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

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 $\text{Top}$, which specializes to $\theta (\text{pt}\to X)={\Omega }^{2}X$ and $\theta (A\to \text{pt})=\Omega A\times \Omega A$.

19 pages, 8 figures. Final version for publication in Higher Structures

References in corpus (1)

• Generating the Fukaya categories of Hamiltonian G-manifolds

Cited by in corpus (1)

• Formal groups and quantum cohomology