Lens put-put laws: monotonic and mixed

Many authors have argued, for good reasons, that in a range of applications the lens put-put law is too strong. On the other hand, the present authors have shown that very well behaved lenses, which do satisfy the put-put law by definition, are algebras for a certain monad, and that this viewpoint admits fruitful generalisations of the lens concept to a variety of base categories. In the algebra approach to lenses, the put-put law corresponds to the associativity axiom, and so is fundamentally important. Thus we have a dilemma. The put-put law seems inappropriate for many applications, but is fundamental to the mathematical development that can support an extended range of applications. In this paper we resolve this dilemma. We outline monotonic put-put laws and introduce a new mixed put-put law that appears to be immune to many of the objections to the classical put-put law, and still supports a very satisfactory mathematical foundation.