nLab comonad




A comonad (or cotriple) on a category AA is a comonoid in the monoidal category of endofunctors AAA \to A. More generally, a comonad in a 2-category EE is a comonoid in the monoidal category E(X,X)E(X,X) for some object XKX\in K.

Just as a monad may be defined for any 2-category, EE, as a lax 2-functor from 1\mathbf{1} to EE, so a comonad in EE is an oplax 2-functor from 1E\mathbf{1} \to E.

See at monad for more.



A coalgebra over a comonad (or comodule) over a comonad CC on a category AA is an object aAa\in A with a map aCaa\to C a 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 AA is called a comonadic functor. Similarly, a comonad also has a co-Kleisli category.

Comonadic homology and descent

Any comonad on AA induces an augmented simplicial endofunctor of AA consisting of its iterates. If AA 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 RM R RSM_R\mapsto M_R\otimes_R S in RR-ModMod for the extension of rings RSR\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 compcomp (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).




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)

As contexts in computer science

On comonads in computer science:

Last revised on May 10, 2024 at 13:04:49. See the history of this page for a list of all contributions to it.