Contents

category theory

# Contents

## 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 $Grp$. 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 $Grp$, but in the larger category $TopGrp$ 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 $[0,\alpha]\to \mathbb{R}$. For $\alpha\lt \beta$, 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 $Grp$, hence also in $TopGrp$, 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) 1. $G$ is a cofiltered limit of discrete groups with surjective transition maps. 1. 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 $f_{ij}(x)=y$. 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

$Hom_{LocGrp}(G,H) \cong \lim_j \colim_i Hom_{Grp}(G_i,H_j).$

But since $H = \lim_j H_j$, we have $Hom_{LocGrp}(G,H) \cong \lim_j Hom_{LocGrp}(G,H_j)$. 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 $f\colon G\to K$ is such a map, then $ker(f)$ is an open normal subgroup of $G$. And if $p_i\colon G\to G_i$ 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 $p_i\colon G\to G_i$ 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

$LocGrp \to Topos$

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).