The Morse complex is an $\infty$-functor

TL;DR

This paper demonstrates that Morse complexes of certain mathematical structures can be organized into an $ abla$-functor, revealing deep algebraic structures and functorial properties in Morse theory.

Contribution

It introduces an $ abla$-functor from Morse complexes of compact Lie monoids to $f$-bialgebras, extending to manifolds and group actions with new algebraic structures.

Findings

Morse complex of a compact Lie monoid forms an $f$-bialgebra.

The assignment defines an $ abla$-functor.

Constructs $ abla$-functors for manifolds and group actions.

Abstract

We show that the Morse complex of a compact Lie monoid can be given the structure of an $f$-bialgebra, a chain-level version of bialgebras introduced in [CHM24]; and that this assignment defines an $\infty$-functor. As a consequence, we obtain two other $\infty$-functors mapping closed smooth manifolds to their Morse complexes with their $A_\infty$-coalgebra structures; and closed smooth manifolds with compact Lie group actions to their Morse complexes, with a ``$u$-bimodule'' structure (a bimodule version for $f$-bialgebras).