Non-linear homomorphisms of algebras of functions are induced by thick morphisms

TL;DR

This paper proves that non-linear algebra homomorphisms of smooth functions are generated by thick morphisms, extending classical duality concepts to a non-linear context within the framework of supermanifolds.

Contribution

It confirms the conjecture that all non-linear homomorphisms are induced by thick morphisms, broadening the understanding of algebraic structures in supergeometry.

Findings

Non-linear homomorphisms are generated by thick morphisms in the class of formal functionals.

Extends classical duality to non-linear algebra homomorphisms.

Provides a new perspective on the structure of function algebras in supergeometry.

Abstract

In 2014, Voronov introduced the notion of thick morphisms of (super)manifolds as a tool for constructing $L_{\infty}$-morphisms of homotopy Poisson algebras. Thick morphisms generalise ordinary smooth maps, but are not maps themselves. Nevertheless, they induce pull-backs on $C^{\infty}$ functions. These pull-backs are in general non-linear maps between the algebras of functions which are so-called 'non-linear homomorphisms'. By definition, this means that their differentials are algebra homomorphisms in the usual sense. The following conjecture was formulated: an arbitrary non-linear homomorphism of algebras of smooth functions is generated by some thick morphism. We prove here this conjecture in the class of formal functionals. In this way, we extend the well-known result for smooth maps of manifolds and algebra homomorphisms of $C^{\infty}$ functions and, more generally, provide an analog of classical 'functional-algebraic duality' in the non-linear setting.