nLab accessible monad

Definition

Definition

An accessible monad is a monad on an accessible category whose underlying functor is an accessible functor.

Properties

Category of algebras over an accessible monad

Proposition

The Eilenberg-Moore category of a $\kappa$-accessible monad, def. , is a $\kappa$-accessible category. If in addition the category on which the monad acts is a $\kappa$-locally presentable category then so is the EM-category.

(Adamek-Rosicky, 2.78)

Moreover, let $C$ be a topos. Then

• if a monad $T : C \to C$ has a right adjoint then $T Alg(C)= C^T$ is itself a topos;

• if a comonad $T : C \to C$ is left exact, then $T CoAlg(C) = C_T$ is itself a topos.

See at topos of algebras over a monad for details.

References

• Jiří Adámek, Jiří Rosický, Locally presentable and accessible categories, Cambridge University Press, (1994)