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.
- Pub Date:
- December 2014
- Mathematics - Category Theory;
- Mathematics - General Topology;
- Mathematics - Logic;
- This new version noticeably differs from the previous one. Many changes, both stylistic and mathematical, were introduced to improve readability. Some sections were cut and new ones have been introduced to compare with the existing literature