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.

中文翻译：

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一般仿射附加、Nullstellensätze 和对偶

我们介绍并研究仿射代数几何中标准“系统-解决方案”附加的范畴论抽象。然后，我们在不同的一般性级别上进一步研究这些几何附加，从句法类别到（可能是无限的）代数方程类。在这样做的过程中，我们讨论了由我们的框架引起的二元性与成熟的具体对偶附加理论之间的关系。在一般代数的上下文中，我们证明了 Hilbert 的 Nullstellensatz 的类似物，从而实现了附加代数侧不动点的完整表征。