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 ...

Following work of Ehrhard and Regnier, we introduce the notion of a differential category: an additive symmetric monoidal category with a comonad (a " coalgebra modality ") and a differential combinator, satisfying a number of coherence conditions. In such a category, one should imagine the morphisms in the base category as being linear maps and the morphisms in the coKleisli category as being ...

Infinite triangular matrices and, in particular, the redecoration operation on them, were studied by Matthes and Picard. In their work, redecoration is characterized as the cobind operation of what the authors call a “weak constructive comonad”. In this work, we identify weak constructive comonads as an instance of the more general notion of relative comonad. Afterwards, building upon the work ...

This paper presents equational-based logics for proving first order properties of programming languages involving effects. We propose two dual inference system patterns that can be instanciated with monads or comonads in order to be used for proving properties of different effects. The first pattern provides inference rules which can be interpreted in the Kleisli category of a monad and the coK...

This paper extends the Day Re ection Theorem to skew monoidal categories. We also provide conditions under which a skew monoidal structure can be lifted to the category of Eilenberg-Moore algebras for a comonad.

We investigate the categories of weak maps associated to an algebraic weak factorisation system (awfs) in the sense of Grandis–Tholen [14]. For any awfs on a category with an initial object, cofibrant replacement forms a comonad, and the category of (left) weak maps associated to the awfs is by definition the Kleisli category of this comonad. We exhibit categories of weak maps as a kind of “hom...