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

Nathaniel Bottman, Katrin Wehrheim

TL;DR

The paper develops a rigorous algebraic and topological framework for a symplectic $(A_ fty,2)$-category, termed $\oldsymbol{\mathsf{Symp}}$, by introducing $(A_ ablafty,2)$-flow categories and linear $(A_ ablafty,2)$-categories. It fixes limitations of earlier Bottman–Carmeli definitions by extending 2-associahedra to include all relevant fiber-product faces and attaching operations to cellular chains, enabling single-arity operations governed by $(A_ ablafty,2)$-equations. A key result is that any regularized $(A_ ablafty,2)$-flow category yields a linear $(A_ ablafty,2)$-category, with the motivating example $\ ext{Symp}$ built from moduli of pseudoholomorphic quilts and regularization data. The framework aims to unify symplectic field theory constructions, Fukaya-categorical functoriality, and mirror-symmetric phenomena, while accounting for figure-eight bubbling and curvature terms via Novikov counts. This provides a principled route to a full symplectic $(A_ ablafty,2)$-category and potentially higher $(A_ ablafty,n)$-categorical structures, with broad implications for Floer theory, topological invariants, and HMS proofs.

Abstract

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.

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

TL;DR

The paper develops a rigorous algebraic and topological framework for a symplectic -category, termed , by introducing -flow categories and linear -categories. It fixes limitations of earlier Bottman–Carmeli definitions by extending 2-associahedra to include all relevant fiber-product faces and attaching operations to cellular chains, enabling single-arity operations governed by -equations. A key result is that any regularized -flow category yields a linear -category, with the motivating example built from moduli of pseudoholomorphic quilts and regularization data. The framework aims to unify symplectic field theory constructions, Fukaya-categorical functoriality, and mirror-symmetric phenomena, while accounting for figure-eight bubbling and curvature terms via Novikov counts. This provides a principled route to a full symplectic -category and potentially higher -categorical structures, with broad implications for Floer theory, topological invariants, and HMS proofs.

Abstract

This paper provides a blueprint for the construction of a symplectic -category, . We develop two ways of encoding the information in -- one topological, one algebraic. The topological encoding is as an -flow category, which we define here. The algebraic encoding is as a linear -category, which we extract from the topological encoding. In upcoming work, we plan to use the adiabatic Fredholm theory developed by us to construct as an -flow category, which thus induces a linear -category. The notion of a linear -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 -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 -equations.

Paper Structure

This paper contains 17 sections, 6 theorems, 71 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $(A_\infty,2)$-flow categories and linear $(A_\infty,2)$-categories
  3. Applications in Symplectic Geometry
  4. From Weinstein's vision via Floer theory to Fukaya $A_\infty$-categories and the monotone symplectic 2-category
  5. Symplectic Examples of $(A_\infty,2)$-categorical structures
  6. Topological invariants constructed via Floer field theories
  7. Functors on Fukaya categories
  8. Generation criteria for Fukaya categories
  9. Proofs of Mirror Symmetry
  10. Monoidal structure on Fukaya categories
  11. Bounding cochains, computations of Floer homology, and Seidel's formal group
  12. Categorification of Dehn twists and (symplectic) Khovanov homology
  13. Preliminaries
  14. $A_\infty$- and $(A_\infty,2)$-flow categories
  15. Extracting linear categorical structures from $A_\infty$- and $(A_\infty,2)$-flow categories
  16. ...and 2 more sections

Key Result

Lemma 2.7

Let $Y_n$ for $1\leq n \leq N$ be $0$-dimensional $\mathcal{C}^0$-manifolds equipped with energy functions $\mathcal{E}_n: Y_n\to\mathbb{R}$ as in Definition def:Novikov count. Define an energy function $Y:=Y_1\times \ldots\times Y_N \to \mathbb{R}$ by addition $(y_1,\ldots,y_N)\mapsto \mathcal{E}_1

Figures (2)

  • Figure 1: Here we depict the codimension-1 degenerations in an element of $\mathscr{O}$ in a heuristic fashion. On the left is a representative element of a fiber product of 2-associahedra. There are three types of codimension-1 degenerations, which are depicted from left to right on the right-hand side: marked points on a single seam on a single sphere can collide; a proper subset of seams on all spheres can collide; or marked points on a single sphere can diverge to infinity -- which is equivalent to all seams on a single sphere colliding. For details see Remark \ref{['rmk:three_types_of_comp_maps']}.
  • Figure :

Theorems & Definitions (41)

  • Remark 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • ...and 31 more