The category of topological abelian groups is not abelian. The reason is not hard to explain. In topological spaces, points are primordial and a monomorphism of points need not be a monomorphism of the attached frames (that is a surjection of the open set lattices). When it is, then ignoring the usual separation axiom, it is a subspace and thereby a kernel. In frames, the reverse is true. Now the open sets are all and you ignore the points. In Chu(Ab,circle), the points and open sets are treated with equal respect. Now a monomorphism must be injective on the points and surjective on the opens and is obviously a kernel. The dual is also true. At another conceptual level, abelianess is defined by certain exactness condition, which concerns canonical arrows, usually between limits and colimits. A category is pointed if the canonical map 0 --> 1 is an isomorphism and there is then a canonical arrow A + B --> A x B and when that is an isomorphism, it is additive. There is a map from the domain of any monomoprhism to the kernel of its cokernel.... The conditions are self dual and a limit is computed in a Chu category as the limit of its first component and colimit of its second and all the required isomoprhisms remain isomorphisms. Of course, this is just as true of Chu(A,_|_) whenever A is abelian (and, of course, closed monoidal). The contrast with the case of topological and that of localic abelian groups is striking. I realized all this as a result of listening to Peter Freyd's lecture in Saint John, NB last week. He was trying to discover the initial abelian category with one object. It is self dual, but not this one since this contains no non-zero bijective (that is objects that are simultaneously injective and projective) while Freyd's category has enough of them. Still it might be interesting. And to anticipate Vaughan's question, no chu(Ab,circle) is not abelian, roughly for the same reasons as topological and localic abelian groups.