2 A. JOYAL $ M. TIERNEY f* is easily calculated as f*(7) = \A* e M|f(x) _ y}. As a right adjoint, f* must preserve infima, and so defines a morphism of sup-lattices f°: - M°. Clearly, (M°)° » M, (f°)° = f, and (fg)° = g°f°. Also, f _ g, iff g* f* iff £° _ g°. Hence Hom(M,N) = Hom(N°,M°). Thus, we have ( )°: si -* Proposition 1. The contravariant functor is a (strong) self-duality. 2• Limits and colimits The calculation of limits in si is very easy: the product I M. iel x is the product over I of the sets M- with the coordinatewise partial order the equalizer of a pair of morphisms f,g: M -* N is the subset {x e M|f(x) - g(x)} with the induced partial order. In short, limits in si are calculated in S. It follows that monomorphisms in si are injective, i.e. monomorphisms in S. In fact, let m: M - * N be a monomorphism of si. We have mm* _ 1, and 1 _ m*m. Hence, m = m(m*m) , and so 1 = m*m. Thus, monomorphisms of si have retractions in S. By duality, epimorphisms have sections. In general, the duality is extremely useful for the calculation of colimits in si - usually much more difficult than limits. For coproducts, consider the product n M. of a family of iel x sup-lattices. The projections p.: I M. M. are calculated pointwise, 1 iel x x as are the infima in I M. . Thus, the p. preserve infima, and therefore iel 1 x have left adjoints u. : M. - I M.. When I is decidable, the u. are 1 1 iel 1 x given by the classical formula: u.(x.) = (x.) where x. = 0 j f i, 1 1 J jel J x. = x. In the general case we have Proposition 1. The product I M. , equipped with the left adjoints iel u.: M. n M. to the projections p., is the coproduct JIM.. 1 x iel x 1 iel 1 Moreover, for each iel p.y. = 1. Proof: It is enough to show that \i : I M. -* M. is the projection. x iel x x
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