General affine adjunctions, Nullstellensätze, and dualities

We develop an abstract categorical framework that generalises the classical "system-solution" adjunction in algebraic geometry, proving that such adjunctions take place in a multitude of contexts. We then look further into these geometric adjunctions at different levels of generality, from syntactic categories to equational classes of algebras. In doing so, we discuss the relationships between the dualities induced by our framework and the theory of dualities generated by a "schizophrenic" object. Notably, classical dualities like Stone duality for Boolean algebras, Gelfand duality for commutative C^∗-algebras, Pontryagin duality for Abelian groups, turn out to be special instances of this framework. To determine how such general adjunctions restrict to dualities we prove abstract analogues of Hilbert's Nullstellensatz and Gelfand-Kolmogorov-Stone lemma, completely characterising the fixed points on one side of the adjunction.