72 For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to each such $B$ and ask that the action of |
72 For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to each such $B$ and ask that the action of |
73 homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. |
73 homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. |
74 The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a |
74 The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a |
75 topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
75 topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
76 |
76 |
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77 In \S \ref{ssec:spherecat} we explain how $n$-categories can be viewed as objects in an $n{+}1$-category |
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78 of sphere modules. |
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79 When $n=1$ this just the familiar 2-category of 1-categories, bimodules and intertwinors. |
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80 |
77 In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category |
81 In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category |
78 (using a colimit along certain decompositions of a manifold into balls). |
82 (using a colimit along certain decompositions of a manifold into balls). |
79 With this in hand, we write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$ |
83 With this in hand, we write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$ |
80 with the system of fields constructed from the $n$-category $\cC$. |
84 with the system of fields constructed from the $n$-category $\cC$. |
81 %\nn{KW: I don't think we use this notational convention any more, right?} |
85 %\nn{KW: I don't think we use this notational convention any more, right?} |