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Groupoidal completely distributive lattices. (English) Zbl 0936.18004
The recently published and refined proof that top\(^{op}\) is a quasi-variety [M. Barr and M. C. Pedicchio, Cah. Topologie Géom. Différ. Catégoriques 36, No. 1, 3-10 (1995; Zbl 0819.18002), and Appl. Categ. Struct. 4, No. 1, 81-85 (1996; Zbl 0879.18008); J. Adámek and M. C. Pedicchio, Cah. Topologie Géom. Différ. Catégoriques 38, No. 3, 217-226 (1997; Zbl 0882.18003)] is based on constructing a variety using the non-full inclusion of the category caba of complete atomic boolean algebras into the category frm of frames. Replacing the latter by the category cd of completely distributive lattices (with morphisms that preserve all infima and suprema) yields a proof that ord\(^{op}\) is a quasi-variety. In order to transfer this result to the constructive setting of an elementary topos \(\mathcal S\), cd needs to be replaced by the category ccd\(\mathcal S\) of constructive completely distributive (CCD) lattices relative to \(\mathcal S\). These were introduced by the second author together with B. Fawcett [Math. Proc. Camb. Philos. Soc. 107, No. 1, 81-89 (1990; Zbl 0694.06008)] and then studied in great detail together with R. Rosebrugh [Math. Proc. Camb. Philos. Soc. 110, No. 2, 245-249 (1991; Zbl 0743.06009); Can. Math. Bull. 35, No. 4, 537-547 (1992; Zbl 0760.06005); and Appl. Categ. Struct. 2, No. 2, 119-144 (1994; Zbl 0804.06013)]. This program raises various interesting questions concerning CCD lattices, in particular how to characterize power object lattices in CCD terms. These turn out to be groupoidal in the sense of A. Carboni and R. F. C. Walters [J. Pure Appl. Algebra 49, 11-32 (1987; Zbl 0637.18003)].

18B25 Topoi
06D10 Complete distributivity
08C15 Quasivarieties
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