ON THE LOCALIC YOSIDA REPRESENTATION OF AN ARCHIMEDEAN LATTICE ORDERED GROUP WITH WEAK ORDER UNIT

[摘要] W is the category of archimedean lattice ordered groups with weak order unit, and F is the category of frames. C:F --> W is the functor which assigns to a given frame F the W-object CF = Hom(F)(OR, F), where OR designates the frame of open sets of the real numbers R, and assigns to the frame homomorphism f:F --> L the W-homomorphism Cf:CF --> CL defined by Cf(g) = fg for all g member of CF. On the other hand, Y:W --> F is the functor which assigns to a given W-object G the frame YG of W-kernels of G, and assigns to the W-homomorphism u:G --> H the frame homomorphism Yu:YG --> YH whose value at K member of YG is the W-kernel of H generated by u(K). We prove in Theorem 2.3.2 that Y is left adjoint to C, and in Theorem 2.4.3 that the adjunction restricts to an equivalence between the full subcategory RL of regular Lindelof frames and the full subcategory C of W-objects of the form CF for some frame F. The equivalence was first established by Madden and Vermeer. The unit mu = (mu-G) is the reflection of W in C, herein termed the (localic) Yosida representation, while the counit lambda = (lambda-F) is the (frame counterpart of the) reflection of locales into regular Lindelof locales. Finally, we note that the sense in which the Yosida representation is unique is precisely that (mu-G, YG) is the C-universal map for G.

[发布日期] 1991-03-15