nLab
accessible monad

Contents

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 κ\kappa-accessible monad, def. 1, is a κ\kappa-accessible category. If in addition the category on which the monad acts is a κ\kappa-locally presentable category then so is the EM-category.

(Adamek-Rosicky, 2.78)

Moreover, let CC be a topos. Then

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

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

See at topos of algebras over a monad for details.

References

Revised on February 12, 2014 06:16:09 by Urs Schreiber (88.128.80.11)