have provided no evidence that blob homology is interesting in degrees |
8 So far we have provided no evidence that blob homology is interesting in degrees |
9 greater than zero. |
9 greater than zero. |
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 |
12 Hochschild complex of the 1-category $\cC$. |
12 Recall (\S \ref{sec:example:traditional-n-categories(fields)}) |
13 (Recall from \S \ref{sec:example:traditional-n-categories(fields)} that a |
13 that from a *-1-category $C$ we can construct a system of fields $\cC$. |
14 $1$-category gives rise to a $1$-dimensional system of fields; as usual, |
14 In this section we prove that $\bc_*(S^1, \cC)$ is homotopy equivalent to the |
15 talking about the blob complex with coefficients in an $n$-category means |
15 Hochschild complex of $C$. |
16 first passing to the corresponding $n$ dimensional system of fields.) |
16 %(Recall from \S \ref{sec:example:traditional-n-categories(fields)} that a |
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17 %$1$-category gives rise to a $1$-dimensional system of fields; as usual, |
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18 %talking about the blob complex with coefficients in an $n$-category means |
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19 %first passing to the corresponding $n$ dimensional system of fields.) |
17 Thus the blob complex is a natural generalization of something already |
20 Thus the blob complex is a natural generalization of something already |
18 known to be interesting in higher homological degrees. |
21 known to be interesting in higher homological degrees. |
19 |
22 |
20 It is also worth noting that the original idea for the blob complex came from trying |
23 It is also worth noting that the original idea for the blob complex came from trying |
21 to find a more ``local" description of the Hochschild complex. |
24 to find a more ``local" description of the Hochschild complex. |
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25 |
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26 \medskip |
22 |
27 |
23 Let $C$ be a *-1-category. |
28 Let $C$ be a *-1-category. |
24 Then specializing the definition of the associated system of fields from \S \ref{sec:example:traditional-n-categories(fields)} above to the case $n=1$ we have: |
29 Then specializing the definition of the associated system of fields from \S \ref{sec:example:traditional-n-categories(fields)} above to the case $n=1$ we have: |
25 \begin{itemize} |
30 \begin{itemize} |
26 \item $\cC(pt) = \ob(C)$ . |
31 \item $\cC(pt) = \ob(C)$ . |
51 Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$. |
56 Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$. |
52 We define a blob-like complex $K_*(S^1, (p_i), (M_i))$. |
57 We define a blob-like complex $K_*(S^1, (p_i), (M_i))$. |
53 The fields have elements of $M_i$ labeling |
58 The fields have elements of $M_i$ labeling |
54 the fixed points $p_i$ and elements of $C$ labeling other (variable) points. |
59 the fixed points $p_i$ and elements of $C$ labeling other (variable) points. |
55 As before, the regions between the marked points are labeled by |
60 As before, the regions between the marked points are labeled by |
56 objects of $\cC$. |
61 objects of $C$. |
57 The blob twig labels lie in kernels of evaluation maps. |
62 The blob twig labels lie in kernels of evaluation maps. |
58 (The range of these evaluation maps is a tensor product (over $C$) of $M_i$'s, |
63 (The range of these evaluation maps is a tensor product (over $C$) of $M_i$'s, |
59 corresponding to the $p_i$'s that lie within the twig blob.) |
64 corresponding to the $p_i$'s that lie within the twig blob.) |
60 Let $K_*(M) = K_*(S^1, (*), (M))$, where $* \in S^1$ is some standard base point. |
65 Let $K_*(M) = K_*(S^1, (*), (M))$, where $* \in S^1$ is some standard base point. |
61 In other words, fields for $K_*(M)$ have an element of $M$ at the fixed point $*$ |
66 In other words, fields for $K_*(M)$ have an element of $M$ at the fixed point $*$ |