36 Thus we can, if we choose, restrict the blob twig labels to things of this form. |
36 Thus we can, if we choose, restrict the blob twig labels to things of this form. |
37 \end{itemize} |
37 \end{itemize} |
38 |
38 |
39 We want to show that $\bc_*(S^1)$ is homotopy equivalent to the |
39 We want to show that $\bc_*(S^1)$ is homotopy equivalent to the |
40 Hochschild complex of $C$. |
40 Hochschild complex of $C$. |
41 Note that both complexes are free (and hence projective), so it suffices to show that they |
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42 are quasi-isomorphic. |
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43 In order to prove this we will need to extend the |
41 In order to prove this we will need to extend the |
44 definition of the blob complex to allow points to also |
42 definition of the blob complex to allow points to also |
45 be labeled by elements of $C$-$C$-bimodules. |
43 be labeled by elements of $C$-$C$-bimodules. |
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44 (See Subsections \ref{moddecss} and \ref{ssec:spherecat} for a more general (i.e.\ $n>1$) |
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45 version of this construction.) |
46 |
46 |
47 Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$. |
47 Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$. |
48 We define a blob-like complex $K_*(S^1, (p_i), (M_i))$. |
48 We define a blob-like complex $K_*(S^1, (p_i), (M_i))$. |
49 The fields have elements of $M_i$ labeling $p_i$ and elements of $C$ labeling |
49 The fields have elements of $M_i$ labeling |
50 other points. |
50 the fixed points $p_i$ and elements of $C$ labeling other (variable) points. |
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51 As before, the regions between the marked points are labeled by |
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52 objects of $\cC$. |
51 The blob twig labels lie in kernels of evaluation maps. |
53 The blob twig labels lie in kernels of evaluation maps. |
52 (The range of these evaluation maps is a tensor product (over $C$) of $M_i$'s.) |
54 (The range of these evaluation maps is a tensor product (over $C$) of $M_i$'s, |
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55 corresponding to the $p_i$'s that lie within the twig blob.) |
53 Let $K_*(M) = K_*(S^1, (*), (M))$, where $* \in S^1$ is some standard base point. |
56 Let $K_*(M) = K_*(S^1, (*), (M))$, where $* \in S^1$ is some standard base point. |
54 In other words, fields for $K_*(M)$ have an element of $M$ at the fixed point $*$ |
57 In other words, fields for $K_*(M)$ have an element of $M$ at the fixed point $*$ |
55 and elements of $C$ at variable other points. |
58 and elements of $C$ at variable other points. |
56 |
59 |
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60 In the theorems, propositions and lemmas below we make various claims |
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61 about complexes being homotopy equivalent. |
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62 In all cases the complexes in question are free (and hence projective), |
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63 so it suffices to show that they are quasi-isomorphic. |
57 |
64 |
58 We claim that |
65 We claim that |
59 \begin{thm} \label{hochthm} |
66 \begin{thm} \label{hochthm} |
60 The blob complex $\bc_*(S^1; C)$ on the circle is quasi-isomorphic to the |
67 The blob complex $\bc_*(S^1; C)$ on the circle is homotopy equivalent to the |
61 usual Hochschild complex for $C$. |
68 usual Hochschild complex for $C$. |
62 \end{thm} |
69 \end{thm} |
63 |
70 |
64 This follows from two results. First, we see that |
71 This follows from two results. First, we see that |
65 \begin{lem} |
72 \begin{lem} |
69 $\bc_*(S^1; C)$. (Proof later.) |
76 $\bc_*(S^1; C)$. (Proof later.) |
70 \end{lem} |
77 \end{lem} |
71 |
78 |
72 Next, we show that for any $C$-$C$-bimodule $M$, |
79 Next, we show that for any $C$-$C$-bimodule $M$, |
73 \begin{prop} \label{prop:hoch} |
80 \begin{prop} \label{prop:hoch} |
74 The complex $K_*(M)$ is quasi-isomorphic to $\HC_*(M)$, the usual |
81 The complex $K_*(M)$ is homotopy equivalent to $\HC_*(M)$, the usual |
75 Hochschild complex of $M$. |
82 Hochschild complex of $M$. |
76 \end{prop} |
83 \end{prop} |
77 \begin{proof} |
84 \begin{proof} |
78 Recall that the usual Hochschild complex of $M$ is uniquely determined, |
85 Recall that the usual Hochschild complex of $M$ is uniquely determined, |
79 up to quasi-isomorphism, by the following properties: |
86 up to quasi-isomorphism, by the following properties: |
89 \item \label{item:hochschild-free}% |
96 \item \label{item:hochschild-free}% |
90 $\HC_*(C\otimes C)$ is contractible. |
97 $\HC_*(C\otimes C)$ is contractible. |
91 (Here $C\otimes C$ denotes |
98 (Here $C\otimes C$ denotes |
92 the free $C$-$C$-bimodule with one generator.) |
99 the free $C$-$C$-bimodule with one generator.) |
93 That is, $\HC_*(C\otimes C)$ is |
100 That is, $\HC_*(C\otimes C)$ is |
94 quasi-isomorphic to its $0$-th homology (which in turn, by \ref{item:hochschild-coinvariants}, is just $C$) via the quotient map $\HC_0 \onto \HH_0$. |
101 quasi-isomorphic to its $0$-th homology (which in turn, by \ref{item:hochschild-coinvariants} |
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102 above, is just $C$) via the quotient map $\HC_0 \onto \HH_0$. |
95 \end{enumerate} |
103 \end{enumerate} |
96 (Together, these just say that Hochschild homology is `the derived functor of coinvariants'.) |
104 (Together, these just say that Hochschild homology is `the derived functor of coinvariants'.) |
97 We'll first recall why these properties are characteristic. |
105 We'll first recall why these properties are characteristic. |
98 |
106 |
99 Take some $C$-$C$ bimodule $M$, and choose a free resolution |
107 Take some $C$-$C$ bimodule $M$, and choose a free resolution |