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.
