Delta Lenses and Opfibrations

We compare the delta lenses, also known as d-lenses, of Diskin et al. with the c-lenses, known to be equivalent to opfibrations, already studied by the authors. Contrary to expectation a c-lens is a d-lens but not conversely. This result is surprising because d-lenses appear to provide the same information as c-lenses, and some more besides, suggesting that the implication would be the reverse – a d-lens would appear to be a special kind of c-lens. The source of the surprise can be traced to the way the two concepts deal differently with morphisms in a certain base comma category (G,1V). Both c-lenses and d-lenses are important because they extend the notion of lens to take account of the information available in known transitions between states and this has important implications in practice.