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Abstract

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 $A \to X$, we can associate an
$(A_\infty,2)$-algebra $\theta(A \to X)$ in $\textsf{Top}$, which specializes
to $\theta(\text{pt} \to X) = \Omega^2 X$ and $\theta(A \to \text{pt}) = \Omega
A \times \Omega A$.