General affine adjunctions, Nullstellens\"atze, and dualities

TL;DR

This paper develops a categorical framework for affine algebraic geometry, generalizing classical dualities and Nullstellensatz, and explores their relationships across various algebraic structures.

Contribution

It introduces a category-theoretic abstraction of the system-solution adjunction and proves an analogue of Hilbert's Nullstellensatz within this framework.

Findings

Established a categorical abstraction of affine adjunctions.

Proved a Nullstellensatz analogue for general algebraic structures.

Characterized fixed points in the algebraic side of the adjunction.

Abstract

We introduce and investigate a category-theoretic abstraction of the standard "system-solution" adjunction in affine algebraic geometry. We then look further into these geometric adjunctions at different levels of generality, from syntactic categories to (possibly infinitary) equational classes of algebras. In doing so, we discuss the relationships between the dualities induced by our framework and the well-established theory of concrete dual adjunctions. In the context of general algebra we prove an analogue of Hilbert's Nullstellensatz, thereby achieving a complete characterisation of the fixed points on the algebraic side of the adjunction.