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 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
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