nLab
progroup

Definition

A progroup is a pro-object in the category Grp of groups. In other words, it is a formal cofiltered limit of groups.

Surjective progroups versus localic groups

Of course, the category Grp is complete, but in general a progroup represented by some cofiltered diagram of groups is not equivalent to the actual limit of that diagram in $\mathrm{Grp}$. However, profinite groups (i.e. cofiltered systems of finite groups) can be identified with actual limits of finite groups if we take those limits, not in $\mathrm{Grp}$, but in the larger category $\mathrm{TopGrp}$ of topological groups. The resulting topological groups are precisely those with Stone topologies.

This is not true for pro-systems of non-finite groups, even if we restrict to those with surjective transition maps. The following counterexample is due to Higman and Stone, and is reproduced in (Moerdijk). Let ${\omega }_1$ be the set of countable ordinals, with the reverse of its usual ordering, and for $\alpha \in {\omega }_1$ let $S_{\alpha }$ be the set of strictly increasing functions $[0,\alpha ]\to ℝ$. For $\alpha <\beta$, let $S_{\beta }\to S_{\alpha }$ be the restriction. Then each such transition map is surjective, but the inverse limit is empty. The sets $S_{\alpha }$ are not groups, but if we take the free vector space on each of them, we obtain a nontrivial pro-group with surjective transition maps whose limit in $\mathrm{Grp}$, hence also in $\mathrm{TopGrp}$, is trivial.

However, we do get an embedding on pro-groups with surjective transition maps if instead of Top we take the limit in the category Loc of locales.

Proposition

The following are equivalent for a localic group $G$:

1. $G$ is a cofiltered limit of discrete groups (considered as discrete localic groups)
2. $G$ is a cofiltered limit of discrete groups with surjective transition maps.
3. The open normal subgroups of $G$ form a neighborhood base at the identity $e\in G$.

Proof

This can be found in (Moerdijk).

Definition

A localic group with these properties is called prodiscrete.

We may as well assume that any surjective progroup is indexed on a directed poset. If $(G_i{)}_{i\in I}$ is such an inverse system, then the localic group $G={\lim }_i{G}_i$ is presented by the following posite. The elements of the underlying poset are pairs $(x,i)$ where $x\in G_i$, with $(x,i)\le (y,j)$ when $i\le j$ and $f_{\mathrm{ij}}(x)=y$. The coverings are given as follows: for any $j$, the element $(x,i)$ is covered by the family of all $(z,k)$ such that $k\le j$ and $(z,k)\le (x,i)$.

Definition

A surjective progroup, also called a strict progroup, is a progroup whose cofiltered diagram consists of surjections.

One can show that a progroup is isomorphic to a surjective one, in the category of pro-groups, if and only if it satisfies the Mittag-Leffler condition: for each $G_i$ the images of the functions $G_j\to G_i$ are eventually constant.

By a fundamental fact about locales, if $G$ is prodiscrete and represented as the limit of a system with surjective transition maps, then the legs $G\to G_i$ of the limiting cone are also surjective (i.e. they are represented by injective frame homomorphisms). This is false for limits of topological spaces.

Theorem

The category of prodiscrete localic groups is equivalent to the category of surjective progroups.

Proof

In view of the above proposition it suffices to show that for surjective progroups $(G_i)$ and $(H_j)$, with prodiscrete localic limits $G$ and $H$, we have

$${\mathrm{Hom}}_{\mathrm{LocGrp}}(G,H)\cong \underset{j}{\lim }{colim}_i{\mathrm{Hom}}_{\mathrm{Grp}}(G_i,H_j).$$Hom_{LocGrp}(G,H) \cong \lim_j \colim_i Hom_{Grp}(G_i,H_j).

But since $H={\lim }_j{H}_j$, we have ${\mathrm{Hom}}_{\mathrm{LocGrp}}(G,H)\cong {\lim }_j{\mathrm{Hom}}_{\mathrm{LocGrp}}(G,H_j)$. Thus it suffices to show that any map from $G$ to a discrete group $K$ (such as $H_j$) factors through some essentially unique $G_i$.

But if $f:G\to K$ is such a map, then $\ker (f)$ is an open normal subgroup of $G$. And if $p_i:G\to G_i$ are the projections, then the kernels $\ker (p_i)$ are a neighborhood base at $e$, so we have $\ker (p_i)\subseteq \ker (f)$ for some $i$, hence $f$ factors through $G/\ker (p_i)$. Finally, this last is isomorphic to $G_i$, since $p_i:G\to G_i$ is an open surjection of locales.

Any localic group $G$ has a classifying topos consisting of continuous $G$-sets, i.e. discrete locales with a $G$-action. In general, the resulting functor

$$\mathrm{LocGrp}\to \mathrm{Topos}$$LocGrp \to Topos

is not an embedding into Topos, but it can be shown to be so when restricted to prodiscrete localic groups. One can also characterize the toposes that are sheaves on a prodiscrete localic group as the Galois toposes.

Most of these results have corresponding facts for pro-groupoids and prodiscrete localic groupoids. However, in full generality, the category of (even surjective) pro-groupoids does not embed into localic groupoids, since the category of pro-sets (= categorically discrete pro-groupoids) does not embed into locales (= categorically discrete localic groupoids).

References

• Ieke Moerdijk, Prodiscrete groups and Galois toposes