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Keywords:
$(\infty,2)$-categories; higher operads; Fukaya categories
$(\infty,2)$-categories; higher operads; Fukaya categories
Summary:
We define the notion of a 2-operad relative to an operad, and prove that the 2-associahedra form a 2-operad relative to the associahedra. Using this structure, we define the notions of an $(A_\infty,2)$-category and $(A_\infty,2)$-algebra in spaces and in chain complexes over a ring. Finally, we show that for any continuous map $A \rightarrow X$, we can associate the related notion of an $\widetilde {(A_\infty,2)}$-algebra $\theta (A \rightarrow X)$ in Top, which specializes to $\theta (pt \rightarrow X)=\Omega^2 X$ and $\theta (A \rightarrow pt)=\Omega A \times \Omega A$.
References:
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