nLab
comonad

Definition

A comonad (or cotriple) on a category $A$ is a monad on its dual category $A^{\mathrm{op}}$. More generally, a comonad in a 2-category $E$ is a comonoid in the monoidal category $E(X,X)$ for some object $X\in K$.

Properties

Coalgebras

A coalgebra (or comodule) over a comonad $C$ on a category $A$ is an object $a\in A$ with a map $a\to Ca$ satisfying dual axioms 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.

Comonadic homology and descent

Any comonad on $A$ induces an augmented simplicial endofunctor of $A$ consisting of its iterates. If $A$ is an abelian category and the comonad is additive, then this is the basis of comonadic homology?. Comodules (= coalgebras) over the comonad with underlying endofunctor $M_R\mapsto M_R{\otimes }_{R}S$ in $R$-$\mathrm{Mod}$ for the extension of rings $R\hookrightarrow S$ correspond to the descent data for that extension. Gluing of categories from localizations may also be formalized via comonads.

Mixed distributive laws

Distributive laws between a monad and a comonad are so-called mixed distributive laws; a special case has been rediscovered in physics under the name entwining structures (Brzeziński, Majid 1997). Their theory is often studied in the connection with the theory of comonads in the bicategory of rings, modules and morphisms of modules, that is corings. There is a homomorphism of bicategories from a bicategory of entwinings to a bicategory of corings (Škoda 2008), which is an analogue of the 2-functor $\mathrm{comp}$ (R. Street, Formal theory of monads, JPAA 1972) of strict 2-categories in the case of distributive laws of monads (recall also that a distributive law among monads corresponds to a monad in the 2-category of monads).