We introduce an algebraic structure with the purpose of modelling an arbitrary space with a suitable notion of geodesic path for every two points in it. We prove that this structure satisfies a weak Mal'tsev property, and show that any smooth surface in which every two points are linked by a unique geodesic path, in the sense of differential geometry, is an example of these kind of structures. The main application is intended to be in the theory of computation where a space with geodesics can now be considered as a simple algebraic structure satisfying some carefully chosen axioms. Joint work with J. P. Fatelo.