Symmetric delta lenses and spans of asymmetric delta lenses

Bidirectional Transformations provide mechanisms for maintaining synchronization between updatable data sources. Lenses are certain mathematically specified bidirectional transformations. As part of a project to unify the treatment of symmetric lenses (of various kinds) as equivalence classes of spans of asymmetric lenses (of corresponding kinds), we relate symmetric delta lenses with spans of asymmetric delta lenses. Because delta lenses are based on state spaces which are categories rather than sets, there is further structure that needs to be accounted for. One of the main findings in this paper is that the required equivalence relation among spans is compatible with, but coarser than, the one expected. The main result is an isomorphism of categories between a category whose morphisms are equivalence classes of symmetric delta lenses (here called fb-lenses) and the category of spans of delta lenses modulo the new equivalence.