nLab
idempotent monad

An idempotent monad is a monad $(T,\mu ,\eta )$ on a category $C$ such that one (hence all) of the following equivalent statements are true:

1. $\mu :TT\to T$ is a natural isomorphism.

2. The maps $T\eta ,\eta T:T\to TT$ are equal.

3. For every $T$-algebra ($T$-module) $(M,u)$, the corresponding $T$-action $u:TM\to M$ is an isomorphism.

4. The forgetful functor $C^T\to C$ (where $C^T$ is the Eilenberg-Moore category of $T$-algebras) is a full and faithful functor.

The last statement means that in such a case $C^T$ is, up to equivalence a full reflective subcategory of $C$. Conversely, the monad induced by any reflective subcategory is idempotent, so giving an idempotent monad on $C$ is equivalent to giving a reflective subcategory of $C$.

In the language of stuff, structure, property, an idempotent monad may be said to equip its algebras with properties only (since $C^T\to C$ is fully faithful), unlike an arbitrary monad, which equips its algebras with at most structure (since $C^T\to C$ is, in general, faithful but not full).

If $T$ is idempotent, then it follows in particular that an object of $C$ admits at most one structure of $T$-algebra, that this happens precisely when the unit ${\eta }_X:X\to TX$ is an isomorphism, and in this case the $T$-algebra structure map is ${\eta }_X^{-1}:TX\to X$. However, it is possible to have a non-idempotent monad for which any object of $C$ admits at most one structure of $T$-algebra, in which case $T$ can be said to equip objects of $C$ with property-like structure; an easy example is the monad on semigroups whose algebras are monoids.

The associated idempotent monad of a monad

Let $C$ be a category with equalizers, and let $(T:C\to C,\mu ,\eta )$ be a monad on $C$. There is an associated idempotent monad $T\prime$ which at the functor level is obtained as the equalizer

$$T\prime \stackrel{e}{\to }T\stackrel{\stackrel{\eta T}{\to }}{\underset{T\eta }{\to }}TT$$T' \overset{e}{\to} T \stackrel{\overset{\eta T}{\to}}{\underset{T \eta}{\to}} T T

The map $\eta :1_C\to T$ equalizes the pair of maps $\eta T$, $T\eta$, so it factors as $e\circ u$ for some unique map $u:1_C\to T\prime$. This defines the unit of the monad $T\prime$.

A diagram chase shows that the composite

$$T\prime T\prime \stackrel{ee}{\to }TT\stackrel{\mu }{\to }T$$T' T' \overset{e e}{\to} T T \overset{\mu}{\to} T

equalizes the pair of maps $\eta T$, $T\eta$; therefore $\mu \circ ee$ factors as $e\circ \mu \prime$ for some unique map $\mu \prime :T\prime T\prime \to T\prime$. This defines the multiplication of the monad $T\prime$. By construction, $e:T\prime \to T$ is a map which preserves the monad structure.

A result due to Fakir is that the monad $T\prime$ is idempotent in the sense given above. In fact, if $C$ has equalizers, then the category of idempotent monads on $C$ is a coreflective subcategory of the category of monads on $C$, meaning that the full embedding

$$i:{\mathrm{IdemMonad}}_C\hookrightarrow {\mathrm{Monad}}_C$$i: IdemMonad_C \hookrightarrow Monad_C

has a right adjoint given by the associated idempotent monad construction.

Mike Shulman: How about some examples of monads and their associated idempotent monads?

Do 2-monads have associated lax-, colax-, or pseudo-idempotent 2-monads?

References

• F. Borceux, Handbook of categorical algebra, vol.2, p. 196.

• P. Gabriel and M. Zisman, Calculus of Fractions and Homotopy Theory