### Context

#### Higher algebra

higher algebra

universal algebra

# 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 $\kappa$-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.

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

Last revised on February 12, 2014 at 06:16:09. See the history of this page for a list of all contributions to it.