nLab coalgebra over a comonad

Idea

A coalgebra or comodule over a comonad $C$ on a category $A$ is an object $a\in A$ with a morphisms $a\to C a$ satisfying axioms formally dual to those for an algebra over a monad. The category of coalgebras is called its (co-)Eilenberg-Moore category and satisfies a universal property dual to that of the Eilenberg-Moore object for a monad; it can thereby be internalized to any 2-category. The forgetful functor from the category of coalgebras to the category $A$ is called a comonadic functor. Similarly, a comonad also has a co-Kleisli category.

Examples

• the modality ! (of course) in linear logic is modeled by a comonad and its coalgebras, which are also comonoids, allow us to model intuitionistic formulas in categorical logic

• partial differential equations are the coalgebras of a jet comonad (see there)

• well-behaved lenses (in computer science) are the coalgebras of the costate comonad (see there)

Related concepts

• topos of coalgebras over a comonad

• coalgebra over an endofunctor

• cocommutative coalgebra

References

Some introductory material on comonads, coalgebras and co-Kleisli morphisms can be found in

• Valeria de Paiva, The Dialectica Categories, Chapter 2. (Ph Thesis)

• Paolo Perrone, Notes on Category Theory with examples from basic mathematics, Chapter 5. (arXiv)

On model category-structures on coalgebras over a comonad:

• Kathryn Hess, Brooke Shipley, The homotopy theory of coalgebras over a comonad, Proceedings of theLondon Mathematical Society 108 2 (2014) [arXiv:1205.3979, doi:10.1112/plms/pdt038]