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\({\mathcal T}op^{op}\) is a quasi-variety. (English) Zbl 0819.18002

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 \(U\)).

MSC:

18B30 Categories of topological spaces and continuous mappings (MSC2010)
06D99 Distributive lattices
08C15 Quasivarieties
18C10 Theories (e.g., algebraic theories), structure, and semantics
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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
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