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The Morse complex is an $\infty$-functor

Guillem Cazassus

TL;DR

This work develops a higher-categorical Morse-theory framework that realizes Morse-theoretic data as $infty$-functors into algebraic structures capturing multiplicative and comultiplicative features. It introduces perturbation-parameterized moduli spaces built from grafted forests and biforests (the $n$-grafted bimultiplihedra) to achieve coherent composition and boundary relations, assembling them into weak Kan models $f$-Bialg, $f$-Alg, and $f$-Coalg. The main results show that, for smooth manifolds, the Morse complex yields a functor to $f$-coalgebras (and dually to $A_infty$-algebras via cochains), while for compact Lie groups (or monoids) the Morse theory produces an $infty$-functor into $f ext{-} ext{Bialg}$ with coherent group-action-induced module structures. This provides a rigorous higher-categorical realization of Morse theory compatible with equivariant settings and suggests deep connections to Fukaya-type categories and extended topological field theories.

Abstract

We provide two versions of the statement in the title. First, when the source is the category of smooth manifolds, we construct a simplicial map to a weak Kan complex whose objects are A-infinity coalgebras. Second, when the source is the category of compact Lie groups (or more generally, compact Lie monoids), the target is a weak Kan complex whose objects consist in $f$-bialgebras, as defined in [CHM24]. We also establish a statement describing module structures induced by group actions, as conjectured in [CHM24].

The Morse complex is an $\infty$-functor

TL;DR

This work develops a higher-categorical Morse-theory framework that realizes Morse-theoretic data as -functors into algebraic structures capturing multiplicative and comultiplicative features. It introduces perturbation-parameterized moduli spaces built from grafted forests and biforests (the -grafted bimultiplihedra) to achieve coherent composition and boundary relations, assembling them into weak Kan models -Bialg, -Alg, and -Coalg. The main results show that, for smooth manifolds, the Morse complex yields a functor to -coalgebras (and dually to -algebras via cochains), while for compact Lie groups (or monoids) the Morse theory produces an -functor into with coherent group-action-induced module structures. This provides a rigorous higher-categorical realization of Morse theory compatible with equivariant settings and suggests deep connections to Fukaya-type categories and extended topological field theories.

Abstract

We provide two versions of the statement in the title. First, when the source is the category of smooth manifolds, we construct a simplicial map to a weak Kan complex whose objects are A-infinity coalgebras. Second, when the source is the category of compact Lie groups (or more generally, compact Lie monoids), the target is a weak Kan complex whose objects consist in -bialgebras, as defined in [CHM24]. We also establish a statement describing module structures induced by group actions, as conjectured in [CHM24].
Paper Structure (7 sections, 4 theorems, 6 equations)

This paper contains 7 sections, 4 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Organization of the paper
  4. Some background
  5. Morse theory
  6. Infinity categories
  7. $f$-bialgebras

Key Result

Theorem 1

(biass) Assume that: Then, at the level of Morse complexes:

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Theorem 3
  • Corollary A
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4