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40 Topological $A_\infty$-$1$-categories are equivalent to the usual notion of |
40 Topological $A_\infty$-$1$-categories are equivalent to the usual notion of |
41 $A_\infty$-$1$-categories. |
41 $A_\infty$-$1$-categories. |
42 \end{thm} |
42 \end{thm} |
43 |
43 |
44 Before proving this theorem, we embark upon a long string of definitions. |
44 Before proving this theorem, we embark upon a long string of definitions. |
45 For expository purposes, we begin with the $n=1$ special cases,\scott{Why are we treating the $n>1$ cases at all?} and define |
45 For expository purposes, we begin with the $n=1$ special cases, |
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46 and define |
46 first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn |
47 first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn |
47 to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules. |
48 to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules. |
48 \nn{Something about duals?} |
49 \nn{Something about duals?} |
49 \todo{Explain that we're not making contact with any previous notions for the general $n$ case?} |
50 \todo{Explain that we're not making contact with any previous notions for the general $n$ case?} |
50 \kevin{probably we should say something about the relation |
51 \kevin{probably we should say something about the relation |