Left adjoint to forgetful functor from topological rings to topological abelian groups

Question

Does the forgetful functor from the category of topological rings to the category of topological abelian groups, $U: \mathbf{TopRing} \to \mathbf{TopAb}$, have a left adjoint and if so, what is it? A reference would be very much appreciated.

Answer 1

Yes.

If you have a functor from a topological category, and that functor has a left adjoint if you forget about the topological structure, then frequently (under quite mild assumptions) you can "lift" that left adjoint to the topological case.

One nice reference is Tholen's On Wyler Taut's Lift Theorem, which proves an even more general version of this result. Here's the special case of interest (originally due to Wyler):

Given a commutative diagram of functors

where $T$ and $T'$ are topological and $\tilde{U}$ preserves initial sources, then if $U$ has a left adjoint, $\tilde{U}$ does too!

For us, $A$ and $A'$ are the categories of topological rings and groups (respesctively). These are both topological over rings, and groups, which are our $X$ and $X'$.

Then $U$ and $\tilde{U}$ should be the forgetful functors, and it's clear that initial topologies get sent to initial topologies by $\tilde{U}$ (since it's not touching the topology at all!).

So the left adjoint to $U$ lifts to a left adjoint of $\tilde{U}$, as desired.

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I hope this helps ^_^