Convergence and numbers

Our paper is devoted to ernbeddings of the rational numbers Q into exotic groups, linear spaces and fields, all of which carry a complete sequential convergence compatible with the algebraic structure. We enlarge the usual metric convergence on Q and study the impact on the categorical sequential group completion of Q. We compare the completion with the real line. In particular, we construct a convergence compatible with the group structure of Q such that the resulting completion is a Q-linear space and the real numbers are a proper subspace of it.