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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