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.

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)

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]

Last revised on August 17, 2022 at 11:12:33. See the history of this page for a list of all contributions to it.