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.

Related concepts

• coalgebra over an endofunctor

References

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

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