Ab is discrete abelian groups. Thus the circle has, for these purposes, the discrete topology. Since you are mapping discrete groups to it, the topology is irrelevant. But the small chu category is much more like hausdorff topological abelian groups (in fact, is equivalent to two full subcategories of them) and is not abelian for similar reasons. One take on this is that separation conditions are more or less incompatible with effective equivalence relations (= monics are kernels in the additive case). And extensionality is incompatible with the dual. On Fri, 19 Jun 1998, Vaughan Pratt wrote:
And to anticipate Vaughan's question, no chu(Ab,circle) is not abelian, roughly for the same reasons as topological ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ and localic abelian groups.
Actually I was going to ask about Ab. I thought it was abelian groups but (maybe I'm just the last to be told) the context suggests Ab = topological abelian groups. Is there life in discrete circles?
(To reconcile the subject line with the underlined line you need to know Mike's usage: the large print giveth and the small print taketh away. Chu is to chu as preordered sets are to posets, or topological spaces to T_0 spaces.)
Vaughan