4

A. JOYAL $ M. TIERNEY

By duality,

M° * Hom(M,Pl°).

4. Sub and Quotient Lattices

As we know from our calculation of limits, all monomorphisms of s£

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