Idempotent monad

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In category theory, an idempotent monad is, roughly speaking, a monad that theorizes a reflective subcategory.

Definition

Let ${\displaystyle (T,\eta ,\mu )}$ be a monad on a category ${\displaystyle {\mathcal {C}}}$. Then the following conditions are equivalent.

• The monad binary operation ${\displaystyle \mu \colon TT\Rightarrow T}$ is a natural isomorphism.
• The forgetful functor $${\displaystyle {\begin{array}{l}{\mathcal {C}}_{T}\to {\mathcal {C}}\\(A,e)\mapsto A\\(f\colon (A,e)\to (A',e'))\mapsto (f\colon A\to A')\end{array}}}$$ for the Eilenberg–Moore category is full and faithful.
• For every ${\displaystyle T}$-algebra ${\displaystyle (A,e)}$ the morphism ${\displaystyle e\colon TA\to A}$ is an isomorphism.

Such a monad is called an idempotent monad.

Properties

The notion of the idempotent monads is equivalent to that of the reflective subcategories. Given an idempotent monad on a category ${\displaystyle {\mathcal {C}}}$, the forgetful functor

${\displaystyle {\mathcal {C}}_{T}\to {\mathcal {C}}}$

from the Eilenberg–Moore category ${\displaystyle {\mathcal {C}}_{T}}$ to ${\displaystyle {\mathcal {C}}}$ is full and faithful. Conversely, every full and faithful right adjoint ${\displaystyle I\colon {\mathcal {D}}\to {\mathcal {C}}}$ is monadic and its corresponding monad

${\displaystyle (I\circ R,\eta ,I\circ \epsilon \circ R)}$

is idempotent.^{: 197, Corollary 4.2.4}

References

External links

• Idempotent monad at the nLab