Taylor Morphisms

TL;DR

This paper develops a generalized framework for Taylor morphisms in differential rings, extending previous constructions to arbitrary differential rings and characterizing all such morphisms.

Contribution

It extends the twisted Taylor morphism construction to all differential rings and provides a comprehensive characterization of generalized Taylor morphisms.

Findings

Constructs a functor as the right adjoint to a forgetful functor.

Generalizes the twisted Taylor morphism to arbitrary differential rings.

Provides a concrete characterization of all generalized Taylor morphisms.

Abstract

We study generalised Taylor morphisms, functors which construct differential ring homomorphisms from ring homomorphisms in a uniform way, analogous to the Taylor expansion for smooth functions. We generalise the construction of the twisted Taylor morphism by Le\'on S\'anchez and Tressl to arbitrary differential rings by `twisting' the ring of Hurwitz series, and prove that this results in a functor which is the right adjoint to a certain forgetful functor. We therefore give a concrete characterisation of all generalised Taylor morphisms over all differential rings with finitely many commuting derivations.