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 . 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 , but in the larger category 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 be the set of countable ordinals, with the reverse of its usual ordering, and for let be the set of strictly increasing functions . For , let be the restriction. Then each such transition map is surjective, but the inverse limit is empty. The sets 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 , hence also in , is trivial.
The following are equivalent for a localic group :
This can be found in (Moerdijk).
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 is such an inverse system, then the localic group is presented by the following posite. The elements of the underlying poset are pairs where , with when and . The coverings are given as follows: for any , the element is covered by the family of all such that and .
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 the images of the functions are eventually constant.
By a fundamental fact about locales, if is prodiscrete and represented as the limit of a system with surjective transition maps, then the legs of the limiting cone are also surjective (i.e. they are represented by injective frame homomorphisms). This is false for limits of topological spaces.
The category of prodiscrete localic groups is equivalent to the category of surjective progroups.
In view of the above proposition it suffices to show that for surjective progroups and , with prodiscrete localic limits and , we have
But since , we have . Thus it suffices to show that any map from to a discrete group (such as ) factors through some essentially unique .
But if is such a map, then is an open normal subgroup of . And if are the projections, then the kernels are a neighborhood base at , so we have for some , hence factors through . Finally, this last is isomorphic to , since is an open surjection of locales.
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).