There are at least three different formal graphical representations of optics on the literature (Hedges, Riley, Boisseau). They do not seem to provide a clear way of composing optics of different kinds in a faithful way: for instance, composing optics as setters is not necessarily faithful. We present a graphical formalism of optics that eases this task by observing that categories of optics are full subcategories of a Kleisli category for a monad.
We can link this to the elegant diagrams proposed by Boisseau, which are able to describe the laws of optics. Because the monad is opmonoidal, its Eilenberg-Moore category is monoidal with the forgetful functor being strong monoidal. We can then use the fully faithful embedding of Kleisli categories into Eilenberg-Moore categories to interpret our diagrams in the promonoidal context. This is work in progress.
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