Page 181 - Textos de Matemática Vol. 46

NOTES ON POINT-FREE REAL FUNCTIONS AND SUBLOCALES 171
of a frame homomorphism h = f ⇤ : M ! L. This can be done since frame ho- momorphisms preserve suprema; but of course not every mapping preserving infima is a localic one. Here is a characterization (see [27] or [26]).
Letf:L!M havealeftadjointf⇤:M!L.Thenitisalocalicmapi↵ (1) f[Lr{1}]✓Mr{1},and
(2) f(f⇤(a)!b)=a!f(b)
(! is the Heyting operation in the frames L resp. M).
2.2. The frame of sublocales. A sublocale of a frame L is a subset S ✓ L such that V
(1) M✓Simplies M2S,and
(2) ifa2Lands2Sthena!s2S.
This concept expresses the intuition of a natural subobject of L understood as a generalized space; in the category of locales and localic maps the inclusion j : S ✓ L is a localic extremal monomorphism, hence indeed a sub-locale (in the frame perspective, it is the image of a nucleus). The set of all sublocales ordered by inclusion, denoted by
S (L), is a co-frame, with the lattice operations
V Si = T Si and W Si ={VA|A✓ S Si}. i2J i2J i2J i2J
We have the closed resp. open sublocales
c(a)="a resp. o(a)={x|a!x=x}={a!x|x2L}
modelling closed resp. open subspaces. They are complements of each other, and the o(a) are in a natural one-one correspondence with the elements of L, preserving joins and finite meets.
We will need, rather, the opposite of S(L), the frame of sublocales, denoted
S(L),
W Si = T Si and V Si ={VA|A✓ S Si}.
withST i↵S◆T and
i2J i2J i2J i2J
Note that S(L) is isomorphic with the frame of congruences on L and we have a natural frame embedding
cL : L ! S(L) (a 7! c(a)). (2.2.1)
This means that there is a one-one correspondence between the elements of L and the closed sublocales of L, agreeing in arbitrary joins and finite meets in