4 A. JOYAL $ M. TIERNEY By duality, * Hom(M,Pl°). 4. Sub and Quotient Lattices As we know from our calculation of limits, all monomorphisms of are injective, and the sub-objects of a sup-lattice M are the subsets S CI M closed under suprema. For any morphism f: M - * N, the set-image of f, f(M) c_ N, is a subobject, since it is closed under suprema. Moreover, the composite a = ff*: N N is a coclosure operator on N, i.e. 1) x _ y = ct(x) _ ot(y) 2) a(x) £ x 2 3) a (x) = a (x) . Also, f(M) = N = {y e N|a(y) = y}. In fact, we have Proposition 1. There is a natural, order preserving isomorphism between the set of subobjects of N and the set of coclosure operators on N. Proof : To a coclosure operator a: N N, we assign the subobject N = {y e N|a(y) = y}. To a subobject S c:N, we assign the coclosure operator a : N - N defined by as(x) =\/{y e S|y _ x} . By duality, given f: M * N, the composite 3 = f*f: M M is a closure operator on M, i.e. 1) x _ y = 3(x) _ 3(y) 2) x £ S(x) 3) B2(x) = B(x). Also, the set M0 = {x e M|3(x) = x} P is canonically isomorphic to f(M). We have Proposition 2. The following are canonically isomorphic: 1) The set of all quotients of M. 2) The set of all subsets QCM , which are closed under infima. 3) The set of all closure operators 3 on M. Here, the isomorphism between 1) and 2) is order preserving, but those between 2) and 3), and 1) and 3), are order reversing. Proof : Let us just remark, that given a subset Q CLM closed under
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