localic group

The notion of locale is a “point-less” version of that of topological space. A *localic group* is much like a topological group, but there are some differences.

A **localic group** is a group object in the category of locales.

Localic groups are similar to topological groups, and many examples can be considered as either one. For instance, the real numbers $\mathbb{R}$ under addition can be considered as either a topological group or a localic group. Since the “space of points” functor $Loc \to Top$ 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 $Top\to Loc$ 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 $\mathbb{Q}$ of rational numbers (with topology induced from that of $\mathbb{R}$) is not a localic group under addition, because the locale product $\mathbb{Q}\times_l \mathbb{Q}$ is “bigger” than the topological-space product (and in particular is not spatial), and the addition map $\mathbb{Q}\times \mathbb{Q}\to \mathbb{Q}$ cannot be extended to the locale product. However, if $G$ is a locally compact topological group (such as $\mathbb{R}$), then the space product $G\times G$ does agree with the locale product (using the ultrafilter principle in the proof), and hence $G$ is also a localic group.

- Another important source of localic groups is from progroups: cofiltered limits of discrete groups.

A remarkable fact about localic groups is the following (which also proves that $\mathbb{Q}$ 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 $H\rightarrowtail G$ is a localic subgroup, we construct its closure $\bar{H}$, which is also a localic subgroup in which $H$ is dense. By pullback, it follows that $H\times \bar{H} \to \bar{H} \times \bar{H}$ is fiberwise dense? over $\bar{H}$ via the second projection. Applying the automorphism $(g,h) \mapsto (g,g^{-1}h)$ of $G\times G$, we conclude that $H\times \bar{H} \to \bar{H} \times \bar{H}$ is also fiberwise dense over $\bar{H}$ via the “composition” map. Dually, $\bar{H}\times H \to \bar{H} \times \bar{H}$ is also fiberwise dense over $\bar{H}$ via the “composition” map, and thus (by the basic fact cited above), so is their intersection, which is $H\times H$. Since $\bar{H}\times \bar{H}\to \bar{H}$ is an epimorphism, so is $H\times H\to\bar{H}$. But this map factors through $H\rightarrowtail \bar{H}$ (since $H$ is itself a subgroup of $G$), so that inclusion is also epic. But it is also a regular monomorphism, and hence an isomorphism; thus $H$ 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.

- The Elephant, chapter C5.

Revised on October 28, 2010 04:29:08
by Mike Shulman
(71.137.3.108)