ABSTRACT
LEMMA 3.2.2. If j is a quantaloidal nucleus on P(stf), th en Oj is a congruence on stf.
Proof: The fact t h a t each (§j)a ,b *s a congruence is imm edia te . Now, suppose j a ,b(f) = Ja.b^g) a n d h: b-»c is a m o rp h ism of srf. Then, h°f € Jb,c(h)oja b(f) = jb ,c( h ) ° j a ,b(g) S j a ,c(h°g). Thus, Ja,c(h°f) C j aiC(hog). The opposite c o n ta in m en t follows similarly, t h u s p rov ing (2) in Def. 3.1.. (3) follows using analogous a rg u m en ts . ■
Let Con(sO denote the lat t ice of congruences on s4 and let Tl(P(9f)) denote the lattice of quanta loidal nuclei on P(stf). Using L em m as 3.2.1. and 3.2.2., we obtain order preserving mappings F: Con(90 —» Tl(P(stf)) and G: Tl(P(stf)) —* Con(s0 given by F(o) = j e and G(j) = 0j. We have the following adjunction.
