nLab localic group

Idea

A localic group is a group object internal to locales.

As the notion of locale is a point-free version of that of topological space, a localic group is much like a topological group, but there are some differences.

The further generalization to groupoids is that of localic groupoids.

Definition

A localic group is a group object in the category of locales. More explicitly, this means we have a locale $G$ with continuous maps $\epsilon:1 \to G$, $\iota:G \to G$, and $\mu:G \times G \to G$, such that $(\mathrm{id}_G, m) \circ m = (m, \mathrm{id}_G) \circ m$, $(\mathrm{id}_G, u_G^1 \circ \epsilon) \circ m = \mathrm{id}_G$, $(u_G^1 \circ \epsilon, \mathrm{id}_G) \circ m = \mathrm{id}_G$, $(\iota, \mathrm{id}_G) \circ m = u_G^1 \circ \epsilon$, and $(\mathrm{id}_G, \iota) \circ m = u_G^1 \circ \epsilon$, where $1$ is the terminal locale and $u_G^1:G \to 1$ is the unique map into $1$.

Localic groups versus topological groups

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, this functor can discard information; for instance, IKPR constructs a nontrivial localic group with only one point.

Moreover, 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.

But 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.

Examples

• Another important source of localic groups is from progroups: cofiltered limits of discrete groups.
• The isotropy group of a topos is naturally regarded as a localic group inside the topos.

Localic subgroups are closed

A remarkable fact about localic groups is the following (Corollary C5.3.2 of the Elephant; this also proves that $\mathbb{Q}$ cannot be a localic group):

Theorem

Any overt localic subgroup of a localic group is weakly closed. If the ambient localic group is in a Boolean topos then any localic subgroup is a closed subgroup.

Sketch of Proof

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.

 See also

• topological group

• localic groupoid

References

• The Elephant, chapter C5.

• John Isbell, Igor Křiž, Aleš Pultr, Jiři Rosický, Remarks on Localic Groups, Lecture Notes in Mathematics 1348 (1988), pp. 154-172: doi

• Graham Manuell, Uniform locales and their constructive aspects, (arXiv:2106.00678)

An expository account of the closed subgroup theorem can be found in

• Gavin Wraith, The Closed Subgroup Theorem