  This paper gives two new categorical characterisations of lenses: one as a
coalgebra of the store comonad, and the other as a monoidal natural
transformation on a category of a certain class of coalgebras. The store
comonad of the first characterisation can be generalized to a Cartesian store
comonad, and the coalgebras of this Cartesian store comonad turn out to be
exactly the Biplates of the Uniplate generic programming library. On the other
hand, the monoidal natural transformations on functors can be generalized to
work on a category of more specific coalgebras. This generalization turns out
to be the type of compos from the Compos generic programming library. A
theorem, originally conjectured by van Laarhoven, proves that these two
generalizations are isomorphic, thus the core data types of the Uniplate and
Compos libraries supporting generic program on single recursive types are the
same. Both the Uniplate and Compos libraries generalize this core functionality
to support mutually recursive types in different ways. This paper proposes a
third extension to support mutually recursive data types that is as powerful as
Compos and as easy to use as Uniplate. This proposal, called Multiplate, only
requires rank 3 polymorphism in addition to the normal type class mechanism of
Haskell.
