Frames and Grids

M. Barr and M.C. Pedicchio (4) introduced the category Grids of grids in order to show the opposite of the category Top of topological spaces is a quasivariety. J. Adamek and M.c. Pedicchio proved in (2) that there exist a duality D between the category TopSys of topological systems (defined by S.Vickers (11) and the category Grids. In both papers (4) and (2) a description of the full subcategory D(Top) of the category Grids is given. In this paper we describe internally all grids isomorphic to the objects of the full coreflective subcategory D(Loc) of the category Grids, i.e. we characterize internally all grids of the form D(C), where C is a localis topological system (here Loc is the category of locales regarded as a full subcategory of TopSys (see 11). Since, obviously, the category Frm of frames is equivalent to D(Loc), we can say that in this paper those grids which could be called frames are characterized internally. An internal chacterization of all grids which correspond (in the above sense) to the frames having T1 spectra and a generalization of the well-known facty that the spectrum of a locale is a sober space are obtained as well.