Categories of algebraic sets
Guided by the theory of commutative algebras over a field \(K\), the author introduces for every algebraic theory \(T\) in the sense of Lawvere-Linton the notion of \(T\)-set, as a set \(X\) together with a \(T\)-subalgebra of \(K^X\), where \(K\) is the initial algebra. There is a dual adjunction between the emerging topological category of \(T\)-sets and the category of \(T\)-algebras, which induces a concrete duality between the fixed subcategories. With a suitable Zariski closure for subsets of \(T\)-sets, the author obtains natural notions of separated and algebraic \(T\)-sets, which are dually equivalent to the functional \(T\)-algebras, i.e., the \(T\)-algebras which are subobjects of powers of \(K\). In his guiding example, the author's setting provides a very natural notion of morphism of algebraic sets over \(K\). He describes the duality not only in this case, but shows that other classical dualities (such as Gelfand-Naimark duality, (spatial frames)\(\sim\) (sober spaces)), fit into his framework.
scientific article; zbMATH DE number 934063
scientific article