Lie monoid actions in Morse theory

Let $G,G',H,H'$ be compact Lie monoids, let $X,X'$ be closed smooth manifolds acted on by $G\times H$ and $G'\times H'$ respectively, let $\varphi\colon G\to G'$ and $\psi\colon H\to H'$ be morphisms of Lie monoids, and let $f\colon X\to X'$ be $(\varphi,\psi)$ bi-equivariant. Write $CM(-)$ for the corresponding Morse complexes. Lie monoid actions in Morse theory. The complexes $CM(G),CM(G'),CM(H),CM(H')$ are f-bialgebras; $CM(X),CM(X')$ are $u$-bimodules of f-bialgebras; $\varphi$ and $\psi$ induce morphisms of f-bialgebras

$$\varphi_*\colon CM(G)\to CM(G'),\qquad \psi_*\colon CM(H)\to CM(H');$$

and $f$ induces a $(\varphi_*,\psi_*)$ bi-equivariant morphism of $u$-bimodules

$$f_*\colon CM(X)\to CM(X').$$

This is the proposed chain-level compatibility between Lie monoid actions and the multiplicative structures of Morse theory; the paper states that the relevant notions are developed in its algebraic framework and that a proof is planned in forthcoming work.