Page 102 - Textos de Matemática Vol. 41

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Chapter 12. Entourages and (quasi-)uniformities
G⊕P idG⊕ε HHHHHHHHH
// G⊕G oo ε⊕idG P⊕G m vvvvvvvv
(identity),
(inverses)
p1 HHHHH vvvvv p2 ## G {{v
G⊕G ı⊕idG <<
// G⊕G m
∆ DDDD ""
∆ zzzzzz zzzzzz
z // ε // GPG
DDDD DDDDD
OO
m
//
G⊕G idG⊕ı G⊕G
By drawing diagrams it is easy to see that the usual properties for groups
ı2 =idG, iε=ε, m(ı⊕ı)r=ım (12.8.1)
are also valid here, where r is the automorphism G ⊕ G → G ⊕ G interchanging the two product projections G ⊕ G → G.
A homomorphism f : G → H of localic groups is a localic map of the underlying locales compatible with the operations of G and H, that is, such that
mH(f⊕f)=fmG, ιHf=fιG, εH =fεG.
12.9. Localic subgroups. When does a sublocale S of a localic group G give a subgroup of G, that is, when does S have a group structure such that the embedding jS : S → G is a group homomorphism?
We need
˜ι:S→S, m˜:S⊕S→S and ε˜:P→S
s u c h t h a t j S · ˜ι = ι · j S , j S · ε˜ = ε a n d j S · m˜ = m ( j S ⊕ j S ) . I t i s e a s y t o s e e t h a t the conditions for S to be a subgroup of G are precisely
jS∗ ·ι·jS =ι·jS, jS∗ ·ε=ε and jS∗ ·m(jS ⊕jS)=m(jS ⊕jS) (12.9.1)
(indeed, given these properties, the group axioms follow for S).
It is now easy to check that the closure of a subgroup is a subgroup.