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°: N° - 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 -* » s£

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 T 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 I M. -» • M. are calculated pointwise,

1 iel x x

as are the infima in I I M. . Thus, the p. preserve infima, and therefore

iel 1 x

have left adjoints u. : M. - I 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 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 I M. -* • M. is the projection.

x iel x x