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progroup

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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 GrpGrp. 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 GrpGrp, but in the larger category TopGrpTopGrp 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 ω 1\omega_1 be the set of countable ordinals, with the reverse of its usual ordering, and for αω 1\alpha\in\omega_1 let S αS_\alpha be the set of strictly increasing functions [0,α][0,\alpha]\to \mathbb{R}. For α<β\alpha\lt \beta, let S βS αS_\beta \to S_\alpha be the restriction. Then each such transition map is surjective, but the inverse limit is empty. The sets S αS_\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 GrpGrp, hence also in TopGrpTopGrp, 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 GG:

  1. GG is a cofiltered limit of discrete groups (considered as discrete localic groups)
  2. GG is a cofiltered limit of discrete groups with surjective transition maps.
  3. The open normal subgroups of GG form a neighborhood base at the identity eGe\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) iI(G_i)_{i\in I} is such an inverse system, then the localic group G=lim iG iG=\lim_i G_i is presented by the following posite. The elements of the underlying poset are pairs (x,i)(x,i) where xG ix\in G_i, with (x,i)(y,j)(x,i)\le (y,j) when iji\le j and f ij(x)=yf_{ij}(x)=y. The coverings are given as follows: for any jj, the element (x,i)(x,i) is covered by the family of all (z,k)(z,k) such that kjk\le j and (z,k)(x,i)(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 iG_i the images of the functions G jG iG_j\to G_i are eventually constant.

By a fundamental fact about locales, if GG is prodiscrete and represented as the limit of a system with surjective transition maps, then the legs GG iG\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)(G_i) and (H j)(H_j), with prodiscrete localic limits GG and HH, we have

Hom LocGrp(G,H)lim jcolim iHom Grp(G i,H j).Hom_{LocGrp}(G,H) \cong \lim_j \colim_i Hom_{Grp}(G_i,H_j).

But since H=lim jH jH = \lim_j H_j, we have Hom LocGrp(G,H)lim jHom LocGrp(G,H j)Hom_{LocGrp}(G,H) \cong \lim_j Hom_{LocGrp}(G,H_j). Thus it suffices to show that any map from GG to a discrete group KK (such as H jH_j) factors through some essentially unique G iG_i.

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

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

LocGrpTopos 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

Revised on February 5, 2011 23:01:26 by Mike Shulman (71.136.232.119)