Foundations of $(A_\infty,2)$-categories: from flow to linear

This paper provides a blueprint for the construction of a symplectic $(A_\infty,2)$-category, $\mathsf{Symp}$. We develop two ways of encoding the information in $\mathsf{Symp}$ -- one topological, one algebraic. The topological encoding is as an $(A_\infty,2)$-flow category, which we define here. The algebraic encoding is as a linear $(A_\infty,2)$-category, which we extract from the topological encoding. In upcoming work, we plan to use the adiabatic Fredholm theory developed by us to construct $\mathsf{Symp}$ as an $(A_\infty,2)$-flow category, which thus induces a linear $(A_\infty,2)$-category. The notion of a linear $(A_\infty,2)$-category developed here goes beyond the proposal of Bottman and Carmeli. The recursive structure of the 2-associahedra identifies faces with fiber products of 2-associahedra over associahedra, which led Bottman and Carmeli to associate operations to singular chains on 2-associahedra. The innovation in our new definition of linear $(A_\infty,2)$-category is to extend the family of 2-associahedra to include all fiber products of 2-associahedra over associahedra. This allows us to associate operations to cellular chains, which in particular enables us to produce a definition that involves only one operation in each arity, governed by a collection of $(A_\infty,2)$-equations.