ABSTRACT
COROLLARY 2.3.1. The c a teg o ry 0 r d of o rde red se ts a nd ideals is a q u a n ta lo id a n d we h a v e b ie q u iv a le n c e s (of bicategories) 0 rd s Mon(Kel) = Bim(!Rel).
Proof: (1) Since 0 rd = Mon(!Rel), by Prop. 2.3.3. it is a quanta lo id an d by th e w ork in [CKW], Bim(ftel) = Mon(Matr(!Rel)) = Mon(Matr(Matr(2)) = Mon(Matr(2) = Qrd ■
M ore exa m p les : (1) Let stf be a locally small c a teg o ry and consider the quantaloid P(s4). Then, a m onad in P(stf) is j u s t an object a € $4, toge ther w i th a submonoid S of stf(a,a). If S is a m o n a d on a and T is a n o th e r m o n a d on b, t h e n a m o n a d m o rp h ism A: S —» T is a set of m aps A C P(a,b), sa tisfying t h a t S°A c A and A°T c A.
