Localic groups are similar to topological groups, and many examples can be considered as either one. For instance, the real numbers under addition can be considered as either a topological group or a localic group. Since the “space of points” functor is a right adjoint, it preserves limits and hence group objects, so every localic group has an underlying topological group.
However, the “locale of opens” functor does not preserve products, so not every topological group is a localic group—even if its underlying topological space is sober (hence is the space of points of some locale). In particular, the locale of rational numbers (with topology induced from that of ) is not a localic group under addition, because the locale product is “bigger” than the topological-space product (and in particular is not spatial), and the addition map cannot be extended to the locale product. However, if is a locally compact topological group (such as ), then the space product does agree with the locale product (using the ultrafilter principle in the proof), and hence is also a localic group.
A remarkable fact about localic groups is the following (which also proves that cannot be a localic group):
Any subgroup of a localic group is closed.
Details can be found in C5.3.1 of the Elephant, in the more general case of localic groupoids. The basic idea of the proof is to use the fact that the intersection of any two dense sublocales is again dense (a fact which very much fails for topological spaces).
If is a localic subgroup, we construct its closure , which is also a localic subgroup in which is dense. By pullback, it follows that is fiberwise dense? over via the second projection. Applying the automorphism of , we conclude that is also fiberwise dense over via the “composition” map. Dually, is also fiberwise dense over via the “composition” map, and thus (by the basic fact cited above), so is their intersection, which is . Since is an epimorphism, so is . But this map factors through (since is itself a subgroup of ), so that inclusion is also epic. But it is also a regular monomorphism, and hence an isomorphism; thus is closed.
An important generalization of localic groups is to localic groupoids , i.e. internal groupoids in the category of locales. Localic groupoids are important, among other reasons, because every Grothendieck topos can be presented as the topos of equivariant sheaves on some localic groupoid. This fact is due to Joyal and Tierney. For more see classifying topos of a localic groupoid.