Barr, Michael; Pedicchio, M. Christina \({\mathcal T}op^{op}\) is a quasi-variety. (English) Zbl 0819.18002 Cah. Topologie Géom. Différ. Catég. 36, No. 1, 3-10 (1995). The authors show that the dual of the category of topological spaces and continuous maps is a quasi-variety, that is a regular-epi-reflective subcategory of a variety, or equivalently a category of algebras definable by (infinitary) Horn sentences. This result is in principle well-known, since \({\mathcal T}op^{\text{op}}\) is a cocomplete regular category with a regular projective generator, and such categories are known to be quasivarieties. However, the main interest of the paper lies in the construction of a new variety of “grids”, which are frames with an extra unary operation satisfying certain equations, and in a proof that \({\mathcal T}op^{\text{op}}\) is regular-epi-reflective in the variety of grids. (The grid corresponding to a space \(X\) is the lattice of pairs \((U,A)\), where \(U\) is an open subset of \(X\) and \(A\) an arbitrary subset of \(U\)). Reviewer: P.T.Johnstone (Cambridge) Cited in 4 ReviewsCited in 4 Documents MSC: 18B30 Categories of topological spaces and continuous mappings (MSC2010) 06D99 Distributive lattices 08C15 Quasivarieties 18C10 Theories (e.g., algebraic theories), structure, and semantics Keywords:dual of category of topological spaces; infinitary Horn sentences; category of algebras; cocomplete regular category; quasivarieties; frames PDFBibTeX XMLCite \textit{M. Barr} and \textit{M. C. Pedicchio}, Cah. Topologie Géom. Différ. Catégoriques 36, No. 1, 3--10 (1995; Zbl 0819.18002) Full Text: Numdam EuDML References: [1] M. Barr ( 1989 ), Models of Horn theories . In J. W. Gray, ed., Categories in Computer Science and Logic , Contemporary Math. 92 , 1 - 7 , Amer. Math. Soc. MR 1003191 | Zbl 0676.03022 · Zbl 0676.03022 [2] P.T. Johnstone ( 1982 ), Stone Spaces . Cambridge University Press . MR 698074 | Zbl 0499.54001 · Zbl 0499.54001 [3] M.C. Pedicchio (to appear), On k-pcrintitability for categories of P-algebras . · Zbl 0863.08010 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.