Frames and Grids

M. Barr and M.-C. Pedicchio introduced the category Grids of grids in order to show that the opposite of the category Top of topological spaces is a quasivariety. J. Adamek and M.-C. Pedicchio proved that there exists a duality D between the category TopSys of topological systems (defined by S. Vickers) and the category Grids. In both papers 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 localic topological system (here Loc is the category of locales regarded as a full subcategory of TopSys). 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 characterization of all grids which correspond (in the above sense) to the frames having T1 spectra and a generalization of the well-known fact that the spectrum of a locale is a sober space are obtained as well.