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If $(G_i,f_{ij})$ is a direct system of topological groups, is it always the case that the topological\group-theoretical direct limit $G:=\varinjlim_iG_i$ is a topological group? (The topology on $G$ is the final topology with respect to the canonical homomorphisms $\psi_i:G_i\rightarrow G$). It is immediate from the group structure on $G$ and the definition of its topology that the inversion map is continuous. Moreover, I'm pretty certain that if the transition maps $f_{ij}$ are open, then the $\psi_i$ are open, and this gives continuity of multiplication. It's not at all clear to me that this condition is necessary though.

I have always sort of assumed that this was true, but it's never really been an issue because I never start with a system of topological groups and take the direct limit; I always have a topological group and express it as a direct limit (e.g. the idele group of a global field, or the Cartier dual of a finite free $\mathbb{Z}_p$-module).

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3 Answers

up vote 5 down vote accepted

You can construct colimits by a rather general procedure. See my answer here.

The topologies mentioned by Agusti are not correct. The continuity of the group laws does not hold in general. The problem is that quotient topologies do not commute with product topologies.

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Ok. You're right. Nobody defines the coproduct topology as I do in my second attempt and for a reason: if I'm not wrong, when I said, up there, "identify $\psi_i \times \psi_i$ with...", I assumed, without noticing it, that the product topology on $G\times G$ was the same as the final topology induced by the $\psi_i\times \psi_i$'s. But the last one seems to be strictly finer. –  a.r. Sep 21 '10 at 10:50
Thank you Martin. –  Keenan Kidwell Sep 21 '10 at 15:44
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Examples of direct systems $(G_i,f_{i,j})$ of topological groups for which the direct limit topology DOES NOT make $G:=\lim_{i \to \infty} G_i$ a topological group are described in the following paper. N. Tatsuma, H. Shimomura and T. Hirai, On group topologies and unitary representations of inductive limits of topological groups and the case of the groups of diffeomorphisms, J. Math. Kyoto Univ., 38-3 (1998) 551-578. However, it is established in the same paper (Theorem 2.7) that, if the index set is countable and the groups $G_i$ locally compact, then $G$ IS a topological group.

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The following paper (available here)

R. Brown and J.P.L. Hardy, `Topological groupoids I: universal constructions'', Math. Nachr. 71 (1976) 273-286.

deals with the more general case of topological groupoids, and constructs colimits by an adjoint functor type method. Similar methods apply for topological groups. This method gives very little hold on the actual topology, but it just shows that it exists. On the other hand, the property one often wants is just the universal property.

What is true that in general the topology is not given by an identification topology, because the product of identification maps is in general not an identification map.

An alternative procedure is work in what is called a convenient category of topological spaces, i.e. one which is cartesian closed. An example is the category of compactly generated spaces. The cartesisn closed property implies that the product of identification maps in this category is an identification map.

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