Spans of lenses

Corresponding to the variety of notions of asymmetric lens, various notions of symmetric lens have been proposed. A common theory of the various asymmetric and symmetric lenses should result from a study of spans of asymmetric lenses. In order to define a category whose arrows are spans of asymmetric lenses, the fact that a cospan of asymmetric lenses may not have a pullback must be dealt with. In this article, after resolving that problem we develop the functors which exhibit a category whose arrows are spans of wellbehaved lenses as a retract of a category whose arrows are the corresponding symmetric lenses. We relate them to the symmetric lenses of Hofmann, Pierce and Wagner.