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.
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)
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:
Last revised on August 14, 2023 at 15:39:58. See the history of this page for a list of all contributions to it.