Categories of algebraic sets
type: Item
instance of: scholarly article
title: Categories of algebraic sets
zbMATH Open document ID: 0859.18006
DOI: 10.1007/BF00122260
published in: Applied Categorical Structures
publication date: 1996-10-13T00:00:00Z
review text: 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.
reviewed by: Walter Tholen
Mathematics Subject Classification ID: 18C10
Mathematics Subject Classification ID: 14A99
Mathematics Subject Classification ID: 18B99
zbMATH DE Number: 934063
zbMATH Keywords: concrete duality
zbMATH Keywords: algebraic theory
zbMATH Keywords: dual adjunction
zbMATH Keywords: Zariski closure
zbMATH Keywords: algebraic sets
author: Yves Diers
MaRDI profile type: Publication
cites work: Q5639839
cites work: Duality in analysis from the point of view of triples
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description: scientific article; zbMATH DE number 934063