internalization and categorical algebra
algebra object (associative, Lie, …)
internal category ($\to$ more)
Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
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.
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)
Some introductory material on comonads, coalgebras and co-Kleisli morphisms can be found in
On model category-structures on coalgebras over a comonad:
Last revised on November 5, 2022 at 09:56:56. See the history of this page for a list of all contributions to it.