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

TL;DR

This paper introduces the concept of 2-operads relative to operads, constructs an $(A_ fty,2)$-category framework, and applies it to associate specific $(A_ abla,2)$-algebras to continuous maps, extending the algebraic structures in topology.

Contribution

It defines 2-operads relative to operads, constructs $(A_ abla,2)$-categories and algebras, and links these structures to continuous maps in topology.

Findings

The 2-associahedra form a 2-operad relative to the associahedra.

Defined $(A_ abla,2)$-categories and algebras in spaces and chain complexes.

Established a correspondence between continuous maps and $(A_ abla,2)$-algebras in topology.

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 $\theta(A \to \text{pt}) = \Omega A \times \Omega A$.