$(A_{\mathrm{\infty }},2)$-categories and relative 2-operads

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 $(A_{\mathrm{\infty }},2)$-category and $(A_{\mathrm{\infty }},2)$-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 $(A_{\mathrm{\infty }},2)$-algebra $\theta (A\to X)$ in $\mathsf{\text{Top}}$, which specializes to $\theta (\text{pt}\to X)={\mathrm{\Omega }}^{2}X$ and $\theta(A \to $\text{pt}$) = \Omega A \times \Omega A$.